Asian Pacific Mathematics Olympiad · 2008

Problems

Problem 1Let $ABC$ be a triangle with $\angle A<60^\circ$ . Let $X$ and $Y$ be the points on the sides $AB$ and $AC$ , respectively, such that $CA+AX=CB+BX$ and $BA+AY=BC+CY$ . Let $P$ be the point in the plane such that the lines $PX$ and $PY$ are perpendicular to $AB$ and $AC$ , respectively. Prove that $\angle BPC<120^\circ$ .Solutions: 1
Problem 2Students in a class form groups, each of which contains exactly three members, such that any two distinct groups have at most one member in common. Prove that, when the class size is 46, there is a set of 10 students in which no group is properly contained.Solutions: 1
Problem 3Let $\Gamma$ be the circumcircle of a triangle $ABC$ . A circle passing through points $A$ and $C$ meets the sides $BC$ and $BA$ at $D$ and $E$ , respectively. The lines $AD$ and $CE$ meet $\Gamma$ again at $G$ and $H$ , respectively. The tangent lines of $\Gamma$ at $A$ and $C$ meet the line $DE$ at $L$ and $M$ , respectively. Prove that the lines $LH$ and $MG$ meet at a point on $\Gamma$ .Solutions: 1
Problem 4Consider the function $f:\mathbb N_0\to\mathbb N_0$ , where $\mathbb N_0$ is the set of all non-negative integers, defined by $f(0)=0$ , $f(2n)=2f(n)$ and $f(2n+1)=n+2f(n)$ for all $n\ge0$ . (a) Determine the three sets $L=\{n\mid f(n)<f(n+1)\}$ , $E=\{n\mid f(n)=f(n+1)\}$ , and $G=\{n\mid f(n)>f(n+1)\}$ . (b) For each $k\ge0$ , find a formula for $a_k=\max\{f(n):0\le n\le2^k\}$ in terms of $k$ .Solutions: 1
Problem 5Let $a,b,c$ be integers satisfying $0<a<c-1$ and $1<b<c$ . For each $k$ , $0\le k\le a$ , let $r_k$ , $0\le r_k<c$ , be the remainder of $kb$ when divided by $c$ . Prove that the two sets $\{r_0,r_1,r_2,\ldots,r_a\}$ and $\{0,1,2,\ldots,a\}$ are different.Solutions: 1