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美国奥数训练题(国家奥数队领队出题)6


1 In five years, Tom will be twice as old as Cindy. Thirteen years ago, Tom was three times as old as Cindy. How many years ago was Tom four times as old as Cindy? 2 Find the least positive integer such that for every prime number is never prime. is a regular heptagon 3 In the diagram (a sided polygon). Shown is the star . The degree measure of the obtuse angle formed by and is where m and n are relatively prime positive integers. Find .

4 There are three bags of marbles. Bag two has twice as many marbles as bag one. Bag three has three times as many marbles as bag one. Half the marbles in bag one, one third the marbles in bag two, and one fourth the marbles in bag three are green. If all three bags of marbles are dumped into a single pile, of the marbles in the pile would be green where and are relatively prime positive integers. Find

5 Find so that

6 Wiles county contains eight townships as shown on the map. If there are four colors available, in how many ways can the the map be colored so that each township is colored with one color and no two townships that share a border are colored with the same color?

is bounded by a semicircle 7 The figure and a quarter circle . Given that the distance from to is , find the area of the figure.

8 Find the least positive integer that has exactly positive integer divisors.

9 Bill bought 13 notebooks, 26 pens, and 19 markers for 25 dollars. Paula bought 27 notebooks, 18 pens, and 31 markers for 31 dollars. How many dollars would it cost Greg to buy 24 notebooks, 120 pens, and 52 markers? 10 Towers grow at points along a line. All towers start with height 0 and grow at the rate of one meter per second. As soon as any two adjacent towers are each at least 1 meter tall, a new tower begins to grow at a point along the line exactly half way between those two adjacent towers. Before time 0 there are no towers, but at time 0 the first two towers begin to grow at two points along the line. Find the total of all the heights of all towers at time 10 seconds.

and 11 The four points lie in the coordinate plane. Compute the minimum possible value of over all points P . 12 What is the least possible sum of two positive integers and where 13 Greta is completing an art project. She has twelve sheets of paper: four red, four white, and four blue. She also has twelve paper stars: four red, four white, and four blue. She randomly places one star on each sheet of paper. The probability that no star will be placed on a sheet of paper that is the same color as the star is where and are relatively prime positive integers. Find 14 Let points be a trapezoid with parallel to has length and has length Let and lie on sides and

respectively, such that is parallel to and and has length Let and be two relatively prime positive integers such that the ratio of the area of to the area of is Find

15 What is the remainder when

is divided by

16 Let the complex number Find the smallest positive integer so that imaginary part which exceeds

has an

17 How many ordered triples integers satisfy

of odd positive

let be the point on so that 18 On triangle is an altitude of the triangle, and be the point on so that bisects angle Let be the intersection of and and let point be the intersection of side and the ray If has length has length and has length then the length of can be written as where and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find 19 If and are complex numbers such that and , then their sum, , is real. The greatest possible value for the sum is where and are integers. Find

20 Five men and seven women stand in a line in random order. Let m and n be relatively prime

positive integers so that is the probability that each man stands next to at least one woman. Find

21 A cylinder radius and a cylinder radius are held tangent to each other with a tight band. The length of the band is where , , and are positive integers, and is not divisible by the square of any prime. Find .

22 The diagram shows a parabola, a line perpendicular to the parabola's axis of symmetry, and three similar isosceles triangles each with a base on the line and vertex on the parabola. The two smaller triangles are congruent and each have one base vertex on the parabola and one base vertex shared with the larger triangle. The ratio of the height of the larger triangle to the height of the smaller triangles is where , , and are positive integers, and and are relatively prime. Find .

has side length . Points and 23 Square are the midpoints of sides and , respectively. Eight by rectangles are placed inside the square so that no two of the eight rectangles overlap (see diagram). If the arrangement of eight rectangles is chosen randomly, then there are relatively prime positive integers and so that is the probability that none of the rectangles crosses the line segment (as in the arrangement on the right). Find .

24 A right circular cone pointing downward forms an angle of at its vertex. Sphere with radius is set into the cone so that it is tangent to the side of the cone. Three congruent spheres are placed in the cone on top of S so that they are all tangent to each other, to sphere , and to the side of the cone. The radius of these congruent spheres can be written as where , , and are positive integers such

that and are relatively prime. Find

.

25 The polynomial has the property that prime positive integers . Find for . There are relatively and such that .

Middle School The pentagon below has three right angles. Find its area.

1

2

Let be the sequence of prime numbers. Find the least positive even integer so that is not prime. The Purple Comet! Math Meet runs from April 27 through May 3, so the sum of the calendar dates for these seven days is What is the largest sum of the calendar dates for seven consecutive Fridays occurring at any time in any year? John, Paul, George, and Ringo baked a circular pie. Each cut a piece that was a sector of the circle. John took one-third of the whole pie. Paul took one-fourth of the whole pie. George took one-fifth of the whole pie. Ringo took one-sixth of the whole pie. At the end the pie had one sector remaining. Find the measure in degrees of the angle formed by this remaining sector. A train car held pounds of mud which was percent water. Then the train car sat in the sun, and some of the water evaporated so that now the mud

3

4

5

is only percent water. How many pounds does the mud weigh now? Find such that .

6

7

How many distinct four letter arrangements can be formed by rearranging the letters found in the word FLUFFY? For example, FLYF and ULFY are two possible arrangements. Find the number of non-congruent scalene triangles whose sides all have integral length, and the longest side has length . One plant is now centimeters tall and will grow at a rate of centimeters every years. A second plant is now centimeters tall and will grow at a rate of centimeters every years. In how many years will the plants be the same height?

8

9

. The 10 The diagram shows a by square points , , and are equally spaced on side . The points , , , and on side are placed so that the triangles , , , and are isosceles. Points and are midpoints of the sides and , respectively. Find the total area of the shaded regions.

11 Aisha went shopping. At the first store she spent percent of her money plus four dollars. At the second store she spent percent of her remaining money plus dollars. At the third store she spent percent of her remaining money plus six dollars. When Aisha was done shopping at the three stores, she had two dollars left. How many dollars did she have with her when she started shopping? sides and have 12 In isosceles triangle length while side has length . Point is the midpoint of side . is on side so that and are perpendicular. Similarly, is on side so that and are perpendicular. Find the area of triangle .

13 How many subsets of the set contain exactly one or two prime numbers? measures by . Eighteen 14 Rectangle points (including and ) are marked on the diagonal dividing the diagonal into congruent pieces. Twenty-two points (including A

and B) are marked on the side dividing the side into congruent pieces. Seventeen nonoverlapping triangles are constructed as shown. Each triangle has two vertices that are two of these adjacent marked points on the side of the rectangle, and one vertex that is one of the marked points along the diagonal of the rectangle. Only the left of the congruent pieces along the side of the rectangle are used as bases of these triangles. Find the sum of the areas of these triangles.

15 We have twenty-seven by cubes. Each face of every cube is marked with a natural number so that two opposite faces (top and bottom, front and back, left and right) are always marked with an even number and an odd number where the even number is twice that of the odd number. The twenty-seven cubes are put together to form one by cube as shown. When two cubes are placed face-to-face, adjoining faces are always marked with an odd number and an even number where the even number is one greater than the odd number. Find the sum of all of the numbers on all of the faces of

all the by cubes.


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