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The 40th British Mathematical Olympiad
2004年第40届英国奥林匹克数学竞赛
  1. ABC is an equilateral triangle. D is a point on the side BC (not at the endpoints). A circle touches BC at D and meets the side AB at M and N, and the side AC at P and Q. Show that BD + AM + AN = CD + AP + AQ .
  2. Show that there is a multiple of 2004 whose binary expression has exactly 2004 0s and 2004 1s .
  3. a, b, c are reals with sum zero. Show that a3 + b3 + c3 > 0 iff a5 + b5 + c5 > 0. Prove the same result for 4 reals .
  4. The decimal 0.a1a2a3a4... hs the property that there are at most 2004 distinct blocks akak+1...ak+2003 in the expansion. Show that the decimal must be rational .
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