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The 6th Asian Pacific Mathematical Olympiad
1994年第6届亚太地区数学奥林匹克
  1. Find all real-valued functions f on the reals such that (1) f(1) = 1, (2) f(-1) = -1, (3) f(x) ≤ f(0) for 0 < x < 1, (4) f(x + y) ≥ f(x) + f(y) for all x, y, (5) f(x + y) ≤ f(x) + f(y) + 1 for all x, y.
  2. Given a nondegenerate triangle ABC, with circumcentre O, orthocentre H, and circumradius R, prove that |OH|< 3R.
  3. Let n be an integer of the form a2 +b2, where a and b are relatively prime integers and such
    that if p is a prime, p ≤ √n, then p divides ab. Determine all such n.
  4. Is there an infinite set of points in the plane such that no three points are collinear, and the
    distance between any two points is rational?
  5. You are given three lists A, B, and C. List A contains the numbers of the form 10k in base
    10, with k any integer greater than or equal to 1. Lists B and C contain the same numbers
    translated into base 2 and 5 respectively:
                A     B           C
                10    1010        20
                100   1100100     400
                1000  1111101000  13000
                ...
    Prove that for every integer n > 1, there is exactly one number in exactly one of the lists Bor C that has exactly n digits.
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