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52nd International Mathematical Olympiad

12 – 24 July 2011 Amsterdam The Netherlands

Problem Shortlist with Solutions

International Mathematical Olympiad Am sterdam 2011

IMO2011

Amsterdam

52nd International Mathematical Olympiad 12-24 July 2011 Amsterdam The Netherlands

Problem shortlist with solutions

IMPORTANT

IMO regulation: these shortlist problems have to be kept strictly con?dential until IMO 2012.

The problem selection committee Bart de Smit (chairman), Ilya Bogdanov, Johan Bosman, Andries Brouwer, Gabriele Dalla Torre, G? eza K? os, Hendrik Lenstra, Charles Leytem, Ronald van Luijk, Christian Reiher, Eckard Specht, Hans Sterk, Lenny Taelman

The committee gratefully acknowledges the receipt of 142 problem proposals by the following 46 countries: Armenia, Australia, Austria, Belarus, Belgium, Bosnia and Herzegovina, Brazil, Bulgaria, Canada, Colombia, Cyprus, Denmark, Estonia, Finland, France, Germany, Greece, Hong Kong, Hungary, India, Islamic Republic of Iran, Ireland, Israel, Japan, Kazakhstan, Republic of Korea, Luxembourg, Malaysia, Mexico, Mongolia, Montenegro, Pakistan, Poland, Romania, Russian Federation, Saudi Arabia, Serbia, Slovakia, Slovenia, Sweden, Taiwan, Thailand, Turkey, Ukraine, United Kingdom, United States of America

Algebra

Problem shortlist

52nd IMO 2011

A1

Algebra

A1

For any set A = {a1 , a2 , a3 , a4 } of four distinct positive integers with sum sA = a1 + a2 + a3 + a4 , let pA denote the number of pairs (i, j ) with 1 ≤ i < j ≤ 4 for which ai + aj divides sA . Among all sets of four distinct positive integers, determine those sets A for which pA is maximal.

A2

A2

Determine all sequences (x1 , x2 , . . . , x2011 ) of positive integers such that for every positive integer n there is an integer a with

n n n+1 xn + 1. 1 + 2x2 + · · · + 2011x2011 = a

A3

A3

Determine all pairs (f, g ) of functions from the set of real numbers to itself that satisfy g (f (x + y )) = f (x) + (2x + y )g (y ) for all real numbers x and y . A4

A4

Determine all pairs (f, g ) of functions from the set of positive integers to itself that satisfy f g(n)+1 (n) + g f (n) (n) = f (n + 1) ? g (n + 1) + 1 for every positive integer n. Here, f k (n) means f (f (. . . f (n) . . .)).

k

A5

A5

Prove that for every positive integer n, the set {2, 3, 4, . . . , 3n + 1} can be partitioned into n triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle.

4

52nd IMO 2011

Problem shortlist

Algebra

A6

A6

Let f be a function from the set of real numbers to itself that satis?es f (x + y ) ≤ yf (x) + f (f (x)) for all real numbers x and y . Prove that f (x) = 0 for all x ≤ 0. A7

A7

Let a, b, and c be positive real numbers satisfying min(a + b, b + c, c + a) > Prove that √ 2 and a2 + b2 + c2 = 3.

a b c 3 + + ≥ . 2 2 2 (b + c ? a) (c + a ? b) ( a + b ? c) (abc)2

5

Combinatorics

Problem shortlist

52nd IMO 2011

C1

Combinatorics

C1

Let n > 0 be an integer. We are given a balance and n weights of weight 20 , 21 , . . . , 2n?1 . In a sequence of n moves we place all weights on the balance. In the ?rst move we choose a weight and put it on the left pan. In each of the following moves we choose one of the remaining weights and we add it either to the left or to the right pan. Compute the number of ways in which we can perform these n moves in such a way that the right pan is never heavier than the left pan.

C2

C2

Suppose that 1000 students are standing in a circle. Prove that there exists an integer k with 100 ≤ k ≤ 300 such that in this circle there exists a contiguous group of 2k students, for which the ?rst half contains the same number of girls as the second half. C3

C3

Let S be a ?nite set of at least two points in the plane. Assume that no three points of S are collinear. By a windmill we mean a process as follows. Start with a line ? going through a point P ∈ S . Rotate ? clockwise around the pivot P until the line contains another point Q of S . The point Q now takes over as the new pivot. This process continues inde?nitely, with the pivot always being a point from S . Show that for a suitable P ∈ S and a suitable starting line ? containing P , the resulting windmill will visit each point of S as a pivot in?nitely often. C4

C4

Determine the greatest positive integer k that satis?es the following property: The set of positive integers can be partitioned into k subsets A1 , A2 , . . . , Ak such that for all integers n ≥ 15 and all i ∈ {1, 2, . . . , k } there exist two distinct elements of Ai whose sum is n.

6

52nd IMO 2011

Problem shortlist

Combinatorics

C5

C5

Let m be a positive integer and consider a checkerboard consisting of m by m unit squares. At the midpoints of some of these unit squares there is an ant. At time 0, each ant starts moving with speed 1 parallel to some edge of the checkerboard. When two ants moving in opposite directions meet, they both turn 90? clockwise and continue moving with speed 1. When more than two ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls o? and will not re-appear. Considering all possible starting positions, determine the latest possible moment at which the last ant falls o? the checkerboard or prove that such a moment does not necessarily exist. C6

C6

Let n be a positive integer and let W = . . . x?1 x0 x1 x2 . . . be an in?nite periodic word consisting of the letters a and b. Suppose that the minimal period N of W is greater than 2n . A ?nite nonempty word U is said to appear in W if there exist indices k ≤ ? such that U = xk xk+1 . . . x? . A ?nite word U is called ubiquitous if the four words Ua, Ub, aU , and bU all appear in W . Prove that there are at least n ubiquitous ?nite nonempty words. C7

C7

On a square table of 2011 by 2011 cells we place a ?nite number of napkins that each cover a square of 52 by 52 cells. In each cell we write the number of napkins covering it, and we record the maximal number k of cells that all contain the same nonzero number. Considering all possible napkin con?gurations, what is the largest value of k ?

7

Geometry

Problem shortlist

52nd IMO 2011

G1

Geometry

G1

Let ABC be an acute triangle. Let ω be a circle whose center L lies on the side BC . Suppose that ω is tangent to AB at B ′ and to AC at C ′ . Suppose also that the circumcenter O of the triangle ABC lies on the shorter arc B ′ C ′ of ω . Prove that the circumcircle of ABC and ω meet at two points.

G2

G2

Let A1 A2 A3 A4 be a non-cyclic quadrilateral. Let O1 and r1 be the circumcenter and the circumradius of the triangle A2 A3 A4 . De?ne O2 , O3 , O4 and r2 , r3 , r4 in a similar way. Prove that 1 1 1 1 + + + = 0. 2 2 2 2 2 2 2 2 O1 A1 ? r1 O2 A2 ? r2 O3 A3 ? r3 O4 A4 ? r4 G3

G3

Let ABCD be a convex quadrilateral whose sides AD and BC are not parallel. Suppose that the circles with diameters AB and CD meet at points E and F inside the quadrilateral. Let ωE be the circle through the feet of the perpendiculars from E to the lines AB , BC , and CD . Let ωF be the circle through the feet of the perpendiculars from F to the lines CD , DA, and AB . Prove that the midpoint of the segment EF lies on the line through the two intersection points of ωE and ωF . G4

G4

Let ABC be an acute triangle with circumcircle ?. Let B0 be the midpoint of AC and let C0 be the midpoint of AB . Let D be the foot of the altitude from A, and let G be the centroid of the triangle ABC . Let ω be a circle through B0 and C0 that is tangent to the circle ? at a point X = A. Prove that the points D , G, and X are collinear. G5

G5

Let ABC be a triangle with incenter I and circumcircle ω . Let D and E be the second intersection points of ω with the lines AI and BI , respectively. The chord DE meets AC at a point F , and BC at a point G. Let P be the intersection point of the line through F parallel to AD and the line through G parallel to BE . Suppose that the tangents to ω at A and at B meet at a point K . Prove that the three lines AE , BD , and KP are either parallel or concurrent.

8

52nd IMO 2011

Problem shortlist

Geometry

G6

G6

Let ABC be a triangle with AB = AC , and let D be the midpoint of AC . The angle bisector of ∠BAC intersects the circle through D , B , and C in a point E inside the triangle ABC . The line BD intersects the circle through A, E , and B in two points B and F . The lines AF and BE meet at a point I , and the lines CI and BD meet at a point K . Show that I is the incenter of triangle KAB . G7

G7

Let ABCDEF be a convex hexagon all of whose sides are tangent to a circle ω with center O . Suppose that the circumcircle of triangle ACE is concentric with ω . Let J be the foot of the perpendicular from B to CD . Suppose that the perpendicular from B to DF intersects the line EO at a point K . Let L be the foot of the perpendicular from K to DE . Prove that DJ = DL. G8

G8

Let ABC be an acute triangle with circumcircle ω . Let t be a tangent line to ω . Let ta , tb , and tc be the lines obtained by re?ecting t in the lines BC , CA, and AB , respectively. Show that the circumcircle of the triangle determined by the lines ta , tb , and tc is tangent to the circle ω .

9

Number Theory

Problem shortlist

52nd IMO 2011

N1

Number Theory

N1

For any integer d > 0, let f (d) be the smallest positive integer that has exactly d positive divisors (so for example we have f (1) = 1, f (5) = 16, and f (6) = 12). Prove that for every integer k ≥ 0 the number f (2k ) divides f (2k+1).

N2

N2

Consider a polynomial P (x) = (x + d1 )(x + d2 ) · . . . · (x + d9 ), where d1 , d2 , . . . , d9 are nine distinct integers. Prove that there exists an integer N such that for all integers x ≥ N the number P (x) is divisible by a prime number greater than 20. N3

N3

Let n ≥ 1 be an odd integer. Determine all functions f from the set of integers to itself such that for all integers x and y the di?erence f (x) ? f (y ) divides xn ? y n . N4

N4

For each positive integer k , let t(k ) be the largest odd divisor of k . Determine all positive integers a for which there exists a positive integer n such that all the di?erences t(n + a) ? t(n), are divisible by 4. N5 t(n + a + 1) ? t(n + 1), ..., t(n + 2a ? 1) ? t(n + a ? 1)

N5

Let f be a function from the set of integers to the set of positive integers. Suppose that for any two integers m and n, the di?erence f (m) ? f (n) is divisible by f (m ? n). Prove that for all integers m, n with f (m) ≤ f (n) the number f (n) is divisible by f (m). N6

N6

Let P (x) and Q(x) be two polynomials with integer coe?cients such that no nonconstant polynomial with rational coe?cients divides both P (x) and Q(x). Suppose that for every positive integer n the integers P (n) and Q(n) are positive, and 2Q(n) ? 1 divides 3P (n) ? 1. Prove that Q(x) is a constant polynomial.

10

52nd IMO 2011

Problem shortlist

Number Theory

N7

N7

Let p be an odd prime number. For every integer a, de?ne the number Sa = Let m and n be integers such that S3 + S4 ? 3 S2 = Prove that p divides m. N8 m . n ap?1 a a2 + +···+ . 1 2 p?1

N8

Let k be a positive integer and set n = 2k + 1. Prove that n is a prime number if and only if the following holds: there is a permutation a1 , . . . , an?1 of the numbers 1, 2, . . . , n ? 1 and a ai sequence of integers g1 , g2 , . . . , gn?1 such that n divides gi ? ai+1 for every i ∈ {1, 2, . . . , n ? 1}, where we set an = a1 .

11

A1

A1

Algebra – solutions

52nd IMO 2011

For any set A = {a1 , a2 , a3 , a4 } of four distinct positive integers with sum sA = a1 + a2 + a3 + a4 , let pA denote the number of pairs (i, j ) with 1 ≤ i < j ≤ 4 for which ai + aj divides sA . Among all sets of four distinct positive integers, determine those sets A for which pA is maximal.

Answer. The sets A for which pA is maximal are the sets the form {d, 5d, 7d, 11d} and {d, 11d, 19d, 29d}, where d is any positive integer. For all these sets pA is 4. Solution. Firstly, we will prove that the maximum value of pA is at most 4. Without loss of generality, we may assume that a1 < a2 < a3 < a4 . We observe that for each pair of indices (i, j ) with 1 ≤ i < j ≤ 4, the sum ai + aj divides sA if and only if ai + aj divides sA ? (ai + aj ) = ak + al , where k and l are the other two indices. Since there are 6 distinct pairs, we have to prove that at least two of them do not satisfy the previous condition. We claim that two such pairs are (a2 , a4 ) and (a3 , a4 ). Indeed, note that a2 + a4 > a1 + a3 and a3 + a4 > a1 + a2 . Hence a2 + a4 and a3 + a4 do not divide sA . This proves pA ≤ 4. Now suppose pA = 4. By the previous argument we have a1 + a4 a2 + a3 a1 + a2 a3 + a4 a1 + a3 a2 + a4 and and and a2 + a3 a1 + a4 , a3 + a4 a2 + a4 a1 + a2 , a1 + a3 .

Hence, there exist positive integers m and n with m > n ≥ 2 such that ? ? ? ? ?a1 + a4 = a2 + a3 Adding up the ?rst equation and the third one, we get n(a1 + a3 ) = 2a2 + a3 ? a1 . If n ≥ 3, then n(a1 + a3 ) > 3a3 > 2a2 + a3 > 2a2 + a3 ? a1 . This is a contradiction. Therefore n = 2. If we multiply by 2 the sum of the ?rst equation and the third one, we obtain 6a1 + 2a3 = 4a2 , while the sum of the ?rst one and the second one is (m + 1)a1 + (m ? 1)a2 = 2a3 . Adding up the last two equations we get (m + 7)a1 = (5 ? m)a2 . 12 m(a1 + a2 ) = a3 + a4 ? ? ? ?n(a1 + a3 ) = a2 + a4 .

52nd IMO 2011

Algebra – solutions

A1

It follows that 5 ? m ≥ 1, because the left-hand side of the last equation and a2 are positive. Since we have m > n = 2, the integer m can be equal only to either 3 or 4. Substituting (3, 2) and (4, 2) for (m, n) and solving the previous system of equations, we ?nd the families of solutions {d, 5d, 7d, 11d} and {d, 11d, 19d, 29d}, where d is any positive integer.

13

A2

A2

Algebra – solutions

52nd IMO 2011

Determine all sequences (x1 , x2 , . . . , x2011 ) of positive integers such that for every positive integer n there is an integer a with

n n n+1 xn + 1. 1 + 2x2 + · · · + 2011x2011 = a

Answer. The only sequence that satis?es the condition is (x1 , . . . , x2011 ) = (1, k, . . . , k ) with k = 2 + 3 + · · · + 2011 = 2023065. Solution. Throughout this solution, the set of positive integers will be denoted by Z+ . Put k = 2 + 3 + · · · + 2011 = 2023065. We have 1n + 2k n + · · · 2011k n = 1 + k · k n = k n+1 + 1 for all n, so (1, k, . . . , k ) is a valid sequence. We shall prove that it is the only one. Let a valid sequence (x1 , . . . , x2011 ) be given. For each n ∈ Z+ we have some yn ∈ Z+ with

n n n+1 xn + 1. 1 + 2x2 + · · · + 2011x2011 = yn n n n+1 Note that xn , which implies that 1 + 2x2 + · · · + 2011x2011 < (x1 + 2x2 + · · · + 2011x2011 ) the sequence (yn ) is bounded. In particular, there is some y ∈ Z+ with yn = y for in?nitely many n.

Let m be the maximum of all the xi . Grouping terms with equal xi together, the sum xn 1 + n 2xn 2 + · · · + 2011x2011 can be written as

n n n n xn 1 + 2x2 + · · · + x2011 = am m + am?1 (m ? 1) + · · · + a1

with ai ≥ 0 for all i and a1 + · · · + am = 1 + 2 + · · · + 2011. So there exist arbitrarily large values of n, for which am mn + · · · + a1 ? 1 ? y · y n = 0. (1) The following lemma will help us to determine the ai and y : Lemma. Let integers b1 , . . . , bN be given and assume that there are arbitrarily large positive integers n with b1 + b2 2n + · · · + bN N n = 0. Then bi = 0 for all i. Proof. Suppose that not all bi are zero. We may assume without loss of generality that bN = 0.

14

52nd IMO 2011 Dividing through by N n gives |bN | = bN ?1 N ?1 N

n n

Algebra – solutions

A2

n

+ · · · + b1

1 N

n

≤ (|bN ?1 | + · · · + |b1 |)

N ?1 N

.

?1 The expression NN can be made arbitrarily small for n large enough, contradicting the assumption that bN be non-zero.

We obviously have y > 1. Applying the lemma to (1) we see that am = y = m, a1 = 1, and all the other ai are zero. This implies (x1 , . . . , x2011 ) = (1, m, . . . , m). But we also have 1 + m = a1 + · · · + am = 1 + · · · + 2011 = 1 + k so m = k , which is what we wanted to show.

15

A3

A3

Algebra – solutions

52nd IMO 2011

Determine all pairs (f, g ) of functions from the set of real numbers to itself that satisfy g (f (x + y )) = f (x) + (2x + y )g (y ) for all real numbers x and y .

