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The 15th Asian Pacific Mathematical Olympiad
2003年第15届亚太地区数学奥林匹克
  1. Let a, b, c, d, e, f be real numbers such that the polynomial
    p(x) = x8 -4x7 + 7x6 + ax5 + bx4 + cx3 + dx2 + ex + f
    factorises into eight linear factors x - xi, with xi > 0 for i = 1, 2, ... , 8. Determine all possible values of f.
  2. Suppose ABCD is a square piece of cardboard with side length a. On a plane are two parallel lines l1 and l2 , which are also a units apart. The square ABCD is placed on the plane so that sides AB and AD intersect l1 at E and F respectively. Also, sides CB and CD intersect l2 at G and H respectively. Let the perimeters of △AEF and △CGH be m1 and m2 respectively. Prove that no matter how the square was placed, m1 + m2 remains constant.
  3. Let k ≥ 14 be an integer, and let pk be the largest prime number which is strictly less than k. You may assume that pk ≥ 3k/4. Let n be a composite integer. Prove:
    (a) if n = 2pk, then n does not divide (n - k)! ;
    (b) if n > 2pk, then n divides (n - k)! .
  4. Let a, b, c be the sides of a triangle, with a + b + c = 1, and let n ≥ 2 be an integer. Show that (an + bn)1/n + (bn + cn)1/n + (cn + an)1/n < 1 +(21/n)/2 .
  5. Given two positive integers m and n, find the smallest positive integer k such that among any k people, either there are 2m of them who form m pairs of mutually acquainted people or there are 2n of them forming n pairs of mutually unacquainted people.
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