Asian Pacific Mathematics Olympiad · 2005
Problems
- Problem 1Prove that for every irrational real number a, there are irrational real numbers b and b' such that a+b and ab' are rational, while ab and a+b' are irrational.Solutions: 1
- Problem 2Let a,b,c be positive real numbers with abc=8. Prove that a squared divided by the square root of (1+a cubed)(1+b cubed), plus the two cyclic analogues, is at least 4/3.Solutions: 1
- Problem 3Prove that there exists a triangle which can be cut into 2005 congruent triangles.Solutions: 1
- Problem 4In an n by n town, houses are indexed by (i,j), with (1,1) at the top left. Initially the house (1,c) burns, where c is at most n/2. During each unit interval firefighters defend one unburned house, then fire spreads to every undefended neighbor of each house burning at the start of the interval. Defended houses remain defended. What is the maximum number of houses that can be saved? Neighbors differ by one in exactly one coordinate.Solutions: 1
- Problem 5In triangle ABC, points M and N lie on AB and AC, respectively, and MB=BC=CN. If R and r are the circumradius and inradius of ABC, express MN/BC in terms of R and r.Solutions: 1