40th British Mathematical Olympiad 2004 Problems

40th British Mathematical Olympiad 2004 Problems

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 a_ka_k+1...a_k+2003 in the expansion. Show that the decimal must be rational.