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The 18th Indian National Mathematical Olympiad
2003年第18届印度奥林匹克数学竞赛
  1. ABC is acute-angled. P is an interior point. The line BP meets AC at E, and the line CP meets AB at F. AP meets EF at D. K is the foot of the perpendicular from D to BC. Show that KD bisects ∠EKF.
  2. Find all primes p, q and even n > 2 such that pn + pn-1 + ... + p + 1 = q2 + q + 1.
  3. Show that 8x4 - 16x3 + 16x2 - 8x + k = 0 has at least one real root for all real k. Find the sum of the non-real roots.
  4. Find all 7-digit numbers which use only the digits 5 and 7 and are divisible by 35.
  5. ABC has sides a, b, c. The triangle A'B'C' has sides a + b/2, b + c/2, c + a/2. Show that its area is at least (9/4) area ABC.
  6. Each lottery ticket has a 9-digit numbers, which uses only the digits 1, 2, 3. Each ticket is colored red, blue or green. If two tickets have numbers which differ in all nine places, then the tickets have different colors. Ticket 122222222 is red, and ticket 222222222 is green. What color is ticket 123123123 ?
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