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The 1st Asian Pacific Mathematical Olympiad
1989年第1届亚太地区数学奥林匹克
  1. Let x1, x2, . . . , xn be positive real numbers, and S =x1+ x2+ . . . + xn . Prove that
    (1 + x1) ... (1 + xn) ≤ 1 + S + S2/2! + ... + Sn/n! .
  2. Prove that the equation: 6(6a^2 + 3b^2 + c^2) = 5n^2
    has no solutions in integers except a = b = c = n = 0.
  3. Let A1, A2, A3 be three points in the plane, and for convenience, let A4 = A1, A5 = A2. For n = 1, 2, and 3, suppose that Bn is the midpoint of AnAn+1, and suppose that Cn is the midpoint of AnBn. Suppose that AnCn+1 and BnAn+2 meet at Dn, and that AnBn+1 and CnAn+2 meet at En. Calculate the ratio of the area of triangle D1D2D3 to the area of triangle E1E2E3.
  4. Let S be a set consisting of m pairs (a, b) of positive integers with the property that 1 ≤ a < b ≤ n. Show that there are at least m(4m-n^2)/(3n) triples (a, b, c) such that (a, b), (a, c), and (b, c) belong to S.
  5. Determine all functions f from the reals to the reals for which
    (1) f(x) is strictly increasing,
    (2) f(x) + g(x) = 2x for all real x,
    where g(x) is the composition inverse function to f(x).
    (Note: f and g are said to be composition inverses if f(g(x)) = x and g(f(x)) = x for all real x.)
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