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The 12th Asian Pacific Mathematical Olympiad
2000年第12届亚太地区数学奥林匹克
  1. Find x13/(1 - 3x1 + 3x12) + x23/(1 - 3x2 + 3x22) + ... + x1013/(1 - 3x101 + 3x1012), where xn = n/101.
  2. Find all permutations a1, a2, ... , a9 of 1, 2, ... , 9 such that a1 + a2 + a3 + a4 = a4 + a5 + a6 + a7 = a7 + a8 + a9 + a1 and a12 + a22 + a32 + a42 = a42 + a52 + a62 + a72 = a72 + a82 + a92 + a12.
  3. Let ABC be a triangle. Let M and N be the points in which the median and the angle bisector, respectively, at A meet the side BC. Let Q and P be the points in which the perpendicular at N to NA meets MA and BA, respectively, and O the point in which the perpendicular at P to BA meets AN produced. Prove that QO is perpendicular to BC.
  4. Let n, k be given positive integers with n > k. Prove that
    nn/(kk (n-k)n-k) > n!/( k! (n-k)! ) > nn/( kk (n-k)n-k )/(n+1).
  5. Given a permutation (a0 ,a1 , ... ,an ) of the sequence 0, 1, …, n. A transposition of ai with aj is called legal if ai = 0 for i > 0, and ai-1 + 1= aj. The permutation (a0 ,a1 , ... ,an ) is called regular if after a number of legal transpositions it becomes (1, 2, ... , n, 0) . For which numbers n is the permutation (1,n , n-1, ... , 3, 2, 0) regular?
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