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第13届哈佛-MIT校际数学竞赛代数试题


13th Annual Harvard-MIT Mathematics Tournament
Saturday 20 February 2010

Algebra Subject Test
1. [3] Suppose that x and y are positive reals such that x ? y 2 = 3, x2 + y 4 = 13. Find x.
1 2. [3] The rank of a rational number q is the unique k for which q = a1 + · · · + a1 , where each ai is the k 1 1 smallest positive integer such that q ≥ a1 + · · · + ai . Let q be the largest rational number less than 1 4 1 1 1 with rank 3, and suppose the expression for q is a1 + a2 + a3 . Find the ordered triple (a1 , a2 , a3 ).

3. [4] Let S0 = 0 and let Sk equal a1 + 2a2 + . . . + kak for k ≥ 1. De?ne ai to be 1 if Si?1 < i and -1 if Si?1 ≥ i. What is the largest k ≤ 2010 such that Sk = 0? 4. [4] Suppose that there exist nonzero complex numbers a, b, c, and d such that k is a root of both the equations ax3 + bx2 + cx + d = 0 and bx3 + cx2 + dx + a = 0. Find all possible values of k (including complex values). 5. [5] Suppose that x and y are complex numbers such that x + y = 1 and that x20 + y 20 = 20. Find the sum of all possible values of x2 + y 2 . 6. [5] Suppose that a polynomial of the form p(x) = x2010 ± x2009 ± · · · ± x ± 1 has no real roots. What is the maximum possible number of coe?cients of ?1 in p? 7. [5] Let a, b, c, x, y, and z be complex numbers such that a= c+a a+b b+c , b= , c= . x?2 y?2 z?2

If xy + yz + zx = 67 and x + y + z = 2010, ?nd the value of xyz. 8. [6] How many polynomials of degree exactly 5 with real coe?cients send the set {1, 2, 3, 4, 5, 6} to a permutation of itself? 9. [7] Let f (x) = cx(x ? 1), where c is a positive real number. We use f n (x) to denote the polynomial obtained by composing f with itself n times. For every positive integer n, all the roots of f n (x) are real. What is the smallest possible value of c? 10. [8] Let p(x) and q(x) be two cubic polynomials such that p(0) = ?24, q(0) = 30, and p(q(x)) = q(p(x)) for all real numbers x. Find the ordered pair (p(3), q(6)).



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