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The 16th United States of America Mathematics Olympiad
1987年第16届美国数学奥林匹克
  1. Find all solutions to (m2 + n)(m + n2) = (m - n)3, where m and n are non-zero integers.
  2. The feet of the angle bisectors of the triangle ABC form a right-angled triangle. If the right-angle is at X, where AX is the bisector of angle A, find all possible values for angle A.
  3. X is the smallest set of polynomials p(x) such that: (1) p(x) = x belongs to X; and (2) if r(x) belongs to X, then x r(x) and (x + (1 - x) r(x) ) both belong to X. Show that if r(x) and s(x) are distinct elements of X, then r(x) ≠ s(x) for any 0 < x < 1.
  4. M is the midpoint of XY. The points P and Q lie on a line through Y on opposite sides of Y, such that |XQ| = 2|MP| and |XY|/2 < |MP| < 3|XY|/2. For what value of |PY|/|QY| is |PQ| a minimum ?
  5. a1, a2, ... , an is a sequence of 0s and 1s. T is the number of triples (ai, aj, ak) with i < j < k which are not equal to (0, 1, 0) or (1, 0, 1). For 1 ≤ i ≤ n, f(i) is the number of j < i with aj = ai plus the number of j > i with aj ≠ ai. Show that T = f(1) (f(1) - 1)/2 + f(2) (f(2) - 1)/2 + ... + f(n) (f(n) - 1)/2. If n is odd, what is the smallest value of T ?
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