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The 15th Indian National Mathematical Olympiad
2000年第15届印度奥林匹克数学竞赛
  1. The incircle of ABC touches BC, CA, AB at K, L, M respectively. The line through A parallel to LK meets MK at P, and the line through A parallel to MK meets LK at Q. Show that the line PQ bisects AB and bisects AC.
  2. Find the integer solutions to a + b = 1 - c, a3 + b3 = 1 - c2.
  3. a, b, c are non-zero reals, and x is real and satisfies [bx + c(1-x)]/a = [cx + a(1-x)]/b = [ax + b(1-x)]/b. Show that a = b = c.
  4. In a convex quadrilateral PQRS, PQ = RS, SP = (√3 + 1)QR, and ∠RSP - ∠SQP = 30o. Show that ∠PQR - ∠QRS = 90o.
  5. a, b, c are reals such that 0 ≤ c ≤ b ≤ a ≤ 1. Show that if α is a root of z3 + az2 + bz + c = 0, then |α| ≤ 1.
  6. Let f(n) be the number of incongruent triangles with integral sides and perimeter n, eg f(3) = 1, f(4) = 0, f(7) = 2. Show that f(1999) > f(1996) and f(2000) = f(1997).
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