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The Fifteenth British Mathematical Olympiad
1979年第十五届英国奥林匹克数学竞赛
  1. Find all triangles ABC such that AB + AC = 2 and AD + BD = √5, where AD is the altitude .
  2. Three rays in space have endpoints at O. The angles between the pairs are α, β, γ, where 0 < α < β < γ. Show that there are unique points A, B, C, one on each ray, so that the triangles OAB, OBC, OCA all have perimeter 2s. Find their distances from 0 .
  3. Show that the sum of any n distinct positive odd integers whose pairs all have different differences is at least n(n2 + 2)/3 .
  4. f(x) is defined on the rationals and takes rational values. f(x + f(y) ) = f(x) f(y) for all x, y. Show that f must be constant .
  5. Let p(n) be the number of partitions of n. For example, p(4) = 5: 1 + 1 + 1 + 1, 1 + 1 + 2, 2 + 2, 1 + 3, 4. Show that p(n+1) ≥ 2p(n) - p(n-1) .
  6. Show that the number 1 + 104 + 108 + ... + 104n is not prime for n > 0 .
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