Answer. Either both f and g vanish identically, or there exists a real number C such that f (x) = x2 + C and g (x) = x for all real numbers x.

Solution. Clearly all these pairs of functions satisfy the functional equation in question, so it su?ces to verify that there cannot be any further ones. Substituting ?2x for y in the given functional equation we obtain g (f (?x)) = f (x). (1) Using this equation for ?x ? y in place of x we obtain f (?x ? y ) = g (f (x + y )) = f (x) + (2x + y )g (y ). Now for any two real numbers a and b, setting x = ?b and y = a + b we get f (?a) = f (?b) + (a ? b)g (a + b). If c denotes another arbitrary real number we have similarly f (?b) = f (?c) + (b ? c)g (b + c) as well as f (?c) = f (?a) + (c ? a)g (c + a). Adding all these equations up, we obtain (a + c) ? (b + c) g (a + b) + (a + b) ? (a + c) g (b + c) + (b + c) ? (a + b) g (a + c) = 0. Now given any three real numbers x, y , and z one may determine three reals a, b, and c such that x = b + c, y = c + a, and z = a + b, so that we get (y ? x)g (z ) + (z ? y )g (x) + (x ? z )g (y ) = 0. This implies that the three points (x, g (x)), (y, g (y )), and (z, g (z )) from the graph of g are collinear. Hence that graph is a line, i.e., g is either a constant or a linear function. (2)

16

52nd IMO 2011

Algebra – solutions

A3

Let us write g (x) = Ax + B , where A and B are two real numbers. Substituting (0, ?y ) for (x, y ) in (2) and denoting C = f (0), we have f (y ) = Ay 2 ? By + C . Now, comparing the coe?cients of x2 in (1) we see that A2 = A, so A = 0 or A = 1. If A = 0, then (1) becomes B = ?Bx + C and thus B = C = 0, which provides the ?rst of the two solutions mentioned above. Now suppose A = 1. Then (1) becomes x2 ? Bx + C + B = x2 ? Bx + C , so B = 0. Thus, g (x) = x and f (x) = x2 + C , which is the second solution from above.

Comment. Another way to show that g(x) is either a constant or a linear function is the following. If we interchange x and y in the given functional equation and subtract this new equation from the given one, we obtain f (x) ? f (y ) = (2y + x)g(x) ? (2x + y )g(y ). Substituting (x, 0), (1, x), and (0, 1) for (x, y ), we get f (x) ? f (0) = xg (x) ? 2xg (0), f (1) ? f (x) = (2x + 1)g(1) ? (x + 2)g(x), f (0) ? f (1) = 2g(0) ? g(1).

Taking the sum of these three equations and dividing by 2, we obtain g(x) = x g(1) ? g(0) + g(0). This proves that g(x) is either a constant of a linear function.

17

A4

A4

Algebra – solutions

52nd IMO 2011

Determine all pairs (f, g ) of functions from the set of positive integers to itself that satisfy f g(n)+1 (n) + g f (n) (n) = f (n + 1) ? g (n + 1) + 1 for every positive integer n. Here, f k (n) means f (f (. . . f (n) . . .)).

k

Answer. The only pair (f, g ) of functions that satis?es the equation is given by f (n) = n and g (n) = 1 for all n. Solution. The given relation implies f f g(n) (n) < f (n + 1) for all n, which will turn out to be su?cient to determine f . Let y1 < y2 < . . . be all the values attained by f (this sequence might be either ?nite or in?nite). We will prove that for every positive n the function f attains at least n values, and we have (i)n : f (x) = yn if and only if x = n, and (ii)n : yn = n. The proof will follow the scheme (i)1 , (ii)1 , (i)2 , (ii)2 , . . . , (i)n , (ii)n , . . . (2) To start, consider any x such that f (x) = y1 . If x > 1, then (1) reads f f g(x?1) (x ? 1) < y1 , contradicting the minimality of y1 . So we have that f (x) = y1 is equivalent to x = 1, establishing (i)1 . Next, assume that for some n statement (i)n is established, as well as all the previous statements in (2). Note that these statements imply that for all k ≥ 1 and a < n we have f k (x) = a if and only if x = a. Now, each value yi with 1 ≤ i ≤ n is attained at the unique integer i, so yn+1 exists. Choose an arbitrary x such that f (x) = yn+1 ; we necessarily have x > n. Substituting x ? 1 into (1) we have f f g(x?1) (x ? 1) < yn+1, which implies f g(x?1) (x ? 1) ∈ {1, . . . , n} (3) (1)

Set b = f g(x?1) (x ? 1). If b < n then we would have x ? 1 = b which contradicts x > n. So b = n, and hence yn = n, which proves (ii)n . Next, from (i)n we now get f (k ) = n ?? k = n, so removing all the iterations of f in (3) we obtain x ? 1 = b = n, which proves (i)n+1 . So, all the statements in (2) are valid and hence f (n) = n for all n. The given relation between f and g now reads n + g n (n) = n + 1 ? g (n + 1) + 1 or g n (n) + g (n + 1) = 2, from which it

18

52nd IMO 2011

Algebra – solutions

A4

immediately follows that we have g (n) = 1 for all n.

Comment. Several variations of the above solution are possible. For instance, one may ?rst prove by induction that the smallest n values of f are exactly f (1) < · · · < f (n) and proceed as follows. We certainly have f (n) ≥ n for all n. If there is an n with f (n) > n, then f (x) > x for all x ≥ n. From this we conclude f g(n)+1 (n) > f g(n) (n) > · · · > f (n). But we also have f g(n)+1 < f (n + 1). Having squeezed in a function value between f (n) and f (n + 1), we arrive at a contradiction. In any case, the inequality (1) plays an essential r? ole.

19

A5

A5

Algebra – solutions

52nd IMO 2011

Prove that for every positive integer n, the set {2, 3, 4, . . . , 3n + 1} can be partitioned into n triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle.

Solution. Throughout the solution, we denote by [a, b] the set {a, a + 1, . . . , b}. We say that {a, b, c} is an obtuse triple if a, b, c are the sides of some obtuse triangle. We prove by induction on n that there exists a partition of [2, 3n + 1] into n obtuse triples Ai (2 ≤ i ≤ n + 1) having the form Ai = {i, ai , bi }. For the base case n = 1, one can simply set A2 = {2, 3, 4}. For the induction step, we need the following simple lemma. Lemma. Suppose that the numbers a < b < c form an obtuse triple, and let x be any positive number. Then the triple {a, b + x, c + x} is also obtuse. Proof. The numbers a < b + x < c + x are the sides of a triangle because (c + x) ? (b + x) = c ? b < a. This triangle is obtuse since (c + x)2 ? (b + x)2 = (c ? b)(c + b +2x) > (c ? b)(c + b) > a2 . Now we turn to the induction step. Let n > 1 and put t = ?n/2? < n. By the induction hypothesis, there exists a partition of the set [2, 3t + 1] into t obtuse triples A′i = {i, a′i , b′i } (i ∈ [2, t + 1]). For the same values of i, de?ne Ai = {i, a′i + (n ? t), b′i + (n ? t)}. The constructed triples are obviously disjoint, and they are obtuse by the lemma. Moreover, we have

t+1 i=2

Ai = [2, t + 1] ∪ [n + 2, n + 2t + 1].

Next, for each i ∈ [t + 2, n + 1], de?ne Ai = {i, n + t + i, 2n + i}. All these sets are disjoint, and

n+1

i=t+2

Ai = [t + 2, n + 1] ∪ [n + 2t + 2, 2n + t + 1] ∪ [2n + t + 2, 3n + 1],

n+1

so

Ai = [2, 3n + 1].

i=2

Thus, we are left to prove that the triple Ai is obtuse for each i ∈ [t + 2, n + 1]. Since (2n + i) ? (n + t + i) = n ? t < t + 2 ≤ i, the elements of Ai are the sides of a triangle. Next, we have (2n + i)2 ? (n + t + i)2 = (n ? t)(3n + t + 2i) ≥ n 9n n · (3n + 3(t + 1) + 1) > · ≥ (n + 1)2 ≥ i2 , 2 2 2

so this triangle is obtuse. The proof is completed.

20

52nd IMO 2011

Algebra – solutions

A6

A6

Let f be a function from the set of real numbers to itself that satis?es f (x + y ) ≤ yf (x) + f (f (x)) for all real numbers x and y . Prove that f (x) = 0 for all x ≤ 0. (1)

Solution 1. Substituting y = t ? x, we rewrite (1) as f (t) ≤ tf (x) ? xf (x) + f (f (x)). (2)

Consider now some real numbers a, b and use (2) with t = f (a), x = b as well as with t = f (b), x = a. We get f (f (a)) ? f (f (b)) ≤ f (a)f (b) ? bf (b),

f (f (b)) ? f (f (a)) ≤ f (a)f (b) ? af (a). Adding these two inequalities yields 2f (a)f (b) ≥ af (a) + bf (b). Now, substitute b = 2f (a) to obtain 2f (a)f (b) ≥ af (a) + 2f (a)f (b), or af (a) ≤ 0. So, we get f (a) ≥ 0 for all a < 0. (3)

Now suppose f (x) > 0 for some real number x. From (2) we immediately get that for every xf (x) ? f (f (x)) t< we have f (t) < 0. This contradicts (3); therefore f (x) f (x) ≤ 0 for all real x, and by (3) again we get f (x) = 0 for all x < 0. We are left to ?nd f (0). Setting t = x < 0 in (2) we get 0 ≤ 0 ? 0 + f (0), so f (0) ≥ 0. Combining this with (4) we obtain f (0) = 0. Solution 2. We will also use the condition of the problem in form (2). For clarity we divide the argument into four steps. (4)

21

A6

Algebra – solutions

52nd IMO 2011

Step 1. We begin by proving that f attains nonpositive values only. Assume that there exist some real number z with f (z ) > 0. Substituting x = z into (2) and setting A = f (z ), B = ?zf (z ) ? f (f (z )) we get f (t) ≤ At + B for all real t. Hence, if for any positive real number t we substitute x = ?t, y = t into (1), we get f (0) ≤ tf (?t) + f (f (?t)) ≤ t(?At + B ) + Af (?t) + B ≤ ?t(At ? B ) + A(?At + B ) + B = ?At2 ? (A2 ? B )t + (A + 1)B.

But surely this cannot be true if we take t to be large enough. This contradiction proves that we have indeed f (x) ≤ 0 for all real numbers x. Note that for this reason (1) entails f (x + y ) ≤ yf (x) for all real numbers x and y . Step 2. We proceed by proving that f has at least one zero. If f (0) = 0, we are done. Otherwise, in view of Step 1 we get f (0) < 0. Observe that (5) tells us now f (y ) ≤ yf (0) for all real numbers y . Thus we can specify a positive real number a that is so large that f (a)2 > ?f (0). Put b = f (a) and substitute x = b and y = ?b into (5); we learn ?b2 < f (0) ≤ ?bf (b), i.e. b < f (b). Now we apply (2) to x = b and t = f (b), which yields f (f (b)) ≤ f (b) ? b f (b) + f (f (b)), i.e. f (b) ≥ 0. So in view of Step 1, b is a zero of f . Step 3. Next we show that if f (a) = 0 and b < a, then f (b) = 0 as well. To see this, we just substitute x = b and y = a ? b into (5), thus getting f (b) ≥ 0, which su?ces by Step 1. Step 4. By Step 3, the solution of the problem is reduced to showing f (0) = 0. Pick any zero r of f and substitute x = r and y = ?1 into (1). Because of f (r ) = f (r ? 1) = 0 this gives f (0) ≥ 0 and hence f (0) = 0 by Step 1 again.

Comment 1. Both of these solutions also show f (x) ≤ 0 for all real numbers x. As one can see from Solution 1, this task gets much easier if one already knows that f takes nonnegative values for su?ciently small arguments. Another way of arriving at this statement, suggested by the proposer, is as follows: Put a = f (0) and substitute x = 0 into (1). This gives f (y ) ≤ ay + f (a) for all real numbers y . Thus if for any real number x we plug y = a ? x into (1), we obtain f (a) ≤ (a ? x)f (x) + f (f (x)) ≤ (a ? x)f (x) + af (x) + f (a) and hence 0 ≤ (2a ? x)f (x). In particular, if x < 2a, then f (x) ≥ 0. Having reached this point, one may proceed almost exactly as in the ?rst solution to deduce f (x) ≤ 0 for all x. Afterwards the problem can be solved in a few lines as shown in steps 3 and 4 of the second

(5)

22

52nd IMO 2011

Algebra – solutions

A6

solution. Comment 2. The original problem also contained the question whether a nonzero function satisfying the problem condition exists. Here we present a family of such functions. Notice ?rst that if g : (0, ∞) ?→ [0, ∞) denotes any function such that g(x + y ) ≥ yg(x) for all positive real numbers x and y , then the function f given by ? ??g(x) if x > 0 f (x) = ?0 if x ≤ 0 (6)

(7)

automatically satis?es (1). Indeed, we have f (x) ≤ 0 and hence also f (f (x)) = 0 for all real numbers x. So (1) reduces to (5); moreover, this inequality is nontrivial only if x and y are positive. In this last case it is provided by (6). Now it is not hard to come up with a nonzero function g obeying (6). E.g. g(z ) = Cez (where C is a positive constant) ?ts since the inequality ey > y holds for all (positive) real numbers y . One may also consider the function g(z ) = ez ? 1; in this case, we even have that f is continuous.

23

A7

A7

Algebra – solutions

52nd IMO 2011

Let a, b, and c be positive real numbers satisfying min(a + b, b + c, c + a) > Prove that

√

2 and a2 + b2 + c2 = 3.

b c 3 a + + ≥ . 2 2 2 (b + c ? a) (c + a ? b) ( a + b ? c) (abc)2

(1)

Throughout both solutions, we denote the sums of the form f (a, b, c) + f (b, c, a) + f (c, a, b) by f (a, b, c). √ Solution 1. The condition b + c > 2 implies b2 + c2 > 1, so a2 = 3 ? (b2 + c2 ) < 2, i.e. √ a < 2 < b + c. Hence we have b + c ? a > 0, and also c + a ? b > 0 and a + b ? c > 0 for similar reasons. ¨ lder’s inequality We will use the variant of Ho

+1 +1 +1 xp (x1 + x2 + . . . + xn )p+1 xp xp 1 1 n ≥ , + + . . . + p p p y1 y1 yn (y 1 + y 2 + . . . + y n )p

which holds for all positive real numbers p, x1 , x2 , . . . , xn , y1, y2 , . . . , yn . Applying it to the left-hand side of (1) with p = 2 and n = 3, we get a = (b + c ? a)2 (a2 )3 ≥ a5 (b + c ? a)2 (a2 + b2 + c2 )3 a5/2 (b + c ? a)

2

=

27 a5/2 (b + c ? a)

2.

(2)

To estimate the denominator of the right-hand part, we use an instance of Schur’s inequality, namely a3/2 (a ? b)(a ? c) ≥ 0, which can be rewritten as √ √ √ a5/2 (b + c ? a) ≤ abc( a + b + c). Moreover, by the inequality between the arithmetic mean and the fourth power mean we also have √ √ 4 √ a+ b+ c a2 + b2 + c2 = 1, ≤ 3 3 √ √ √ i.e., a + b + c ≤ 3. Hence, (2) yields 27 a ≥ √ √ √ 2 (b + c ? a) abc( a + b + c) thus solving the problem.

2

≥

3 a2 b2 c2

,

24

52nd IMO 2011

Algebra – solutions

A7

3

¨ lder’s inequality Comment. In this solution, one may also start from the following version of Ho

n n n n

a3 i

i=1 i=1

b3 i

i=1

c3 i

≥

ai bi ci

i=1

applied as

a · (b + c ? a)2

a3 (b + c ? a) ·

a2 (b + c ? a) ≥ 27.

After doing that, one only needs the slightly better known instances a3 (b + c ? a) ≤ (a + b + c)abc of Schur’s Inequality. and a2 (b + c ? a) ≤ 3abc

Solution 2. As in Solution 1, we mention that all the numbers b + c ? a, a + c ? b, a + b ? c are positive. We will use only this restriction and the condition a5 + b5 + c5 ≥ 3, which is weaker than the given one. Due to the symmetry we may assume that a ≥ b ≥ c. In view of (3), it su?ces to prove the inequality a3 b2 c2 ≥ (b + c ? a)2 or, moving all the terms into the left-hand part, a3 (bc)2 ? (a(b + c ? a))2 ≥ 0. 2 (b + c ? a) (4) a5 , (3)

Note that the signs of the expressions (yz )2 ?(x(y + z ? x))2 and yz ?x(y + z ?x) = (x?y )(x?z ) are the same for every positive x, y , z satisfying the triangle inequality. So the terms in (4) corresponding to a and c are nonnegative, and hence it is su?cient to prove that the sum of the terms corresponding to a and b is nonnegative. Equivalently, we need the relation b3 a3 ( a ? b )( a ? c )( bc + a ( b + c ? a )) ≥ (a ? b)(b ? c)(ac + b(a + c ? b)). (b + c ? a)2 (a + c ? b)2 Obviously, we have a3 ≥ b3 ≥ 0, 0 < b + c ? a ≤ a + c ? b, and a ? c ≥ b ? c ≥ 0,

hence it su?ces to prove that ab + ac + bc ? b2 ab + ac + bc ? a2 ≥ . b+c?a c+a?b 25

A7

Algebra – solutions

52nd IMO 2011

Since all the denominators are positive, it is equivalent to (c + a ? b)(ab + ac + bc ? a2 ) ? (ab + ac + bc ? b2 )(b + c ? a) ≥ 0, or (a ? b)(2ab ? a2 ? b2 + ac + bc) ≥ 0. Since a ≥ b, the last inequality follows from c(a + b) > (a ? b)2 which holds since c > a ? b ≥ 0 and a + b > a ? b ≥ 0.

