Asian Pacific Mathematics Olympiad · 2007

Problems

Problem 1Let $S$ be a set of 9 distinct integers all of whose prime factors are at most 3. Prove that $S$ contains 3 distinct integers whose product is a perfect cube.Solutions: 1
Problem 2Let $ABC$ be an acute triangle with $\angle BAC=60^\circ$ and $AB>AC$ . Let $I$ be its incenter and $H$ its orthocenter. Prove that $2\angle AHI=3\angle ABC$ .Solutions: 1
Problem 3Consider $n$ disks $C_1,C_2,\ldots,C_n$ in the plane such that for each $1\le i<n$ , the center of $C_i$ lies on the circumference of $C_{i+1}$ , and the center of $C_n$ lies on the circumference of $C_1$ . Define the score to be the number of pairs $(i,j)$ for which $C_i$ properly contains $C_j$ . Determine the maximum possible score.Solutions: 1
Problem 4Let $x,y,z$ be positive real numbers such that $x+y+z=1$ . Prove that $\frac{x^2+yz}{\sqrt{2x^2(y+z)}}+\frac{y^2+zx}{\sqrt{2y^2(z+x)}}+\frac{z^2+xy}{\sqrt{2z^2(x+y)}}\ge1$ .Solutions: 1
Problem 5A regular $5\times5$ array of lights is defective: toggling the switch for one light causes each adjacent light in the same row and in the same column, as well as the light itself, to change state. Initially all lights are off. After some toggles, exactly one light is on. Find all possible positions of this light.Solutions: 1