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The 27th Canadian Mathematical Olympiads
1995年第二十七届加拿大数学奥林匹克
  1. Find g(1/1996) + g(2/1996) + g(3/1996) + ... + g(1995/1996) where g(x) = 9x/(3 + 9x).
  2. Show that xxyyzz >= (xyz)(x+y+z)/3 for positive reals x , y , z .
  3. A convex n-gon is divided into m quadrilaterals. Show that at most m - n/2 + 1 of the quadrilaterals have an angle exceeding 180o.
  4. Show that for any n > 0 and k ≥ 0 we can find infinitely many solutions in positive integers to x13 + x23 + ... + xn3 = y3k+2 .
  5. 0 < k < 1 is a real number. Define f: [0, 1] → [0, 1] by f(x) = 0 for x ≤ k, 1 - (√(kx) + √( (1-k)(1-x) ) )2 for x > k. Show that the sequence 1 , f(1) , f( f(1) ) , f( f( f(1) ) ) , ... eventually becomes zero .
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