26

52nd IMO 2011

Combinatorics – solutions

C1

C1

Let n > 0 be an integer. We are given a balance and n weights of weight 20 , 21 , . . . , 2n?1 . In a sequence of n moves we place all weights on the balance. In the ?rst move we choose a weight and put it on the left pan. In each of the following moves we choose one of the remaining weights and we add it either to the left or to the right pan. Compute the number of ways in which we can perform these n moves in such a way that the right pan is never heavier than the left pan.

Answer. The number f (n) of ways of placing the n weights is equal to the product of all odd positive integers less than or equal to 2n ? 1, i.e. f (n) = (2n ? 1)!! = 1 · 3 · 5 · . . . · (2n ? 1). Solution 1. Assume n ≥ 2. We claim f (n) = (2n ? 1)f (n ? 1). (1)

Firstly, note that after the ?rst move the left pan is always at least 1 heavier than the right one. Hence, any valid way of placing the n weights on the scale gives rise, by not considering weight 1, to a valid way of placing the weights 2, 22 , . . . , 2n?1 . If we divide the weight of each weight by 2, the answer does not change. So these n ? 1 weights can be placed on the scale in f (n ? 1) valid ways. Now we look at weight 1. If it is put on the scale in the ?rst move, then it has to be placed on the left side, otherwise it can be placed either on the left or on the right side, because after the ?rst move the di?erence between the weights on the left pan and the weights on the right pan is at least 2. Hence, there are exactly 2n ? 1 di?erent ways of inserting weight 1 in each of the f (n ? 1) valid sequences for the n ? 1 weights in order to get a valid sequence for the n weights. This proves the claim. Since f (1) = 1, by induction we obtain for all positive integers n f (n) = (2n ? 1)!! = 1 · 3 · 5 · . . . · (2n ? 1).

Comment 1. The word “compute” in the statement of the problem is probably too vague. An alternative but more arti?cial question might ask for the smallest n for which the number of valid ways is divisible by 2011. In this case the answer would be 1006. Comment 2. It is useful to remark that the answer is the same for any set of weights where each weight is heavier than the sum of the lighter ones. Indeed, in such cases the given condition is equivalent to asking that during the process the heaviest weight on the balance is always on the left pan. Comment 3. Instead of considering the lightest weight, one may also consider the last weight put on the balance. If this weight is 2n?1 then it should be put on the left pan. Otherwise it may be put on

27

C1

Combinatorics – solutions

52nd IMO 2011

any pan; the inequality would not be violated since at this moment the heaviest weight is already put onto the left pan. In view of the previous comment, in each of these 2n ? 1 cases the number of ways to place the previous weights is exactly f (n ? 1), which yields (1).

Solution 2. We present a di?erent way of obtaining (1). Set f (0) = 1. Firstly, we ?nd a recurrent formula for f (n). Assume n ≥ 1. Suppose that weight 2n?1 is placed on the balance in the i-th move with ?1 1 ≤ i ≤ n. This weight has to be put on the left pan. For the previous moves we have n i? 1 choices of the weights and from Comment 2 there are f (i ? 1) valid ways of placing them on the balance. For later moves there is no restriction on the way in which the weights are to be put on the pans. Therefore, all (n ? i)!2n?i ways are possible. This gives

n

f (n) =

i=1

n?1 f (i ? 1)(n ? i)!2n?i = i?1

n

i=1

(n ? 1)!f (i ? 1)2n?i . (i ? 1)!

(2)

Now we are ready to prove (1). Using n ? 1 instead of n in (2) we get

n ?1

f (n ? 1) = Hence, again from (2) we get

n ?1

i=1

(n ? 2)!f (i ? 1)2n?1?i . (i ? 1)!

f (n) = 2(n ? 1)

i=1

(n ? 2)!f (i ? 1)2n?1?i + f (n ? 1) (i ? 1)! = (2n ? 2)f (n ? 1) + f (n ? 1) = (2n ? 1)f (n ? 1),

QED.

Comment. There exist di?erent ways of obtaining the formula (2). Here we show one of them. Suppose that in the ?rst move we use weight 2n?i+1 . Then the lighter n ? i weights may be put on the balance at any moment and on either pan. This gives 2n?i · (n ? 1)!/(i ? 1)! choices for the moves (moments and choices of pan) with the lighter weights. The remaining i ? 1 moves give a valid sequence for the i ? 1 heavier weights and this is the only requirement for these moves, so there are f (i ? 1) such sequences. Summing over all i = 1, 2, . . . , n we again come to (2).

28

52nd IMO 2011

Combinatorics – solutions

C2

C2

Suppose that 1000 students are standing in a circle. Prove that there exists an integer k with 100 ≤ k ≤ 300 such that in this circle there exists a contiguous group of 2k students, for which the ?rst half contains the same number of girls as the second half.

Solution. Number the students consecutively from 1 to 1000. Let ai = 1 if the ith student is a girl, and ai = 0 otherwise. We expand this notion for all integers i by setting ai+1000 = ai?1000 = ai . Next, let Sk (i) = ai + ai+1 + · · · + ai+k?1 . Now the statement of the problem can be reformulated as follows: There exist an integer k with 100 ≤ k ≤ 300 and an index i such that Sk (i) = Sk (i + k ). Assume now that this statement is false. Choose an index i such that S100 (i) attains the maximal possible value. In particular, we have S100 (i ? 100) ? S100 (i) < 0 and S100 (i) ? S100 (i + 100) > 0, for if we had an equality, then the statement would hold. This means that the function S (j ) ? S (j + 100) changes sign somewhere on the segment [i ? 100, i], so there exists some index j ∈ [i ? 100, i ? 1] such that S100 (j ) ≤ S100 (j + 100) ? 1, but S100 (j + 1) ≥ S100 (j + 101) + 1. (1)

Subtracting the ?rst inequality from the second one, we get aj +100 ? aj ≥ aj +200 ? aj +100 + 2, so aj = 0, aj +100 = 1, aj +200 = 0.

Substituting this into the inequalities of (1), we also obtain S99 (j +1) ≤ S99 (j +101) ≤ S99 (j +1), which implies S99 (j + 1) = S99 (j + 101). (2) Now let k and ? be the least positive integers such that aj ?k = 1 and aj +200+? = 1. By symmetry, we may assume that k ≥ ?. If k ≥ 200 then we have aj = aj ?1 = · · · = aj ?199 = 0, so S100 (j ?199) = S100 (j ?99) = 0, which contradicts the initial assumption. Hence ? ≤ k ≤ 199. Finally, we have S100+? (j ? ? + 1) = (aj ??+1 + · · · + aj ) + S99 (j + 1) + aj +100 = S99 (j + 1) + 1,

S100+? (j + 101) = S99 (j + 101) + (aj +200 + · · · + aj +200+??1 ) + aj +200+? = S99 (j + 101) + 1. Comparing with (2) we get S100+? (j ? ? + 1) = S100+? (j + 101) and 100 + ? ≤ 299, which again contradicts our assumption.

Comment. It may be seen from the solution that the number 300 from the problem statement can be

29

C2

Combinatorics – solutions

52nd IMO 2011

replaced by 299. Here we consider some improvements of this result. Namely, we investigate which interval can be put instead of [100, 300] in order to keep the problem statement valid. First of all, the two examples 1, 1, . . . , 1, 0, 0, . . . , 0, 1, 1, . . . , 1, 0, 0, . . . , 0, 1, 1, . . . , 1, 0, 0, . . . , 0

167 167 167 167 167 165

and 1, 1, . . . , 1, 0, 0, . . . , 0, 1, 1, . . . , 1, 0, 0, . . . , 0

249 251 249 251

show that the interval can be changed neither to [84, 248] nor to [126, 374]. On the other hand, we claim that this interval can be changed to [125, 250]. Note that this statement is invariant under replacing all 1’s by 0’s and vice versa. Assume, to the contrary, that there is no admissible k ∈ [125, 250]. The arguments from the solution easily yield the following lemma. Lemma. Under our assumption, suppose that for some indices i < j we have S125 (i) ≤ S125 (i + 125) but S125 (j ) ≥ S125 (j +125). Then there exists some t ∈ [i, j ? 1] such that at = at?1 = · · · = at?125 = 0 and at+250 = at+251 = · · · = at+375 = 0. Let us call a segment [i, j ] of indices a crowd, if (a) ai = ai+1 = · · · = aj , but ai?1 = ai = aj +1 , and (b) j ? i ≥ 125. Now, using the lemma, one can get in the same way as in the solution that there exists some crowd. Take all the crowds in the circle, and enumerate them in cyclic order as A1 , . . . , Ad . We also assume always that As+d = As?d = As . Consider one of the crowds, say A1 . We have A1 = [i, i + t] with 125 ≤ t ≤ 248 (if t ≥ 249, then ai = ai+1 = · · · = ai+249 and therefore S125 (i) = S125 (i + 125), which contradicts our assumption). We may assume that ai = 1. Then we have S125 (i + t ? 249) ≤ 125 = S125 (i + t ? 124) and S125 (i) = 125 ≥ S125 (i + 125), so by the lemma there exists some index j ∈ [i + t ? 249, i ? 1] such that the segments [j ? 125, j ] and [j + 250, j + 375] are contained in some crowds. Let us ?x such j and denote the segment [j + 1, j + 249] by B1 . Clearly, A1 ? B1 . Moreover, B1 cannot contain any crowd other than A1 since |B1 | = 249 < 2 · 126. Hence it is clear that j ∈ Ad and j + 250 ∈ A2 . In particular, this means that the genders of Ad and A2 are di?erent from that of A1 . Performing this procedure for every crowd As , we ?nd segments Bs = [js + 1, js + 249] such that |Bs | = 249, As ? Bs , and js ∈ As?1 , js + 250 ∈ As+1 . So, Bs covers the whole segment between As?1 and As+1 , hence the sets B1 , . . . , Bd cover some 1000 consecutive indices. This implies 249d ≥ 1000, and d ≥ 5. Moreover, the gender of Ai is alternating, so d is even; therefore d ≥ 6.

′ Consider now three segments A1 = [i1 , i′ 1 ], B2 = [j2 + 1, j2 + 249], A3 = [i3 , i3 ]. By construction, we have [j2 ? 125, j2 ] ? A1 and [j2 + 250, j2 + 375] ? A3 , whence i1 ≤ j2 ? 125, i′ 3 ≥ j2 + 375. Therefore ′ ′ ′ i′ 3 ? i1 ≥ 500. Analogously, if A4 = [i4 , i4 ], A6 = [i6 , i6 ] then i6 ? i4 ≥ 500. But from d ≥ 6 we get ′ ′ ′ i1 < i′ 3 < i4 < i6 < i1 + 1000, so 1000 > (i3 ? i1 ) + (i6 ? i4 ) ≥ 500 + 500. This ?nal contradiction shows that our claim holds.

One may even show that the interval in the statement of the problem may be replaced by [125, 249] (both these numbers cannot be improved due to the examples above). But a proof of this fact is a bit messy, and we do not present it here.

30

52nd IMO 2011

Combinatorics – solutions

C3

C3

Let S be a ?nite set of at least two points in the plane. Assume that no three points of S are collinear. By a windmill we mean a process as follows. Start with a line ? going through a point P ∈ S . Rotate ? clockwise around the pivot P until the line contains another point Q of S . The point Q now takes over as the new pivot. This process continues inde?nitely, with the pivot always being a point from S . Show that for a suitable P ∈ S and a suitable starting line ? containing P , the resulting windmill will visit each point of S as a pivot in?nitely often.

Solution. Give the rotating line an orientation and distinguish its sides as the oranje side and the blue side. Notice that whenever the pivot changes from some point T to another point U , after the change, T is on the same side as U was before. Therefore, the number of elements of S on the oranje side and the number of those on the blue side remain the same throughout the whole process (except for those moments when the line contains two points).

U

U

U

T

T

T

First consider the case that |S| = 2n + 1 is odd. We claim that through any point T ∈ S , there is a line that has n points on each side. To see this, choose an oriented line through T containing no other point of S and suppose that it has n + r points on its oranje side. If r = 0 then we have established the claim, so we may assume that r = 0. As the line rotates through 180? around T , the number of points of S on its oranje side changes by 1 whenever the line passes through a point; after 180?, the number of points on the oranje side is n ? r . Therefore there is an intermediate stage at which the oranje side, and thus also the blue side, contains n points. Now select the point P arbitrarily, and choose a line through P that has n points of S on each side to be the initial state of the windmill. We will show that during a rotation over 180? , the line of the windmill visits each point of S as a pivot. To see this, select any point T of S and select a line ? through T that separates S into equal halves. The point T is the unique point of S through which a line in this direction can separate the points of S into equal halves (parallel translation would disturb the balance). Therefore, when the windmill line is parallel to ?, it must be ? itself, and so pass through T . Next suppose that |S| = 2n. Similarly to the odd case, for every T ∈ S there is an oriented

31

C3

Combinatorics – solutions

52nd IMO 2011

line through T with n ? 1 points on its oranje side and n points on its blue side. Select such an oriented line through an arbitrary P to be the initial state of the windmill. We will now show that during a rotation over 360? , the line of the windmill visits each point of S as a pivot. To see this, select any point T of S and an oriented line ? through T that separates S into two subsets with n ? 1 points on its oranje and n points on its blue side. Again, parallel translation would change the numbers of points on the two sides, so when the windmill line is parallel to ? with the same orientation, the windmill line must pass through T .

Comment. One may shorten this solution in the following way. Suppose that |S| = 2n + 1. Consider any line ? that separates S into equal halves; this line is unique given its direction and contains some point T ∈ S . Consider the windmill starting from this line. When the line has made a rotation of 180? , it returns to the same location but the oranje side becomes blue and vice versa. So, for each point there should have been a moment when it appeared as pivot, as this is the only way for a point to pass from on side to the other. Now suppose that |S| = 2n. Consider a line having n ? 1 and n points on the two sides; it contains some point T . Consider the windmill starting from this line. After having made a rotation of 180? , the windmill line contains some di?erent point R, and each point di?erent from T and R has changed the color of its side. So, the windmill should have passed through all the points.

32

52nd IMO 2011

Combinatorics – solutions

C4

C4

Determine the greatest positive integer k that satis?es the following property: The set of positive integers can be partitioned into k subsets A1 , A2 , . . . , Ak such that for all integers n ≥ 15 and all i ∈ {1, 2, . . . , k } there exist two distinct elements of Ai whose sum is n.

Answer. The greatest such number k is 3. Solution 1. There are various examples showing that k = 3 does indeed have the property under consideration. E.g. one can take A1 = {1, 2, 3} ∪ {3m | m ≥ 4},

A3 = {7, 8, 9} ∪ {3m ? 2 | m ≥ 4}.

A2 = {4, 5, 6} ∪ {3m ? 1 | m ≥ 4},

To check that this partition ?ts, we notice ?rst that the sums of two distinct elements of Ai obviously represent all numbers n ≥ 1 + 12 = 13 for i = 1, all numbers n ≥ 4 + 11 = 15 for i = 2, and all numbers n ≥ 7 + 10 = 17 for i = 3. So, we are left to ?nd representations of the numbers 15 and 16 as sums of two distinct elements of A3 . These are 15 = 7 + 8 and 16 = 7 + 9.

Let us now suppose that for some k ≥ 4 there exist sets A1 , A2 , . . . , Ak satisfying the given property. Obviously, the sets A1 , A2 , A3 , A4 ∪ · · · ∪ Ak also satisfy the same property, so one may assume k = 4. Put Bi = Ai ∩ {1, 2, . . . , 23} for i = 1, 2, 3, 4. Now for any index i each of the ten numbers 15, 16, . . . , 24 can be written as sum of two distinct elements of Bi . Therefore this set needs to contain at least ?ve elements. As we also have |B1 | + |B2 | + |B3 | + |B4 | = 23, there has to be some index j for which |Bj | = 5. Let Bj = {x1 , x2 , x3 , x4 , x5 }. Finally, now the sums of two distinct elements of Aj representing the numbers 15, 16, . . . , 24 should be exactly all the pairwise sums of the elements of Bj . Calculating the sum of these numbers in two di?erent ways, we reach 4(x1 + x2 + x3 + x4 + x5 ) = 15 + 16 + . . . + 24 = 195. Thus the number 195 should be divisible by 4, which is false. This contradiction completes our solution.

