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The Twentieth British Mathematical Olympiad
Further International Selection Test
1984年第二十届英国奥林匹克数学竞赛
  1. In the triangle ABC, ∠C = 90o. Find all points D such that AD·BC = AC·BD = AB·CD/√2.
  2. ABCD is a tetrahedron such that DA = DB = DC = d and AB = BC = CA = e. M and N are the midpoints of AB and CD. A variable plane through MN meets AD at P and BC at Q. Show that AP/AD = BQ/BC. Find the value of this ratio in terms of d and e which minimises the area of MQNP.
  3. Find the maximum and minimum values of cos x + cos y + cos z, where x, y, z are non-negative reals with sum 4π/3.
  4. Let bn be the number of partitions of n into non-negative powers of 2. For example b4 = 4: 1 + 1 + 1 + 1, 1 + 1 + 2, 2 + 2, 4. Let cn be the number of partitions which include at least one of every power of 2 from 1 up to the highest in the partition. For example, c4 = 2: 1 + 1 + 1 + 1, 1 + 1 + 2. Show that bn+1 = 2cn.
  5. Show that for any positive integers m, n we can find a polynomial p(x) with integer coefficients such that | p(x) - m/n | ≤ 1/n2 for all x in some interval of length 1/n.
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