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The 16th Indian National Mathematical Olympiad
2001年第16届印度奥林匹克数学竞赛
  1. ABC is a triangle which is not right-angled. P is a point in the plane. A', B', C' are the reflections of P in BC, CA, AB. Show that [incomplete].
  2. Show that a2 + b2 + c2 = (a-b)(b-c)(c-a) has infinitely many integral solutions.
  3. a, b, c are positive reals with product 1. Show that ab+cbc+aca+b ≤ 1.
  4. Show that given any nine integers, we can find four, a, b, c, d such that a + b - c - d is divisible by 20. Show that this is not always true for eight integers.
  5. ABC is a triangle. M is the midpoint of BC. ∠MAB = ∠C, and ∠MAC = 15 o. Show that ∠AMC is obtuse. If O is the circumcenter of ADC, show that AOD is equilateral.
  6. Find all real-valued functions f on the reals such that f(x+y) = f(x) f(y) f(xy) for all x, y.
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