Comment. There are several variation of the proof that k should not exceed 3. E.g., one may consider the sets Ci = Ai ∩ {1, 2, . . . , 19} for i = 1, 2, 3, 4. As in the previous solution one can show that for some index j one has |Cj | = 4, and the six pairwise sums of the elements of Cj should represent all numbers 15, 16, . . . , 20. Let Cj = {y1 , y2 , y3 , y4 } with y1 < y2 < y3 < y4 . It is not hard to deduce

33

C4

Combinatorics – solutions

52nd IMO 2011

Cj = {7, 8, 9, 11}, so in particular we have 1 ∈ Cj . Hence it is impossible to represent 21 as sum of two distinct elements of Aj , which completes our argument.

Solution 2. Again we only prove that k ≤ 3. Assume that A1 , A2 , . . . , Ak is a partition satisfying the given property. We construct a graph G on the set V = {1, 2, . . . , 18} of vertices as follows. For each i ∈ {1, 2, . . . , k } and each d ∈ {15, 16, 17, 19} we choose one pair of distinct elements a, b ∈ Ai with a + b = d, and we draw an edge in the ith color connecting a with b. By hypothesis, G has exactly 4 edges of each color. Claim. The graph G contains at most one circuit. Proof. Note that all the connected components of G are monochromatic and hence contain at most four edges. Thus also all circuits of G are monochromatic and have length at most four. Moreover, each component contains at most one circuit since otherwise it should contain at least ?ve edges. Suppose that there is a 4-cycle in G , say with vertices a, b, c, and d in order. Then {a + b, b + c, c + d, d + a} = {15, 16, 17, 19}. Taking sums we get 2(a + b + c + d) = 15 + 16 + 17 + 19 which is impossible for parity reasons. Thus all circuits of G are triangles. Now if the vertices a, b, and c form such a triangle, then by a similar reasoning the set {a + b, b + c, c + a} coincides with either {15, 16, 17}, or {15, 16, 19}, or {16, 17, 19}, or {15, 17, 19}. The last of these alternatives can be excluded for parity reasons again, whilst in the ?rst three cases the set {a, b, c} appears to be either {7, 8, 9}, or {6, 9, 10}, or {7, 9, 10}, respectively. Thus, a component containing a circuit should contain 9 as a vertex. Therefore there is at most one such component and hence at most one circuit. By now we know that G is a graph with 4k edges, at least k components and at most one circuit. Consequently, G must have at least 4k + k ? 1 vertices. Thus 5k ? 1 ≤ 18, and k ≤ 3.

34

52nd IMO 2011

Combinatorics – solutions

C5

C5

Let m be a positive integer and consider a checkerboard consisting of m by m unit squares. At the midpoints of some of these unit squares there is an ant. At time 0, each ant starts moving with speed 1 parallel to some edge of the checkerboard. When two ants moving in opposite directions meet, they both turn 90? clockwise and continue moving with speed 1. When more than two ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls o? and will not re-appear. Considering all possible starting positions, determine the latest possible moment at which the last ant falls o? the checkerboard or prove that such a moment does not necessarily exist.

Antswer. The latest possible moment for the last ant to fall o? is

3m 2

? 1.

Solution. For m = 1 the answer is clearly correct, so assume m > 1. In the sequel, the word collision will be used to denote meeting of exactly two ants, moving in opposite directions. If at the beginning we place an ant on the southwest corner square facing east and an ant on the southeast corner square facing west, then they will meet in the middle of the bottom row ?1 at time m2 . After the collision, the ant that moves to the north will stay on the board for time units and thus we have established an example in which the last ant falls another m ? 1 2 1 m m?1 = 32 ? 1. So, we are left to prove that this is the latest possible o? at time 2 + m ? 2 moment. Consider any collision of two ants a and a′ . Let us change the rule for this collision, and enforce these two ants to turn anticlockwise. Then the succeeding behavior of all the ants does not change; the only di?erence is that a and a′ swap their positions. These arguments may be applied to any collision separately, so we may assume that at any collision, either both ants rotate clockwise or both of them rotate anticlockwise by our own choice. For instance, we may assume that there are only two types of ants, depending on their initial direction: NE-ants, which move only north or east, and SW-ants, moving only south and west. Then we immediately obtain that all ants will have fallen o? the board after 2m ? 1 time units. However, we can get a better bound by considering the last moment at which a given ant collides with another ant. Choose a coordinate system such that the corners of the checkerboard are (0, 0), (m, 0), (m, m) and (0, m). At time t, there will be no NE-ants in the region {(x, y ) : x + y < t + 1} and no SW-ants in the region {(x, y ) : x + y > 2m ? t ? 1}. So if two ants collide at (x, y ) at time t, we have t + 1 ≤ x + y ≤ 2m ? t ? 1. (1)

35

C5

Combinatorics – solutions

52nd IMO 2011

Analogously, we may change the rules so that each ant would move either alternatingly north and west, or alternatingly south and east. By doing so, we ?nd that apart from (1) we also have |x ? y | ≤ m ? t ? 1 for each collision at point (x, y ) and time t. To visualize this, put B (t) = (x, y ) ∈ [0, m]2 : t + 1 ≤ x + y ≤ 2m ? t ? 1 and |x ? y | ≤ m ? t ? 1 . An ant can thus only collide with another ant at time t if it happens to be in the region B (t). The following ?gure displays B (t) for t = 1 and t = 7 in the case m = 6: 2 2

Now suppose that an NE-ant has its last collision at time t and that it does so at the point (x, y ) m < 32 ?1 time units, so this (if the ant does not collide at all, it will fall o? the board within m? 1 2 case can be ignored). Then we have (x, y ) ∈ B (t) and thus x + y ≥ t +1 and x ? y ≥ ?(m ? t ? 1). So we get (t + 1) ? (m ? t ? 1) m x≥ =t+1? . 2 2 m By symmetry we also have y ≥ t + 1 ? 2 , and hence min{x, y } ≥ t + 1 ? m . After this collision, 2 the ant will move directly to an edge, which will take at most m ? min{x, y } units of time. In sum, the total amount of time the ant stays on the board is at most t + (m ? min{x, y }) ≤ t + m ? t + 1 ? By symmetry, the same bound holds for SW-ants as well. 3m m = ? 1. 2 2

36

52nd IMO 2011

Combinatorics – solutions

C6

C6

Let n be a positive integer and let W = . . . x?1 x0 x1 x2 . . . be an in?nite periodic word consisting of the letters a and b. Suppose that the minimal period N of W is greater than 2n . A ?nite nonempty word U is said to appear in W if there exist indices k ≤ ? such that U = xk xk+1 . . . x? . A ?nite word U is called ubiquitous if the four words Ua, Ub, aU , and bU all appear in W . Prove that there are at least n ubiquitous ?nite nonempty words.

Solution. Throughout the solution, all the words are nonempty. For any word R of length m, we call the number of indices i ∈ {1, 2, . . . , N } for which R coincides with the subword xi+1 xi+2 . . . xi+m of W the multiplicity of R and denote it by ?(R). Thus a word R appears in W if and only if ?(R) > 0. Since each occurrence of a word in W is both succeeded by either the letter a or the letter b and similarly preceded by one of those two letters, we have ?(R) = ?(Ra) + ?(Rb) = ?(aR) + ?(bR) for all words R. We claim that the condition that N is in fact the minimal period of W guarantees that each word of length N has multiplicity 1 or 0 depending on whether it appears or not. Indeed, if the words xi+1 xi+2 . . . xi+N and xj +1 . . . xj +N are equal for some 1 ≤ i < j ≤ N , then we have xi+a = xj +a for every integer a, and hence j ? i is also a period. Moreover, since N > 2n , at least one of the two words a and b has a multiplicity that is strictly larger than 2n?1 . For each k = 0, 1, . . . , n ? 1, let Uk be a subword of W whose multiplicity is strictly larger than 2k and whose length is maximal subject to this property. Note that such a word exists in view of the two observations made in the two previous paragraphs. Fix some index k ∈ {0, 1, . . . , n ? 1}. Since the word Uk b is longer than Uk , its multiplicity can be at most 2k , so in particular ?(Uk b) < ?(Uk ). Therefore, the word Uk a has to appear by (1). For a similar reason, the words Uk b, aUk , and bUk have to appear as well. Hence, the word Uk is ubiquitous. Moreover, if the multiplicity of Uk were strictly greater than 2k+1 , then by (1) at least one of the two words Uk a and Uk b would have multiplicity greater than 2k and would thus violate the maximality condition imposed on Uk . So we have ?(U0 ) ≤ 2 < ?(U1 ) ≤ 4 < . . . ≤ 2n?1 < ?(Un?1 ), which implies in particular that the words U0 , U1 , . . . , Un?1 have to be distinct. As they have been proved to be ubiquitous as well, the problem is solved.

Comment 1. There is an easy construction for obtaining ubiquitous words from appearing words whose multiplicity is at least two. Starting with any such word U we may simply extend one of its occurrences in W forwards and backwards as long as its multiplicity remains ?xed, thus arriving at a

(1)

37

C6

Combinatorics – solutions

52nd IMO 2011

word that one might call the ubiquitous prolongation p(U ) of U . There are several variants of the argument in the second half of the solution using the concept of prolongation. For instance, one may just take all ubiquitous words U1 , U2 , . . . , U? ordered by increasing multiplicity and then prove for i ∈ {1, 2, . . . , ?} that ?(Ui ) ≤ 2i . Indeed, assume that i is a minimal counterexample to this statement; then by the arguments similar to those presented above, the ubiquitous prolongation of one of the words Ui a, Ui b, aUi or bUi violates the de?nition of Ui . Now the multiplicity of one of the two letters a and b is strictly greater than 2n?1 , so passing to ubiquitous prolongations once more we obtain 2n?1 < ?(U? ) ≤ 2? , which entails ? ≥ n, as needed. Comment 2. The bound n for the number of ubiquitous subwords in the problem statement is not optimal, but it is close to an optimal one in the following sense. There is a universal constant C > 0 such that for each positive integer n there exists an in?nite periodic word W whose minimal period is greater than 2n but for which there exist fewer than Cn ubiquitous words.

38

52nd IMO 2011

Combinatorics – solutions

C7

C7

On a square table of 2011 by 2011 cells we place a ?nite number of napkins that each cover a square of 52 by 52 cells. In each cell we write the number of napkins covering it, and we record the maximal number k of cells that all contain the same nonzero number. Considering all possible napkin con?gurations, what is the largest value of k ?

Answer. 20112 ? (522 ? 352 ) · 39 ? 172 = 4044121 ? 57392 = 3986729. Solution 1. Let m = 39, then 2011 = 52m ? 17. We begin with an example showing that there can exist 3986729 cells carrying the same positive number.

To describe it, we number the columns from the left to the right and the rows from the bottom to the top by 1, 2, . . . , 2011. We will denote each napkin by the coordinates of its lowerleft cell. There are four kinds of napkins: ?rst, we take all napkins (52i + 36, 52j + 1) with 0 ≤ j ≤ i ≤ m ? 2; second, we use all napkins (52i + 1, 52j + 36) with 0 ≤ i ≤ j ≤ m ? 2; third, we use all napkins (52i + 36, 52i + 36) with 0 ≤ i ≤ m ? 2; and ?nally the napkin (1, 1). Di?erent groups of napkins are shown by di?erent types of hatchings in the picture. Now except for those squares that carry two or more di?erent hatchings, all squares have the number 1 written into them. The number of these exceptional cells is easily computed to be (522 ? 352 )m ? 172 = 57392. We are left to prove that 3986729 is an upper bound for the number of cells containing the same number. Consider any con?guration of napkins and any positive integer M . Suppose there are g cells with a number di?erent from M . Then it su?ces to show g ≥ 57392. Throughout the solution, a line will mean either a row or a column. Consider any line ?. Let a1 , . . . , a52m?17 be the numbers written into its consecutive cells. For i = 1, 2, . . . , 52, let si = t≡i (mod 52) at . Note that s1 , . . . , s35 have m terms each, while s36 , . . . , s52 have m ? 1 terms each. Every napkin intersecting ? contributes exactly 1 to each si ;

39

C7

Combinatorics – solutions

52nd IMO 2011

hence the number s of all those napkins satis?es s1 = · · · = s52 = s. Call the line ? rich if s > (m ? 1)M and poor otherwise. Suppose now that ? is rich. Then in each of the sums s36 , . . . , s52 there exists a term greater than M ; consider all these terms and call the corresponding cells the rich bad cells for this line. So, each rich line contains at least 17 cells that are bad for this line. If, on the other hand, ? is poor, then certainly s < mM so in each of the sums s1 , . . . , s35 there exists a term less than M ; consider all these terms and call the corresponding cells the poor bad cells for this line. So, each poor line contains at least 35 cells that are bad for this line. Let us call all indices congruent to 1, 2, . . . , or 35 modulo 52 small, and all other indices, i.e. those congruent to 36, 37, . . . , or 52 modulo 52, big. Recall that we have numbered the columns from the left to the right and the rows from the bottom to the top using the numbers 1, 2, . . . , 52m ? 17; we say that a line is big or small depending on whether its index is big or small. By de?nition, all rich bad cells for the rows belong to the big columns, while the poor ones belong to the small columns, and vice versa. In each line, we put a strawberry on each cell that is bad for this line. In addition, for each small rich line we put an extra strawberry on each of its (rich) bad cells. A cell gets the strawberries from its row and its column independently. Notice now that a cell with a strawberry on it contains a number di?erent from M . If this cell gets a strawberry by the extra rule, then it contains a number greater than M . Moreover, it is either in a small row and in a big column, or vice versa. Suppose that it is in a small row, then it is not bad for its column. So it has not more than two strawberries in this case. On the other hand, if the extra rule is not applied to some cell, then it also has not more than two strawberries. So, the total number N of strawberries is at most 2g . We shall now estimate N in a di?erent way. For each of the 2 · 35m small lines, we have introduced at least 34 strawberries if it is rich and at least 35 strawberries if it is poor, so at least 34 strawberries in any case. Similarly, for each of the 2 · 17(m ? 1) big lines, we put at least min(17, 35) = 17 strawberries. Summing over all lines we obtain 2g ≥ N ≥ 2(35m · 34 + 17(m ? 1) · 17) = 2(1479m ? 289) = 2 · 57392, as desired.

Comment. The same reasoning applies also if we replace 52 by R and 2011 by Rm ? H , where m, R, 1 R. More detailed information is provided after the and H are integers with m, R ≥ 1 and 0 ≤ H ≤ 3 next solution.

Solution 2. We present a di?erent proof of the estimate which is the hard part of the problem. Let S = 35, H = 17, m = 39; so the table size is 2011 = Sm + H (m ? 1), and the napkin size is 52 = S + H . Fix any positive integer M and call a cell vicious if it contains a number distinct

40

52nd IMO 2011

Combinatorics – solutions

C7

from M . We will prove that there are at least H 2 (m ? 1) + 2SHm vicious cells. Firstly, we introduce some terminology. As in the previous solution, we number rows and columns and we use the same notions of small and big indices and lines; so, an index is small if it is congruent to one of the numbers 1, 2, . . . , S modulo (S + H ). The numbers 1, 2, . . . , S + H will be known as residues. For two residues i and j , we say that a cell is of type (i, j ) if the index of its row is congruent to i and the index of its column to j modulo (S + H ). The number of vicious cells of this type is denoted by vij . Let s, s′ be two variables ranging over small residues and let h, h′ be two variables ranging over big residues. A cell is said to be of class A, B , C , or D if its type is of shape (s, s′ ), (s, h), (h, s), or (h, h′ ), respectively. The numbers of vicious cells belonging to these classes are denoted in this order by a, b, c, and d. Observe that each cell belongs to exactly one class. Claim 1. We have m≤ a b+c + . 2 S 2SH

(1)

Proof. Consider an arbitrary small row r . Denote the numbers of vicious cells on r belonging to the classes A and B by α and β , respectively. As in the previous solution, we obtain that β α α ≥ S or β ≥ H . So in each case we have S +H ≥ 1. Performing this argument separately for each small row and adding up all the obtained inequalb a c a +H ≥ mS . Interchanging rows and columns we similarly get S +H ≥ mS . ities, we get S Summing these inequalities and dividing by 2S we get what we have claimed. Claim 2. Fix two small residue s, s′ and two big residues h, h′ . Then 2m ? 1 ≤ vss′ + vsh′ + vhh′ . Proof. Each napkin covers exactly one cell of type (s, s′ ). Removing all napkins covering a vicious cell of this type, we get another collection of napkins, which covers each cell of type (s, s′ ) either 0 or M times depending on whether the cell is vicious or not. Hence (m2 ? vss′ )M napkins are left and throughout the proof of Claim 2 we will consider only these remaining napkins. Now, using a red pen, write in each cell the number of napkins covering it. Notice that a cell containing a red number greater than M is surely vicious. We call two cells neighbors if they can be simultaneously covered by some napkin. So, each cell of type (h, h′ ) has not more than four neighbors of type (s, s′), while each cell of type (s, h′ ) has not more than two neighbors of each of the types (s, s′) and (h, h′ ). Therefore, each red number at a cell of type (h, h′ ) does not exceed 4M , while each red number at a cell of type (s, h′ ) does not exceed 2M . Let x, y , and z be the numbers of cells of type (h, h′ ) whose red number belongs to (M, 2M ], (2M, 3M ], and (3M, 4M ], respectively. All these cells are vicious, hence x + y + z ≤ vhh′ . The red numbers appearing in cells of type (h, h′ ) clearly sum up to (m2 ? vss′ )M . Bounding each of these numbers by a multiple of M we get (m2 ? vss′ )M ≤ (m ? 1)2 ? (x + y + z ) M + 2xM + 3yM + 4zM,

41

C7

i.e.

Combinatorics – solutions

52nd IMO 2011

2m ? 1 ≤ vss′ + x + 2y + 3z ≤ vss′ + vhh′ + y + 2z. So, to prove the claim it su?ces to prove that y + 2z ≤ vsh′ . For a cell δ of type (h, h′ ) and a cell β of type (s, h′ ) we say that δ forces β if there are more than M napkins covering both of them. Since each red number in a cell of type (s, h′ ) does not exceed 2M , it cannot be forced by more than one cell. On the other hand, if a red number in a (h, h′ )-cell belongs to (2M, 3M ], then it forces at least one of its neighbors of type (s, h′ ) (since the sum of red numbers in their cells is greater than 2M ). Analogously, an (h, h′ )-cell with the red number in (3M, 4M ] forces both its neighbors of type (s, h′ ), since their red numbers do not exceed 2M . Therefore there are at least y + 2z forced cells and clearly all of them are vicious, as desired. Claim 3. We have 2m ? 1 ≤ b+c d a + + . S 2 2SH H 2

(2)

Proof. Averaging the previous result over all S 2 H 2 possibilities for the quadruple (s, s′ , h, h′ ), b d a we get 2m ? 1 ≤ S 2 + SH + H 2 . Due to the symmetry between rows and columns, the same estimate holds with b replaced by c. Averaging these two inequalities we arrive at our claim. Now let us multiply (2) by H 2 , multiply (1) by (2SH ? H 2 ) and add them; we get H 2 + 2SH ? H 2 2H H 2 + 2SH ? H 2 H (2m?1)+(2SH ?H )m ≤ a· +( b + c ) + d = a · + b+ c + d. S2 2SH S

2 2

The left-hand side is exactly H 2 (m ? 1) + 2SHm, while the right-hand side does not exceed a + b + c + d since 2H ≤ S . Hence we come to the desired inequality.

Comment 1. Claim 2 is the key di?erence between the two solutions, because it allows to get rid of the notions of rich and poor cells. However, one may prove it by the “strawberry method” as well. It su?ces to put a strawberry on each cell which is bad for an s-row, and a strawberry on each cell which is bad for an h′ -column. Then each cell would contain not more than one strawberry. Comment 2. Both solutions above work if the residue of the table size T modulo the napkin size R 2 R, or equivalently if T = Sm + H (m ? 1) and R = S + H for some positive integers S , H , is at least 3 m such that S ≥ 2H . Here we discuss all other possible combinations. Case 1. If 2H ≥ S ≥ H/2, then the sharp bound for the number of vicious cells is mS 2 + (m ? 1)H 2 ; it can be obtained by the same methods as in any of the solutions. To obtain an example showing that the bound is sharp, one may simply remove the napkins of the third kind from the example in Solution 1 (with an obvious change in the numbers). Case 2. If 2S ≤ H , the situation is more di?cult. If (S + H )2 > 2H 2 , then the answer and the example are the same as in the previous case; otherwise the answer is (2m ? 1)S 2 + 2SH (m ? 1), and the example is provided simply by (m ? 1)2 nonintersecting napkins.

42

52nd IMO 2011

Combinatorics – solutions

C7

Now we sketch the proof of both estimates for Case 2. We introduce a more appropriate notation based on that from Solution 2. Denote by a? and a+ the number of cells of class A that contain the number which is strictly less than M and strictly greater than M , respectively. The numbers b± , c± , and d± are de?ned in a similar way. One may notice that the proofs of Claim 1 and Claims 2, 3 lead in fact to the inequalities m?1≤ d+ b? + c? + 2 2SH H and 2m ? 1 ≤ a b+ + c+ d+ + + 2 2 S 2SH H

(to obtain the ?rst one, one needs to look at the big lines instead of the small ones). Combining these inequalities, one may obtain the desired estimates. These estimates can also be proved in some di?erent ways, e.g. without distinguishing rich and poor cells.

43

G1

G1

Geometry – solutions

52nd IMO 2011

Let ABC be an acute triangle. Let ω be a circle whose center L lies on the side BC . Suppose that ω is tangent to AB at B ′ and to AC at C ′ . Suppose also that the circumcenter O of the triangle ABC lies on the shorter arc B ′ C ′ of ω . Prove that the circumcircle of ABC and ω meet at two points.

Solution. The point B ′ , being the perpendicular foot of L, is an interior point of side AB . Analogously, C ′ lies in the interior of AC . The point O is located inside the triangle AB ′ C ′ , hence ∠COB < ∠C ′ OB ′ .

A

α

O C′ B′ B

ω

L

C

O′

Let α = ∠CAB . The angles ∠CAB and ∠C ′ OB ′ are inscribed into the two circles with centers O and L, respectively, so ∠COB = 2∠CAB = 2α and 2∠C ′ OB ′ = 360? ? ∠C ′ LB ′ . From the kite AB ′ LC ′ we have ∠C ′ LB ′ = 180? ? ∠C ′ AB ′ = 180? ? α. Combining these, we get 360? ? ∠C ′ LB ′ 360? ? (180? ? α) α ′ ′ 2α = ∠COB < ∠C OB = = = 90? + , 2 2 2 so α < 60? . Let O ′ be the re?ection of O in the line BC . In the quadrilateral ABO ′ C we have ∠CO ′B + ∠CAB = ∠COB + ∠CAB = 2α + α < 180? , so the point O ′ is outside the circle ABC . Hence, O and O ′ are two points of ω such that one of them lies inside the circumcircle, while the other one is located outside. Therefore, the two circles intersect.

44

52nd IMO 2011

Geometry – solutions

G1

Comment. There are di?erent ways of reducing the statement of the problem to the case α < 60? . E.g., since the point O lies in the interior of the isosceles triangle AB ′ C ′ , we have OA < AB ′ . So, if AB ′ ≤ 2LB ′ then OA < 2LO , which means that ω intersects the circumcircle of ABC . Hence the only interesting case is AB ′ > 2LB ′ , and this condition implies ∠CAB = 2∠B ′ AL < 2 · 30? = 60? .

45

G2

G2

Geometry – solutions

52nd IMO 2011

Let A1 A2 A3 A4 be a non-cyclic quadrilateral. Let O1 and r1 be the circumcenter and the circumradius of the triangle A2 A3 A4 . De?ne O2 , O3 , O4 and r2 , r3 , r4 in a similar way. Prove that 1 1 1 1 + + + = 0. 2 2 2 2 2 2 2 2 O1 A1 ? r1 O2 A2 ? r2 O3 A3 ? r3 O4 A4 ? r4

Solution 1. Let M be the point of intersection of the diagonals A1 A3 and A2 A4 . On each diagonal choose a direction and let x, y , z , and w be the signed distances from M to the points A1 , A2 , A3 , and A4 , respectively. Let ω1 be the circumcircle of the triangle A2 A3 A4 and let B1 be the second intersection point of ω1 and A1 A3 (thus, B1 = A3 if and only if A1 A3 is tangent to ω1 ). Since the expression 2 O1 A2 1 ? r1 is the power of the point A1 with respect to ω1 , we get

2 O1 A2 1 ? r1 = A1 B1 · A1 A3 .

On the other hand, from the equality MB1 · MA3 = MA2 · MA4 we obtain MB1 = yw/z. Hence, we have z?x yw 2 ? x (z ? x) = (yw ? xz ). O1 A2 1 ? r1 = z z Substituting the analogous expressions into the sought sum we get

4

1 ?

2 ri

Oi A2 i i=1 as desired.

=

1 yw ? xz

z w x y ? + ? z?x w?y x?z y?w

= 0,

Comment. One might reformulate the problem by assuming that the quadrilateral A1 A2 A3 A4 is convex. This should not really change the di?culty, but proofs that distinguish several cases may become shorter.

Solution 2. Introduce a Cartesian coordinate system in the plane. Every circle has an equation of the form p(x, y ) = x2 + y 2 + l(x, y ) = 0, where l(x, y ) is a polynomial of degree at most 1. For any point A = (xA , yA ) we have p(xA , yA ) = d2 ? r 2 , where d is the distance from A to the center of the circle and r is the radius of the circle. For each i in {1, 2, 3, 4} let pi (x, y ) = x2 + y 2 + li (x, y ) = 0 be the equation of the circle with center Oi and radius ri and let di be the distance from Ai to Oi . Consider the equation

4

i=1

pi (x, y ) = 1. 2 d2 i ? ri

(1)

46

52nd IMO 2011

Geometry – solutions

G2

Since the coordinates of the points A1 , A2 , A3 , and A4 satisfy (1) but these four points do not lie on a circle or on an line, equation (1) de?nes neither a circle, nor a line. Hence, the equation is an identity and the coe?cient of the quadratic term x2 + y 2 also has to be zero, i.e.

4

d2 i=1 i

1 = 0. 2 ? ri

Comment. Using the determinant form of the equation of the circle through three given points, the same solution can be formulated as follows. For i = 1, 2, 3, 4 let (ui , vi ) be the coordinates of Ai and de?ne

2 u2 1 + v1 2 u2 2 + v2 ?= 2 2 u 3 + v3 2 u2 4 + v4

u1 u2 u3 u4

v1 v2 v3 v4

1 1 1 1

and

ui+1 vi+1 1 ?i = ui+2 vi+2 1 , ui+3 vi+3 1

where i + 1, i + 2, and i + 3 have to be read modulo 4 as integers in the set {1, 2, 3, 4}. u1 u2 Expanding u3 u4 v1 v2 v3 v4 1 1 1 1 1 1 = 0 along the third column, we get ?1 ? ?2 + ?3 ? ?4 = 0. 1 1

The circle through Ai+1 , Ai+2 , and Ai+3 is given by the equation x2 + y 2 x y 2 2 1 ui+1 + vi+1 ui+1 vi+1 2 ?i u2 i+2 + vi+2 ui+2 vi+2 2 u2 i+3 + vi+3 ui+3 vi+3 1 1 = 0. 1 1

(2)

On the left-hand side, the coe?cient of x2 + y 2 is equal to 1. Substituting (ui , vi ) for (x, y ) in (2) we obtain the power of point Ai with respect to the circle through Ai+1 , Ai+2 , and Ai+3 :

2 u2 ui vi i + vi 2 2 1 ui+1 + vi+1 ui+1 vi+1 2 d2 i ? ri = 2 ? i u2 i+2 + vi+2 ui+2 vi+2 2 u2 i+3 + vi+3 ui+3 vi+3

1 1 ? = (?1)i+1 . ?i 1 1

Thus, we have

4

d2 i=1 i

?1 ? ?2 + ?3 ? ?4 1 = = 0. 2 ? ? ri

47

G3

G3

Geometry – solutions

52nd IMO 2011

Let ABCD be a convex quadrilateral whose sides AD and BC are not parallel. Suppose that the circles with diameters AB and CD meet at points E and F inside the quadrilateral. Let ωE be the circle through the feet of the perpendiculars from E to the lines AB , BC , and CD . Let ωF be the circle through the feet of the perpendiculars from F to the lines CD , DA, and AB . Prove that the midpoint of the segment EF lies on the line through the two intersection points of ωE and ωF .

Solution. Denote by P , Q, R, and S the projections of E on the lines DA, AB , BC , and CD respectively. The points P and Q lie on the circle with diameter AE , so ∠QP E = ∠QAE ; analogously, ∠QRE = ∠QBE . So ∠QP E + ∠QRE = ∠QAE + ∠QBE = 90? . By similar reasons, we have ∠SP E + ∠SRE = 90? , hence we get ∠QP S + ∠QRS = 90? + 90? = 180? , and the quadrilateral P QRS is inscribed in ωE . Analogously, all four projections of F onto the sides of ABCD lie on ωF . Denote by K the meeting point of the lines AD and BC . Due to the arguments above, there is no loss of generality in assuming that A lies on segment DK . Suppose that ∠CKD ≥ 90? ; then the circle with diameter CD covers the whole quadrilateral ABCD , so the points E , F cannot lie inside this quadrilateral. Hence our assumption is wrong. Therefore, the lines EP and BC intersect at some point P ′, while the lines ER and AD intersect at some point R′ .

C M′ P′ R N B Q E S

ωE

F K P A M N′

Figure 1 We claim that the points P ′ and R′ also belong to ωE . Since the points R, E , Q, B are concyclic, ∠QRK = ∠QEB = 90? ? ∠QBE = ∠QAE = ∠QP E . So ∠QRK = ∠QP P ′ , which means that the point P ′ lies on ωE . Analogously, R′ also lies on ωE . In the same manner, denote by M and N the projections of F on the lines AD and BC

R′

D

48

52nd IMO 2011

Geometry – solutions

G3

respectively, and let M ′ = F M ∩ BC , N ′ = F N ∩ AD . By the same arguments, we obtain that the points M ′ and N ′ belong to ωF .

M′ P′ R E N V U F K P M N′

ωF

g

ωE

R′

Figure 2 Now we concentrate on Figure 2, where all unnecessary details are removed. Let U = NN ′ ∩ P P ′ , V = MM ′ ∩ RR′ . Due to the right angles at N and P , the points N , N ′ , P , P ′ are concyclic, so UN · UN ′ = UP · UP ′ which means that U belongs to the radical axis g of the circles ωE and ωF . Analogously, V also belongs to g . Finally, since EUF V is a parallelogram, the radical axis UV of ωE and ωF bisects EF .

49

G4

G4

Geometry – solutions

52nd IMO 2011

Let ABC be an acute triangle with circumcircle ?. Let B0 be the midpoint of AC and let C0 be the midpoint of AB . Let D be the foot of the altitude from A, and let G be the centroid of the triangle ABC . Let ω be a circle through B0 and C0 that is tangent to the circle ? at a point X = A. Prove that the points D , G, and X are collinear.

Solution 1. If AB = AC , then the statement is trivial. So without loss of generality we may assume AB < AC . Denote the tangents to ? at points A and X by a and x, respectively. Let ?1 be the circumcircle of triangle AB0 C0 . The circles ? and ?1 are homothetic with center A, so they are tangent at A, and a is their radical axis. Now, the lines a, x, and B0 C0 are the three radical axes of the circles ?, ?1 , and ω . Since a B0 C0 , these three lines are concurrent at some point W . The points A and D are symmetric with respect to the line B0 C0 ; hence W X = W A = W D . This means that W is the center of the circumcircle γ of triangle ADX . Moreover, we have ∠W AO = ∠W XO = 90? , where O denotes the center of ?. Hence ∠AW X + ∠AOX = 180? .

a A

?1 ?

T

W

C0 G O

B0

B X

γ

D

A0

ω

C

x

Denote by T the second intersection point of ? and the line DX . Note that O belongs to ?1 . 1 1 (360? ? ∠AW X ) ? 2 ∠AOX = Using the circles γ and ?, we ?nd ∠DAT = ∠ADX ? ∠AT D = 2 1 ? ? 180 ? 2 (∠AW X + ∠AOX ) = 90 . So, AD ⊥ AT , and hence AT BC . Thus, AT CB is an isosceles trapezoid inscribed in ?. Denote by A0 the midpoint of BC , and consider the image of AT CB under the homothety h 1 . We have h(A) = A0 , h(B ) = B0 , and h(C ) = C0 . From the with center G and factor ? 2

50

52nd IMO 2011

Geometry – solutions

G4

symmetry about B0 C0 , we have ∠T CB = ∠CBA = ∠B0 C0 A = ∠DC0 B0 . Using AT DA0 , we conclude h(T ) = D . Hence the points D , G, and T are collinear, and X lies on the same line. Solution 2. We de?ne the points A0 , O , and W as in the previous solution and we concentrate on the case AB < AC . Let Q be the perpendicular projection of A0 on B0 C0 . Since ∠W AO = ∠W QO = ∠OXW = 90? , the ?ve points A, W , X , O , and Q lie on a common circle. Furthermore, the re?ections with respect to B0 C0 and OW map A to D and X , respectively. For these reasons, we have ∠W QD = ∠AQW = ∠AXW = ∠W AX = ∠W QX. Thus the three points Q, D , and X lie on a common line, say ?.

A

a

W

C0 G

Q O

B0

J

B X

D

A0

C

x

To complete the argument, we note that the homothety centered at G sending the triangle ABC to the triangle A0 B0 C0 maps the altitude AD to the altitude A0 Q. Therefore it maps D to Q, so the points D , G, and Q are collinear. Hence G lies on ? as well.

Comment. There are various other ways to prove the collinearity of Q, D , and X obtained in the middle part of Solution 2. Introduce for instance the point J where the lines AW and BC intersect. Then the four points A, D , X , and J lie at the same distance from W , so the quadrilateral ADXJ is cyclic. In combination with the fact that AW XQ is cyclic as well, this implies ∠JDX = ∠JAX = ∠W AX = ∠W QX. Since BC W Q, it follows that Q, D , and X are indeed collinear.

51

G5

G5

Geometry – solutions

52nd IMO 2011

Let ABC be a triangle with incenter I and circumcircle ω . Let D and E be the second intersection points of ω with the lines AI and BI , respectively. The chord DE meets AC at a point F , and BC at a point G. Let P be the intersection point of the line through F parallel to AD and the line through G parallel to BE . Suppose that the tangents to ω at A and at B meet at a point K . Prove that the three lines AE , BD , and KP are either parallel or concurrent.

Solution 1. Since ∠IAF = ∠DAC = ∠BAD = ∠BED = ∠IEF the quadrilateral AIF E is cyclic. Denote its circumcircle by ω1 . Similarly, the quadrilateral BDGI is cyclic; denote its circumcircle by ω2 . The line AE is the radical axis of ω and ω1 , and the line BD is the radical axis of ω and ω2 . Let t be the radical axis of ω1 and ω2 . These three lines meet at the radical center of the three circles, or they are parallel to each other. We will show that t is in fact the line P K . Let L be the second intersection point of ω1 and ω2 , so t = IL. (If the two circles are tangent to each other then L = I and t is the common tangent.)

ω1

E A

ω

t K ′=K I L F P ′=P B G C

ω2

D

Let the line t meet the circumcircles of the triangles ABL and F GL at K ′ = L and P ′ = L, respectively. Using oriented angles we have ∠(AB, BK ′ ) = ∠(AL, LK ′ ) = ∠(AL, LI ) = ∠(AE, EI ) = ∠(AE, EB ) = ∠(AB, BK ),

52

52nd IMO 2011 so BK ′

Geometry – solutions AK , and therefore K ′ = K . Next, we have

G5

BK . Similarly we have AK ′

∠(P ′F, F G) = ∠(P ′L, LG) = ∠(IL, LG) = ∠(ID, DG) = ∠(AD, DE ) = ∠(P F, F G), hence P ′ F P F and similarly P ′ G P G. Therefore P ′ = P . This means that t passes through K and P , which ?nishes the proof. Solution 2. Let M be the intersection point of the tangents to ω at D and E , and let the lines AE and BD meet at T ; if AE and BD are parallel, then let T be their common ideal point. It is well-known that the points K and M lie on the line T I (as a consequence of Pascal’s theorem, applied to the inscribed degenerate hexagons AADBBE and ADDBEE ). The lines AD and BE are the angle bisectors of the angles ∠CAB and ∠ABC , respectively, so D and E are the midpoints of the arcs BC and CA of the circle ω , respectively. Hence, DM is parallel to BC and EM is parallel to AC . Apply Pascal’s theorem to the degenerate hexagon CADDEB . By the theorem, the points CA ∩ DE = F , AD ∩ EB = I and the common ideal point of lines DM and BC are collinear, therefore F I is parallel to BC and DM . Analogously, the line GI is parallel to AC and EM .

ω

E

A T K I H B G D

FG which takes E to G and D to F . Since the Now consider the homothety with scale factor ? ED triangles EDM and GF I have parallel sides, the homothety takes M to I . Similarly, since the triangles DEI and F GP have parallel sides, the homothety takes I to P . Hence, the points M , I , P and the homothety center H must lie on the same line. Therefore, the point P also lies on the line T KIM .

F P C M

Comment. One may prove that IF BC and IG AC in a more elementary way. Since ∠ADE = ∠EDC and ∠DEB = ∠CED , the points I and C are symmetric about DE . Moreover, since the arcs AE and EC are equal and the arcs CD and DB are equal, we have ∠CF G = ∠F GC , so the triangle CF G is isosceles. Hence, the quadrilateral IF CG is a rhombus.

53

G6

G6

Geometry – solutions

52nd IMO 2011

Let ABC be a triangle with AB = AC , and let D be the midpoint of AC . The angle bisector of ∠BAC intersects the circle through D , B , and C in a point E inside the triangle ABC . The line BD intersects the circle through A, E , and B in two points B and F . The lines AF and BE meet at a point I , and the lines CI and BD meet at a point K . Show that I is the incenter of triangle KAB .

Solution 1. Let D ′ be the midpoint of the segment AB , and let M be the midpoint of BC . By symmetry at line AM , the point D ′ has to lie on the circle BCD . Since the arcs D ′ E and ED of that circle are equal, we have ∠ABI = ∠D ′BE = ∠EBD = IBK , so I lies on the angle bisector of ∠ABK . For this reason it su?ces to prove in the sequel that the ray AI bisects the angle ∠BAK . From 1 1 ∠DF A = 180? ? ∠BF A = 180? ? ∠BEA = ∠MEB = ∠CEB = ∠CDB 2 2 we derive ∠DF A = ∠DAF so the triangle AF D is isosceles with AD = DF .

A

E D′

ω1

I K F

D

B

M

ω2

C

Applying Menelaus’s theorem to the triangle ADF with respect to the line CIK , and applying the angle bisector theorem to the triangle ABF , we infer 1= and therefore AC DK F I DK BF DK BF DK BF · · =2· · =2· · = · CD KF IA KF AB KF 2 · AD KF AD BF + F D BF KF DF AD BD = = +1= +1= = . AD AD AD DK DK DK

54

52nd IMO 2011

Geometry – solutions

G6

It follows that the triangles ADK and BDA are similar, hence ∠DAK = ∠ABD . Then ∠IAB = ∠AF D ? ∠ABD = ∠DAF ? ∠DAK = ∠KAI shows that the point K is indeed lying on the angle bisector of ∠BAK . Solution 2. It can be shown in the same way as in the ?rst solution that I lies on the angle bisector of ∠ABK . Here we restrict ourselves to proving that KI bisects ∠AKB .

A

E I

ω1

D K F

O3

B

ω3 ω2

C

O1

Denote the circumcircle of triangle BCD and its center by ω1 and by O1 , respectively. Since the quadrilateral ABF E is cyclic, we have ∠DF E = ∠BAE = ∠DAE . By the same reason, we have ∠EAF = ∠EBF = ∠ABE = ∠AF E . Therefore ∠DAF = ∠DF A, and hence DF = DA = DC . So triangle AF C is inscribed in a circle ω2 with center D . Denote the circumcircle of triangle ABD by ω3 , and let its center be O3 . Since the arcs BE and EC of circle ω1 are equal, and the triangles ADE and F DE are congruent, we have ∠AO1 B = 2∠BDE = ∠BDA, so O1 lies on ω3 . Hence ∠O3 O1 D = ∠O3 DO1 . The line BD is the radical axis of ω1 and ω3 . Point C belongs to the radical axis of ω1 and ω2 , and I also belongs to it since AI · IF = BI · IE . Hence K = BD ∩ CI is the radical center of ω1 , ω2 , and ω3 , and AK is the radical axis of ω2 and ω3 . Now, the radical axes AK , BK and IK are perpendicular to the central lines O3 D , O3 O1 and O1 D , respectively. By ∠O3 O1 D = ∠O3 DO1 , we get that KI is the angle bisector of ∠AKB . Solution 3. Again, let M be the midpoint of BC . As in the previous solutions, we can deduce ∠ABI = ∠IBK . We show that the point I lies on the angle bisector of ∠KAB . Let G be the intersection point of the circles AF C and BCD , di?erent from C . The lines

55

G6

Geometry – solutions

52nd IMO 2011

CG, AF , and BE are the radical axes of the three circles AGF C , CDB , and ABF E , so I = AF ∩ BE is the radical center of the three circles and CG also passes through I .

A

B′

G I F

E D K

B

M

C

The angle between line DE and the tangent to the circle BCD at E is equal to ∠EBD = ∠EAF = ∠ABE = ∠AF E . As the tangent at E is perpendicular to AM , the line DE is perpendicular to AF . The triangle AF E is isosceles, so DE is the perpendicular bisector of AF and thus AD = DF . Hence, the point D is the center of the circle AF C , and this circle passes through M as well since ∠AMC = 90? . Let B ′ be the re?ection of B in the point D , so ABCB ′ is a parallelogram. Since DC = DG we have ∠GCD = ∠DBC = ∠KB ′ A. Hence, the quadrilateral AKCB ′ is cyclic and thus ∠CAK = ∠CB ′ K = ∠ABD = 2∠MAI . Then 1 1 1 ∠IAB = ∠MAB ? ∠MAI = ∠CAB ? ∠CAK = ∠KAB 2 2 2 and therefore AI is the angle bisector of ∠KAB .

56

52nd IMO 2011

Geometry – solutions

G7

G7

Let ABCDEF be a convex hexagon all of whose sides are tangent to a circle ω with center O . Suppose that the circumcircle of triangle ACE is concentric with ω . Let J be the foot of the perpendicular from B to CD . Suppose that the perpendicular from B to DF intersects the line EO at a point K . Let L be the foot of the perpendicular from K to DE . Prove that DJ = DL.

Solution 1. Since ω and the circumcircle of triangle ACE are concentric, the tangents from A, C , and E to ω have equal lengths; that means that AB = BC , CD = DE , and EF = F A. Moreover, we have ∠BCD = ∠DEF = ∠F AB .

A B J O M C E K′

ω

F

L′ B′

B ′′ P

D

Consider the rotation around point D mapping C to E ; let B ′ and L′ be the images of the points B and J , respectively, under this rotation. Then one has DJ = DL′ and B ′ L′ ⊥ DE ; moreover, the triangles B ′ ED and BCD are congruent. Since ∠DEO < 90? , the lines EO and B ′ L′ intersect at some point K ′ . We intend to prove that K ′ B ⊥ DF ; this would imply K = K ′ , therefore L = L′ , which proves the problem statement. Analogously, consider the rotation around F mapping A to E ; let B ′′ be the image of B under this rotation. Then the triangles F AB and F EB ′′ are congruent. We have EB ′′ = AB = BC = EB ′ and ∠F EB ′′ = ∠F AB = ∠BCD = ∠DEB ′ , so the points B ′ and B ′′ are symmetrical with respect to the angle bisector EO of ∠DEF . So, from K ′ B ′ ⊥ DE we get K ′ B ′′ ⊥ EF . From these two relations we obtain K ′D2 ? K ′ E 2 = B ′ D2 ? B ′E 2 and K ′ E 2 ? K ′ F 2 = B ′′ E 2 ? B ′′ F 2 .

Adding these equalities and taking into account that B ′ E = B ′′ E we obtain K ′ D 2 ? K ′ F 2 = B ′ D 2 ? B ′′ F 2 = BD 2 ? BF 2 , 57

G7

Geometry – solutions

52nd IMO 2011

which means exactly that K ′ B ⊥ DF .

Comment. There are several variations of this solution. For instance, let us consider the intersection point M of the lines BJ and OC . De?ne the point K ′ as the re?ection of M in the line DO. Then one has DK ′2 ? DB 2 = DM 2 ? DB 2 = CM 2 ? CB 2 . Next, consider the rotation around O which maps CM to EK ′ . Let P be the image of B under this rotation; so P lies on ED . Then EF ⊥ K ′ P , so CM 2 ? CB 2 = EK ′2 ? EP 2 = F K ′2 ? F P 2 = F K ′2 ? F B 2 , since the triangles F EP and F AB are congruent.

Solution 2. Let us denote the points of tangency of AB , BC , CD , DE , EF , and F A to ω by R, S , T , U , V , and W , respectively. As in the previous solution, we mention that AR = AW = CS = CT = EU = EV . The re?ection in the line BO maps R to S , therefore A to C and thus also W to T . Hence, both lines RS and W T are perpendicular to OB , therefore they are parallel. On the other hand, the lines UV and W T are not parallel, since otherwise the hexagon ABCDEF is symmetric with respect to the line BO and the lines de?ning the point K coincide, which contradicts the conditions of the problem. Therefore we can consider the intersection point Z of UV and W T .

Z R A W F L S C T

ω

B J

V O U E K

D

Next, we recall a well-known fact that the points D , F , Z are collinear. Actually, D is the pole of the line UT , F is the pole of V W , and Z = T W ∩ UV ; so all these points belong to the polar line of T U ∩ V W .

58

52nd IMO 2011

Geometry – solutions

G7

? ? → Now, we put O into the origin, and identify each point (say X ) with the vector OX . So, from now on all the products of points refer to the scalar products of the corresponding vectors. Since OK ⊥ UZ and OB ⊥ T Z , we have K · (Z ? U ) = 0 = B · (Z ? T ). Next, the condition BK ⊥ DZ can be written as K · (D ? Z ) = B · (D ? Z ). Adding these two equalities we get K · (D ? U ) = B · (D ? T ). By symmetry, we have D · (D ? U ) = D · (D ? T ). Subtracting this from the previous equation, we obtain (K ? D ) · (D ? U ) = (B ? D ) · (D ? T ) and rewrite it in vector form as ? ? → ? ? → ? ? → ? → DK · UD = DB · T D. ? ? → ? ? → Finally, projecting the vectors DK and DB onto the lines UD and T D respectively, we can rewrite this equality in terms of segment lengths as DL · UD = DJ · T D , thus DL = DJ .

Comment. The collinearity of Z , F , and D may be shown in various more elementary ways. For insin ∠DZT sin ∠DT Z stance, applying the sine theorem to the triangles DT Z and DU Z , one gets = ; sin ∠DZU sin ∠DU Z sin ∠F W Z sin ∠F ZW = . The right-hand sides are equal, hence so are the left-hand analogously, sin ∠F ZV sin ∠F V Z sides, which implies the collinearity of the points D , F , and Z . There also exist purely synthetic proofs of this fact. E.g., let Q be the point of intersection of the circumcircles of the triangles ZT V and ZW U di?erent from Z . Then QZ is the bisector of ∠V QW 1 since ∠V QZ = ∠V T Z = ∠V U W = ∠ZQW . Moreover, all these angles are equal to 2 ∠V OW , so ∠V QW = ∠V OW , hence the quadrilateral V W OQ is cyclic. On the other hand, the points O, V , W lie on the circle with diameter OF due to the right angles; so Q also belongs to this circle. Since F V = F W , QF is also the bisector of ∠V QW , so F lies on QZ . Analogously, D lies on the same line.

59

G8

G8

Geometry – solutions

52nd IMO 2011

Let ABC be an acute triangle with circumcircle ω . Let t be a tangent line to ω . Let ta , tb , and tc be the lines obtained by re?ecting t in the lines BC , CA, and AB , respectively. Show that the circumcircle of the triangle determined by the lines ta , tb , and tc is tangent to the circle ω .

To avoid a large case distinction, we will use the notion of oriented angles. Namely, for two lines ? and m, we denote by ∠(?, m) the angle by which one may rotate ? anticlockwise to obtain a line parallel to m. Thus, all oriented angles are considered modulo 180? .

K

A

ω

tc

B ′′ C ′′

B′ X B A′′ F tb A′ I T C E C ′=S t ta D

Solution 1. Denote by T the point of tangency of t and ω . Let A′ = tb ∩ tc , B ′ = ta ∩ tc , C ′ = ta ∩ tb . Introduce the point A′′ on ω such that T A = AA′′ (A′′ = T unless T A is a diameter). De?ne the points B ′′ and C ′′ in a similar way. Since the points C and B are the midpoints of arcs T C ′′ and T B ′′ , respectively, we have ∠(t, B ′′ C ′′ ) = ∠(t, T C ′′ ) + ∠(T C ′′ , B ′′ C ′′ ) = 2∠(t, T C ) + 2∠(T C ′′ , BC ′′ ) = 2 ∠(t, T C ) + ∠(T C, BC ) = 2∠(t, BC ) = ∠(t, ta ). It follows that ta and B ′′ C ′′ are parallel. Similarly, tb A′′ C ′′ and tc A′′ B ′′ . Thus, either the triangles A′ B ′ C ′ and A′′ B ′′ C ′′ are homothetic, or they are translates of each other. Now we will prove that they are in fact homothetic, and that the center K of the homothety belongs

60

52nd IMO 2011

Geometry – solutions

G8

to ω . It would then follow that their circumcircles are also homothetic with respect to K and are therefore tangent at this point, as desired. We need the two following claims. Claim 1. The point of intersection X of the lines B ′′ C and BC ′′ lies on ta . Proof. Actually, the points X and T are symmetric about the line BC , since the lines CT and CB ′′ are symmetric about this line, as are the lines BT and BC ′′ . Claim 2. The point of intersection I of the lines BB ′ and CC ′ lies on the circle ω . Proof. We consider the case that t is not parallel to the sides of ABC ; the other cases may be regarded as limit cases. Let D = t ∩ BC , E = t ∩ AC , and F = t ∩ AB . Due to symmetry, the line DB is one of the angle bisectors of the lines B ′ D and F D ; analogously, the line F B is one of the angle bisectors of the lines B ′ F and DF . So B is either the incenter or one of the excenters of the triangle B ′ DF . In any case we have ∠(BD, DF ) + ∠(DF, F B ) + ∠(B ′ B, B ′ D ) = 90? , so ∠(B ′ B, B ′ C ′ ) = ∠(B ′ B, B ′ D ) = 90? ? ∠(BC, DF ) ? ∠(DF, BA) = 90? ? ∠(BC, AB ). Analogously, we get ∠(C ′ C, B ′ C ′ ) = 90? ? ∠(BC, AC ). Hence, ∠(BI, CI ) = ∠(B ′ B, B ′ C ′ ) + ∠(B ′ C ′ , C ′ C ) = ∠(BC, AC ) ? ∠(BC, AB ) = ∠(AB, AC ), which means exactly that the points A, B , I , C are concyclic. Now we can complete the proof. Let K be the second intersection point of B ′ B ′′ and ω . Applying Pascal’s theorem to hexagon KB ′′ CIBC ′′ we get that the points B ′ = KB ′′ ∩ IB and X = B ′′ C ∩ BC ′′ are collinear with the intersection point S of CI and C ′′ K . So S = CI ∩ B ′ X = C ′ , and the points C ′ , C ′′ , K are collinear. Thus K is the intersection point of B ′ B ′′ and C ′ C ′′ which implies that K is the center of the homothety mapping A′ B ′ C ′ to A′′ B ′′ C ′′ , and it belongs to ω . Solution 2. De?ne the points T , A′ , B ′ , and C ′ in the same way as in the previous solution. Let X , Y , and Z be the symmetric images of T about the lines BC , CA, and AB , respectively. Note that the projections of T on these lines form a Simson line of T with respect to ABC , therefore the points X , Y , Z are also collinear. Moreover, we have X ∈ B ′ C ′ , Y ∈ C ′ A′ , Z ∈ A′ B ′ . Denote α = ∠(t, T C ) = ∠(BT, BC ). Using the symmetry in the lines AC and BC , we get ∠(BC, BX ) = ∠(BT, BC ) = α and ∠(XC, XC ′) = ∠(t, T C ) = ∠(Y C, Y C ′ ) = α. Since ∠(XC, XC ′) = ∠(Y C, Y C ′ ), the points X , Y , C , C ′ lie on some circle ωc . De?ne the circles ωa and ωb analogously. Let ω ′ be the circumcircle of triangle A′ B ′ C ′ .

61

G8

Geometry – solutions

52nd IMO 2011

Now, applying Miquel’s theorem to the four lines A′ B ′ , A′ C ′ , B ′ C ′ , and XY , we obtain that the circles ω ′ , ωa , ωb , ωc intersect at some point K . We will show that K lies on ω , and that the tangent lines to ω and ω ′ at this point coincide; this implies the problem statement. Due to symmetry, we have XB = T B = ZB , so the point B is the midpoint of one of the arcs XZ of circle ωb . Therefore ∠(KB, KX ) = ∠(XZ, XB ). Analogously, ∠(KX, KC ) = ∠(XC, XY ). Adding these equalities and using the symmetry in the line BC we get ∠(KB, KC ) = ∠(XZ, XB ) + ∠(XC, XZ ) = ∠(XC, XB ) = ∠(T B, T C ). Therefore, K lies on ω . Next, let k be the tangent line to ω at K . We have ∠(k, KC ′ ) = ∠(k, KC ) + ∠(KC, KC ′ ) = ∠(KB, BC ) + ∠(XC, XC ′) = ∠(KB, BX ) ? ∠(BC, BX ) + α = ∠(KB ′ , B ′ X ) ? α + α = ∠(KB ′ , B ′ C ′ ), which means exactly that k is tangent to ω ′ .

k K

ωb ω ωc

A

ω′

Z B′ B T ta A′ tb X C t tc C′ Y

Comment. There exist various solutions combining the ideas from the two solutions presented above. For instance, one may de?ne the point X as the re?ection of T with respect to the line BC , and then introduce the point K as the second intersection point of the circumcircles of BB ′ X and CC ′ X . Using the fact that BB ′ and CC ′ are the bisectors of ∠(A′ B ′ , B ′ C ′ ) and ∠(A′ C ′ , B ′ C ′ ) one can show successively that K ∈ ω , K ∈ ω ′ , and that the tangents to ω and ω ′ at K coincide.

62

52nd IMO 2011

Number Theory – solutions

N1

N1

For any integer d > 0, let f (d) be the smallest positive integer that has exactly d positive divisors (so for example we have f (1) = 1, f (5) = 16, and f (6) = 12). Prove that for every integer k ≥ 0 the number f (2k ) divides f (2k+1).

Solution 1. For any positive integer n, let d(n) be the number of positive divisors of n. Let n = p pa(p) be the prime factorization of n where p ranges over the prime numbers, the integers a(p) are nonnegative and all but ?nitely many a(p) are zero. Then we have d(n) = p (a(p)+1). Thus, d(n) is a power of 2 if and only if for every prime p there is a nonnegative integer b(p) with a(p) = 2b(p) ? 1 = 1 + 2 + 22 + · · · + 2b(p)?1 . We then have

b(p)?1

n=

p i=0

p2 ,

i

and d(n) = 2k

r

with k =

p

b(p).

Let S be the set of all numbers of the form p2 with p prime and r a nonnegative integer. Then we deduce that d(n) is a power of 2 if and only if n is the product of the elements of some ?nite subset T of S that satis?es the following condition: for all t ∈ T and s ∈ S with s t we have s ∈ T . Moreover, if d(n) = 2k then the corresponding set T has k elements. Note that the set Tk consisting of the smallest k elements from S obviously satis?es the condition above. Thus, given k , the smallest n with d(n) = 2k is the product of the elements of Tk . This n is f (2k ). Since obviously Tk ? Tk+1 , it follows that f (2k ) f (2k+1 ). Solution 2. This is an alternative to the second part of the Solution 1. Suppose k is a nonnegative integer. From the ?rst part of Solution 1 we see that f (2k ) = p pa(p) with a(p) = 2b(p) ? 1 and p b(p) = k . We now claim that for any two distinct primes p, q with b(q ) > 0 we have b(p) b(q )?1 m = p2 > q 2 = ?. (1) To see this, note ?rst that ? divides f (2k ). With the ?rst part of Solution 1 one can see that the integer n = f (2k )m/? also satis?es d(n) = 2k . By the de?nition of f (2k ) this implies that n ≥ f (2k ) so m ≥ ?. Since p = q the inequality (1) follows. Let the prime factorization of f (2k+1) be given by f (2k+1) = p pr(p) with r (p) = 2s(p) ? 1. Since we have p s(p) = k + 1 > k = p b(p) there is a prime p with s(p) > b(p). For any prime q = p with b(q ) > 0 we apply inequality (1) twice and get q2

s(q )

> p2

s(p)?1

≥ p2

b(p)

> q2

b(q )?1

,

which implies s(q ) ≥ b(q ). It follows that s(q ) ≥ b(q ) for all primes q , so f (2k ) f (2k+1 ).

63

N2

N2

Number Theory – solutions

52nd IMO 2011

Consider a polynomial P (x) = (x + d1 )(x + d2 ) · . . . · (x + d9 ), where d1 , d2 , . . . , d9 are nine distinct integers. Prove that there exists an integer N such that for all integers x ≥ N the number P (x) is divisible by a prime number greater than 20.

Solution 1. Note that the statement of the problem is invariant under translations of x; hence without loss of generality we may suppose that the numbers d1 , d2 , . . . , d9 are positive. The key observation is that there are only eight primes below 20, while P (x) involves more than eight factors. We shall prove that N = d8 satis?es the desired property, where d = max{d1 , d2 , . . . , d9 }. Suppose for the sake of contradiction that there is some integer x ≥ N such that P (x) is composed of primes below 20 only. Then for every index i ∈ {1, 2, . . . , 9} the number x + di can be expressed as product of powers of the ?rst 8 primes. Since x + di > x ≥ d8 there is some prime power fi > d that divides x + di . Invoking the pigeonhole principle we see that there are two distinct indices i and j such that fi and fj are powers of the same prime number. For reasons of symmetry, we may suppose that fi ≤ fj . Now both of the numbers x + di and x + dj are divisible by fi and hence so is their di?erence di ? dj . But as 0 < |di ? dj | ≤ max(di , dj ) ≤ d < fi , this is impossible. Thereby the problem is solved.

Solution 2. Observe that for each index i ∈ {1, 2, . . . , 9} the product Di =

1≤j ≤9,j =i

|di ? dj |

is positive. We claim that N = max{D1 ? d1 , D2 ? d2 , . . . , D9 ? d9 } + 1 satis?es the statement of the problem. Suppose there exists an integer x ≥ N such that all primes dividing P (x) are smaller than 20. For each index i we reduce the fraction (x + di )/Di to lowest terms. Since x + di > Di the numerator of the fraction we thereby get cannot be 1, and hence it has to be divisible by some prime number pi < 20. By the pigeonhole principle, there are a prime number p and two distinct indices i and j such that pi = pj = p. Let pαi and pαj be the greatest powers of p dividing x + di and x + dj , respectively. Due to symmetry we may suppose αi ≤ αj . But now pαi divides di ? dj and hence also Di , which means that all occurrences of p in the numerator of the fraction (x + di )/Di cancel out, contrary to the choice of p = pi . This contradiction proves our claim.

64

52nd IMO 2011

Number Theory – solutions

N2

Solution 3. Given a nonzero integer N as well as a prime number p we write vp (N ) for the exponent with which p occurs in the prime factorization of |N |. Evidently, if the statement of the problem were not true, then there would exist an in?nite sequence (xn ) of positive integers tending to in?nity such that for each n ∈ Z+ the integer P (xn ) is not divisible by any prime number > 20. Observe that the numbers ?d1 , ?d2 , . . . , ?d9 do not appear in this sequence. Now clearly there exists a prime p1 < 20 for which the sequence vp1 (xn + d1 ) is not bounded; thinning out the sequence (xn ) if necessary we may even suppose that vp1 (xn + d1 ) ?→ ∞. Repeating this argument eight more times we may similarly choose primes p2 , . . . , p9 < 20 and suppose that our sequence (xn ) has been thinned out to such an extent that vpi (xn + di ) ?→ ∞ holds for i = 2, . . . , 9 as well. In view of the pigeonhole principle, there are distinct indices i and j as well as a prime p < 20 such that pi = pj = p. Setting k = vp (di ? dj ) there now has to be some n for which both vp (xn + di ) and vp (xn + dj ) are greater than k . But now the numbers xn + di and xn + dj are divisible by pk+1 whilst their di?erence di ? dj is not – a contradiction.

Comment. This problem is supposed to be a relatively easy one, so one might consider adding the hypothesis that the numbers d1 , d2 , . . . , d9 be positive. Then certain merely technical issues are not going to arise while the main ideas required to solve the problems remain the same.

65

N3

N3

Number Theory – solutions

52nd IMO 2011

Let n ≥ 1 be an odd integer. Determine all functions f from the set of integers to itself such that for all integers x and y the di?erence f (x) ? f (y ) divides xn ? y n .

Answer. All functions f of the form f (x) = εxd + c, where ε is in {1, ?1}, the integer d is a positive divisor of n, and c is an integer. Solution. Obviously, all functions in the answer satisfy the condition of the problem. We will show that there are no other functions satisfying that condition. Let f be a function satisfying the given condition. For each integer n, the function g de?ned by g (x) = f (x) + n also satis?es the same condition. Therefore, by subtracting f (0) from f (x) we may assume that f (0) = 0. For any prime p, the condition on f with (x, y ) = (p, 0) states that f (p) divides pn . Since the set of primes is in?nite, there exist integers d and ε with 0 ≤ d ≤ n and ε ∈ {1, ?1} such that for in?nitely many primes p we have f (p) = εpd . Denote the set of these primes by P . Since a function g satis?es the given condition if and only if ?g satis?es the same condition, we may suppose ε = 1. The case d = 0 is easily ruled out, because 0 does not divide any nonzero integer. Suppose d ≥ 1 and write n as md + r , where m and r are integers such that m ≥ 1 and 0 ≤ r ≤ d ? 1. Let x be an arbitrary integer. For each prime p in P , the di?erence f (p) ? f (x) divides pn ? xn . Using the equality f (p) = pd , we get pn ? xn = pr (pd )m ? xn ≡ pr f (x)m ? xn ≡ 0 Since we have r < d, for large enough primes p ∈ P we obtain |pr f (x)m ? xn | < pd ? f (x). Hence pr f (x)m ? xn has to be zero. This implies r = 0 and xn = (xd )m = f (x)m . Since m is odd, we obtain f (x) = xd .

Comment. If n is an even positive integer, then the functions f of the form ? ?xd + c for some integers, f (x) = ??xd + c for the rest of integers,

(mod pd ? f (x))

where d is a positive divisor of n/2 and c is an integer, also satisfy the condition of the problem. Together with the functions in the answer, they are all functions that satisfy the condition when n is even.

66

52nd IMO 2011

Number Theory – solutions

N4

N4

For each positive integer k , let t(k ) be the largest odd divisor of k . Determine all positive integers a for which there exists a positive integer n such that all the di?erences t(n + a) ? t(n), are divisible by 4. t(n + a + 1) ? t(n + 1), ..., t(n + 2a ? 1) ? t(n + a ? 1)

Answer. a = 1, 3, or 5. Solution. A pair (a, n) satisfying the condition of the problem will be called a winning pair. It is straightforward to check that the pairs (1, 1), (3, 1), and (5, 4) are winning pairs. Now suppose that a is a positive integer not equal to 1, 3, and 5. We will show that there are no winning pairs (a, n) by distinguishing three cases. Case 1: a is even. In this case we have a = 2α d for some positive integer α and some odd d. Since a ≥ 2α , for each positive integer n there exists an i ∈ {0, 1, . . . , a ? 1} such that n + i = 2α?1 e, where e is some odd integer. Then we have t(n + i) = t(2α?1 e) = e and t(n + a + i) = t(2α d + 2α?1 e) = 2d + e ≡ e + 2 (mod 4). So we get t(n + i) ? t(n + a + i) ≡ 2 (mod 4), and (a, n) is not a winning pair. Case 2: a is odd and a > 8. For each positive integer n, there exists an i ∈ {0, 1, . . . , a ? 5} such that n + i = 2d for some odd d. We get t(n + i) = d ≡ d + 2 = t(n + i + 4) and t(n + a + i) = n + a + i ≡ n + a + i + 4 = t(n + a + i + 4) (mod 4). Therefore, the integers t(n + a + i) ? t(n + i) and t(n + a + i + 4) ? t(n + i + 4) cannot be both divisible by 4, and therefore there are no winning pairs in this case. Case 3: a = 7. For each positive integer n, there exists an i ∈ {0, 1, . . . , 6} such that n + i is either of the form 8k + 3 or of the form 8k + 6, where k is a nonnegative integer. But we have t(8k + 3) ≡ 3 ≡ 1 ≡ 4k + 5 = t(8k + 3 + 7) and t(8k + 6) = 4k + 3 ≡ 3 ≡ 1 ≡ t(8k + 6 + 7) (mod 4). Hence, there are no winning pairs of the form (7, n). (mod 4) (mod 4)

67

N5

N5

Number Theory – solutions

52nd IMO 2011

Let f be a function from the set of integers to the set of positive integers. Suppose that for any two integers m and n, the di?erence f (m) ? f (n) is divisible by f (m ? n). Prove that for all integers m, n with f (m) ≤ f (n) the number f (n) is divisible by f (m).

Solution 1. Suppose that x and y are two integers with f (x) < f (y ). We will show that f (x) f (y ). By taking m = x and n = y we see that f (x ? y ) |f (x) ? f (y )| = f (y ) ? f (x) > 0, so f (x ? y ) ≤ f (y ) ? f (x) < f (y ). Hence the number d = f (x) ? f (x ? y ) satis?es ?f (y ) < ?f (x ? y ) < d < f (x) < f (y ). Taking m = x and n = x ? y we see that f (y ) d, so we deduce d = 0, or in other words f (x) = f (x ? y ). Taking m = x and n = y we see that f (x) = f (x ? y ) f (x) ? f (y ), which implies f (x) f (y ).

Solution 2. We split the solution into a sequence of claims; in each claim, the letters m and n denote arbitrary integers. Claim 1. f (n) f (mn). Proof. Since trivially f (n) f (1 · n) and f (n) f ((k + 1)n) ? f (kn) for all integers k , this is easily seen by using induction on m in both directions. Claim 2. f (n) f (0) and f (n) = f (?n). Proof. The ?rst part follows by plugging m = 0 into Claim 1. Using Claim 1 twice with m = ?1, we get f (n) f (?n) f (n), from which the second part follows. From Claim 1, we get f (1) f (n) for all integers n, so f (1) is the minimal value attained by f . Next, from Claim 2, the function f can attain only a ?nite number of values since all these values divide f (0). Now we prove the statement of the problem by induction on the number Nf of values attained by f . In the base case Nf ≤ 2, we either have f (0) = f (1), in which case these two numbers are the only values attained by f and the statement is clear, or we have f (0) = f (1), in which case we have f (1) f (n) f (0) for all integers n, so f is constant and the statement is obvious again. For the induction step, assume that Nf ≥ 3, and let a be the least positive integer with f (a) > f (1). Note that such a number exists due to the symmetry of f obtained in Claim 2.

68

52nd IMO 2011

Number Theory – solutions

N5

Claim 3. f (n) = f (1) if and only if a n. Proof. Since f (1) = · · · = f (a ? 1) < f (a), the claim follows from the fact that f (n) = f (1) ?? f (n + a) = f (1). So it su?ces to prove this fact. Assume that f (n) = f (1). Then f (n + a) f (a) ? f (?n) = f (a) ? f (n) > 0, so f (n + a) ≤ f (a) ? f (n) < f (a); in particular the di?erence f (n + a) ? f (n) is stricly smaller than f (a). Furthermore, this di?erence is divisible by f (a) and nonnegative since f (n) = f (1) is the least value attained by f . So we have f (n + a) ? f (n) = 0, as desired. For the converse direction we only need to remark that f (n + a) = f (1) entails f (?n ? a) = f (1), and hence f (n) = f (?n) = f (1) by the forward implication. We return to the induction step. So let us take two arbitrary integers m and n with f (m) ≤ f (n). If a m, then we have f (m) = f (1) f (n). On the other hand, suppose that a m; then by Claim 3 a n as well. Now de?ne the function g (x) = f (ax). Clearly, g satis?es the conditions of the problem, but Ng < Nf ? 1, since g does not attain f (1). Hence, by the induction hypothesis, f (m) = g (m/a) g (n/a) = f (n), as desired.

Comment. After the fact that f attains a ?nite number of values has been established, there are several ways of ?nishing the solution. For instance, let f (0) = b1 > b2 > · · · > bk be all these values. One may show (essentially in the same way as in Claim 3) that the set Si = {n : f (n) ≥ bi } consists exactly of all numbers divisible by some integer ai ≥ 0. One obviously has ai ai?1 , which implies f (ai ) f (ai?1 ) by Claim 1. So, bk bk?1 · · · b1 , thus proving the problem statement. Moreover, now it is easy to describe all functions satisfying the conditions of the problem. Namely, all these functions can be constructed as follows. Consider a sequence of nonnegative integers a1 , a2 , . . . , ak and another sequence of positive integers b1 , b2 , . . . , bk such that |ak | = 1, ai = aj and bi = bj for all 1 ≤ i < j ≤ k, and ai ai?1 and bi bi?1 for all i = 2, . . . , k. Then one may introduce the function f (n) = bi(n) , where i(n) = min{i : ai n}.

These are all the functions which satisfy the conditions of the problem.

69

N6

N6

Number Theory – solutions

52nd IMO 2011

Let P (x) and Q(x) be two polynomials with integer coe?cients such that no nonconstant polynomial with rational coe?cients divides both P (x) and Q(x). Suppose that for every positive integer n the integers P (n) and Q(n) are positive, and 2Q(n) ? 1 divides 3P (n) ? 1. Prove that Q(x) is a constant polynomial.

Solution. First we show that there exists an integer d such that for all positive integers n we have gcd P (n), Q(n) ≤ d. Since P (x) and Q(x) are coprime (over the polynomials with rational coe?cients), Euclid’s algorithm provides some polynomials R0 (x), S0 (x) with rational coe?cients such that P (x)R0 (x)? Q(x)S0 (x) = 1. Multiplying by a suitable positive integer d, we obtain polynomials R(x) = d · R0 (x) and S (x) = d · S0 (x) with integer coe?cients for which P (x)R(x) ? Q(x)S (x) = d. Then we have gcd P (n), Q(n) ≤ d for any integer n. To prove the problem statement, suppose that Q(x) is not constant. Then the sequence Q(n) is not bounded and we can choose a positive integer m for which M = 2Q(m) ? 1 ≥ 3max{P (1),P (2),...,P (d)} . (1)

Since M = 2Q(n) ? 1 3P (n) ? 1, we have 2, 3 M . Let a and b be the multiplicative orders of 2 and 3 modulo M , respectively. Obviously, a = Q(m) since the lower powers of 2 do not reach M . Since M divides 3P (m) ? 1, we have b P (m). Then gcd(a, b) ≤ gcd P (m), Q(m) ≤ d. Since the expression ax ? by attains all integer values divisible by gcd(a, b) when x and y run over all nonnegative integer values, there exist some nonnegative integers x, y such that 1 ≤ m + ax ? by ≤ d. By Q(m + ax) ≡ Q(m) (mod a) we have 2Q(m+ax) ≡ 2Q(m) ≡ 1 and therefore M 2Q(m+ax) ? 1 3P (m+ax) ? 1. Then, by P (m + ax ? by ) ≡ P (m + ax) (mod b) we have 3P (m+ax?by) ≡ 3P (m+ax) ≡ 1 (mod M ). Since P (m + ax ? by ) > 0 this implies M ≤ 3P (m+ax?by) ? 1. But P (m + ax ? by ) is listed among P (1), P (2), . . . , P (d), so M < 3P (m+ax?by) ≤ 3max{P (1),P (2),...,P (d)} (mod M )

70

52nd IMO 2011

Number Theory – solutions

N6

which contradicts (1).

Comment. We present another variant of the solution above. Denote the degree of P by k and its leading coe?cient by p. Consider any positive integer n and let a = Q(n). Again, denote by b the multiplicative order of 3 modulo 2a ? 1. Since 2a ? 1 3P (n) ? 1, we have b P (n). Moreover, since 2Q(n+at) ? 1 3P (n+at) ? 1 and a = Q(n) Q(n + at) for each positive integer t, we have 2a ? 1 3P (n+at) ? 1, hence b P (n + at) as well. Therefore, b divides gcd{P (n + at) : t ≥ 0}; hence it also divides the number

k i=0

(?1)k?i

k P (n + ai) = p · k! · ak . i

Finally, we get b gcd P (n), k! · p · Q(n)k , which is bounded by the same arguments as in the beginning of the solution. So 3b ? 1 is bounded, and hence 2Q(n) ? 1 is bounded as well.

71

N7

N7

Number Theory – solutions

52nd IMO 2011

Let p be an odd prime number. For every integer a, de?ne the number Sa = Let m and n be integers such that S3 + S4 ? 3 S2 = Prove that p divides m. m . n ap?1 a a2 + +···+ . 1 2 p?1

Solution 1. For rational numbers p1 /q1 and p2 /q2 with the denominators q1 , q2 not divisible by p, we write p1 /q1 ≡ p2 /q2 (mod p) if the numerator p1 q2 ? p2 q1 of their di?erence is divisible by p. We start with ?nding an explicit formula for the residue of Sa modulo p. Note ?rst that for p every k = 1, . . . , p ? 1 the number k is divisible by p, and 1 p p k = (p ? 1)(p ? 2) · · · (p ? k + 1) (?1) · (?2) · · · (?k + 1) (?1)k?1 ≡ = k! k! k (mod p)

Therefore, we have

p ?1

Sa = ?

k =1

(?a)k (?1)k?1 1 p (?a)k · ≡? k p k k =1

p ?1

(mod p).

The number on the right-hand side is integer. Using the binomial formula we express it as 1 p (?a) · ? p k k =1

k p ?1

1 =? p

p

?1 ? (?a) +

p

k =0

(?a)k

p k

=

(a ? 1)p ? ap + 1 p

since p is odd. So, we have Sa ≡ (a ? 1)p ? ap + 1 p (mod p).

Finally, using the obtained formula we get S3 + S4 ? 3 S2 ≡ (2p ? 3p + 1) + (3p ? 4p + 1) ? 3(1p ? 2p + 1) p p p p 4·2 ?4 ?4 (2 ? 2)2 = =? (mod p). p p

By Fermat’s theorem, p 2p ? 2, so p2 (2p ? 2)2 and hence S3 + S4 ? 3S2 ≡ 0 (mod p). 72

52nd IMO 2011

Number Theory – solutions

N7

Solution 2. One may solve the problem without ?nding an explicit formula for Sa . It is enough to ?nd the following property. Lemma. For every integer a, we have Sa+1 ≡ S?a (mod p). Proof. We expand Sa+1 using the binomial formula as

p ?1 k p ?1 k p ?1 p ?1 p ?1

Sa+1 =

k =1

1 k

j =0

k j a = j

k =1

1 + k

j =1

aj ·

1 k k j

=

k =1

1 + k

aj

j =1 k =j

1 k k a . k j

1 1 Note that k + p? = k(pp ≡ 0 (mod p) for all 1 ≤ k ≤ p ? 1; hence the ?rst sum vanishes k ?k ) k ?1 1 k =1 to obtain modulo p. For the second sum, we use the relation k j j j ?1 p ?1

Sa+1 ≡ Finally, from the relation

p ?1

j =1

aj j

p ?1

k =1

k?1 j?1

(mod p).

k =1

k?1 j ?1

=

p?1 j

=

(p ? 1)(p ? 2) . . . (p ? j ) ≡ (?1)j j!

p ?1

(mod p)

we obtain Sa+1 ≡

j =1

aj (?1)j = S? a . j!

Now we turn to the problem. Using the lemma we get S3 ? 3 S2 ≡ S? 2 ? 3 S2 = The ?rst sum in (1) expands as

(p?1)/2

1≤k ≤p?1 k is even

?2 · 2k + k

1≤k ≤p?1 k is odd

?4 · 2k k

(mod p).

(1)

?=1

?2 · 22? =? 2?

(p?1)/2

?=1

4? . ?

Next, using Fermat’s theorem, we expand the second sum in (1) as

(p?1)/2

?

?=1

22?+1 ≡? 2? ? 1

(p?1)/2

?=1

2p+2? =? p + 2? ? 1

p ?1

m=(p+1)/2

2 · 4m =? 2m

p ?1

m=(p+1)/2

4m m

(mod p)

(here we set m = ? +

p ?1 ). 2

Hence,

(p?1)/2

S3 ? 3 S2 ≡ ?

?=1

4? ? ?

p ?1

m=(p+1)/2

4m = ?S4 m

(mod p).

73

N8

N8

Number Theory – solutions

52nd IMO 2011

Let k be a positive integer and set n = 2k + 1. Prove that n is a prime number if and only if the following holds: there is a permutation a1 , . . . , an?1 of the numbers 1, 2, . . . , n ? 1 and a ai sequence of integers g1 , g2 , . . . , gn?1 such that n divides gi ? ai+1 for every i ∈ {1, 2, . . . , n ? 1}, where we set an = a1 .

Solution. Let N = {1, 2, . . . , n ? 1}. For a, b ∈ N , we say that b follows a if there exists an integer g such that b ≡ g a (mod n) and denote this property as a → b. This way we have a directed graph with N as set of vertices. If a1 , . . . , an?1 is a permutation of 1, 2, . . . , n ? 1 such that a1 → a2 → . . . → an?1 → a1 then this is a Hamiltonian cycle in the graph.

αs 1 Step I. First consider the case when n is composite. Let n = pα 1 . . . ps be its prime factorization. All primes pi are odd.

Suppose that αi > 1 for some i. For all integers a, g with a ≥ 2, we have g a ≡ pi (mod p2 i ), a 2 because g is either divisible by pi or it is not divisible by pi . It follows that in any Hamiltonian cycle pi comes immediately after 1. The same argument shows that 2pi also should come immediately after 1, which is impossible. Hence, there is no Hamiltonian cycle in the graph. Now suppose that n is square-free. We have n = p1 p2 . . . ps > 9 and s ≥ 2. Assume that there 1 exists a Hamiltonian cycle. There are n? even numbers in this cycle, and each number which 2 1 follows one of them should be a quadratic residue modulo n. So, there should be at least n? 2 +1 nonzero quadratic residues modulo n. On the other hand, for each pi there exist exactly pi2 quadratic residues modulo pi ; by the Chinese Remainder Theorem, the number of quadratic +1 +1 p2 +1 · 2 · . . . · ps2 , including 0. Then we have a contradiction residues modulo n is exactly p12 by s p1 + 1 p2 + 1 ps + 1 2 p1 2 p2 2 ps 2 n?1 4n · ·...· ≤ · ·...· = < . n≤ 2 2 2 3 3 3 3 9 2 This proves the “if”-part of the problem. Step II. Now suppose that n is prime. For any a ∈ N , denote by ν2 (a) the exponent of 2 in the prime factorization of a, and let ?(a) = max{t ∈ [0, k ] | 2t → a}. Lemma. For any a, b ∈ N , we have a → b if and only if ν2 (a) ≤ ?(b). Proof. Let ? = ν2 (a) and m = ?(b).

2 Suppose ? ≤ m. Since b follows 2m , there exists some g0 such that b ≡ g0 (mod n). By gcd(a, n ? 1) = 2? there exist some integers p and q such that pa ? q (n ? 1) = 2? . Choosing 2m +2m?? q (n?1) 2m?? p 2m?? pa 2m ≡ b (mod n) by Fermat’s theorem. g = g0 we have g a = g0 = g0 ≡ g0 Hence, a → b.

m

To prove the reverse statement, suppose that a → b, so b ≡ g a (mod n) with some g . Then ? ? b ≡ (g a/2 )2 , and therefore 2? → b. By the de?nition of ?(b), we have ?(b) ≥ ?. The lemma is

74

52nd IMO 2011

Number Theory – solutions

N8

proved. Now for every i with 0 ≤ i ≤ k , let Ai = {a ∈ N | ν2 (a) = i},

and Ci = {a ∈ N | ?(a) ≥ i} = Bi ∪ Bi+1 ∪ . . . ∪ Bk . We claim that |Ai | = |Bi | for all 0 ≤ i ≤ k . Obviously we have |Ai | = 2k?i?1 for all i = 0, . . . , k ? 1, and |Ak | = 1. Now we determine |Ci |. We have |C0 | = n ? 1 and by Fermat’s theorem we also have Ck = {1}, so |Ck | = 1. Next, notice that Ci+1 = {x2 mod n | x ∈ Ci }. For every a ∈ N , the relation x2 ≡ a (mod n) has at most two solutions in N . Therefore we have 2|Ci+1| ≤ |Ci |, with the equality achieved only if for every y ∈ Ci+1 , there exist distinct elements x, x′ ∈ Ci such that x2 ≡ x′2 ≡ y (mod n) (this implies x + x′ = n). Now, since 2k |Ck | = |C0|, we obtain that this equality should be achieved in each step. Hence |Ci | = 2k?i for 0 ≤ i ≤ k , and therefore |Bi | = 2k?i?1 for 0 ≤ i ≤ k ? 1 and |Bk | = 1. From the previous arguments we can see that for each z ∈ Ci (0 ≤ i < k ) the equation x2 ≡ z 2 (mod n) has two solutions in Ci , so we have n ? z ∈ Ci . Hence, for each i = 0, 1, . . . , k ? 1, exactly half of the elements of Ci are odd. The same statement is valid for Bi = Ci \ Ci+1 for 0 ≤ i ≤ k ? 2. In particular, each such Bi contains an odd number. Note that Bk = {1} also contains an odd number, and Bk?1 = {2k } since Ck?1 consists of the two square roots of 1 modulo n. Step III. Now we construct a Hamiltonian cycle in the graph. First, for each i with 0 ≤ i ≤ k , connect the elements of Ai to the elements of Bi by means of an arbitrary bijection. After performing this for every i, we obtain a subgraph with all vertices having in-degree 1 and outdegree 1, so the subgraph is a disjoint union of cycles. If there is a unique cycle, we are done. Otherwise, we modify the subgraph in such a way that the previous property is preserved and the number of cycles decreases; after a ?nite number of steps we arrive at a single cycle. For every cycle C , let λ(C ) = minc∈C ν2 (c). Consider a cycle C for which λ(C ) is maximal. If λ(C ) = 0, then for any other cycle C ′ we have λ(C ′ ) = 0. Take two arbitrary vertices a ∈ C and a′ ∈ C ′ such that ν2 (a) = ν2 (a′ ) = 0; let their direct successors be b and b′ , respectively. Then we can unify C and C ′ to a single cycle by replacing the edges a → b and a′ → b′ by a → b′ and a′ → b. Now suppose that λ = λ(C ) ≥ 1; let a ∈ C ∩ Aλ . If there exists some a′ ∈ Aλ \ C , then a′ lies in another cycle C ′ and we can merge the two cycles in exactly the same way as above. So, the only remaining case is Aλ ? C . Since the edges from Aλ lead to Bλ , we get also Bλ ? C . If λ = k ? 1 then Bλ contains an odd number; this contradicts the assumption λ(C ) > 0. Finally, if λ = k ? 1, then C contains 2k?1 which is the only element of Ak?1 . Since Bk?1 = {2k } = Ak and Bk = {1}, the cycle C contains the path 2k?1 → 2k → 1 and it contains an odd number again. This completes the proof of the “only if”-part of the problem.

Bi = {a ∈ N | ?(a) = i},

75

N8

Number Theory – solutions

52nd IMO 2011

Comment 1. The lemma and the fact |Ai | = |Bi | together show that for every edge a → b of the Hamiltonian cycle, ν2 (a) = ?(b) must hold. After this observation, the Hamiltonian cycle can be built in many ways. For instance, it is possible to select edges from Ai to Bi for i = k, k ? 1, . . . , 1 in such a way that they form disjoint paths; at the end all these paths will have odd endpoints. In the ?nal step, the paths can be closed to form a unique cycle. Comment 2. Step II is an easy consequence of some basic facts about the multiplicative group modulo the prime n = 2k + 1. The Lemma follows by noting that this group has order 2k , so the a-th powers are exactly the 2ν2 (a) -th powers. Using the existence of a primitive root g modulo n one sees that the map from {1, 2, . . . , n ? 1} to itself that sends a to g a mod n is a bijection that sends Ai to Bi for each i ∈ {0, . . . , k}.

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