Asian Pacific Mathematics Olympiad · 2002

Problems

Problem 1Let $a_1,a_2,\dots,a_n$ be a sequence of non-negative integers, and let $A_n=(a_1+a_2+\cdots+a_n)/n$ . Prove that $a_1!a_2!\cdots a_n!\ge (\lfloor A_n\rfloor!)^n$ , where $a!=1\cdot2\cdots a$ for $a\ge1$ and $0!=1$ . When does equality hold?Solutions: 1
Problem 2Find all positive integers $a,b$ such that $\frac{a^2+b}{b^2-a}$ and $\frac{b^2+a}{a^2-b}$ are both integers.Solutions: 1
Problem 3Let $ABC$ be an equilateral triangle. Let $P$ be a point on $AC$ and $Q$ a point on $AB$ so that triangles $ABP$ and $ACQ$ are acute. Let $R$ and $S$ be the orthocentres of triangles $ABP$ and $ACQ$ , respectively. Let $T$ be the common point of the segments $BP$ and $CQ$ . Find all possible values of $\angle CBP$ and $\angle BCQ$ such that triangle $TRS$ is equilateral.Solutions: 1
Problem 4Let $x,y,z$ be positive numbers such that $\frac1x+\frac1y+\frac1z=1$ . Show that $\sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\ge\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z}$ .Solutions: 1
Problem 5Let $\mathbb R$ denote the set of all real numbers. Find all functions $f:\mathbb R\to\mathbb R$ such that (i) only finitely many $s\in\mathbb R$ satisfy $f(s)=0$ , and (ii) $f(x^4+y)=x^3f(x)+f(f(y))$ for all $x,y\in\mathbb R$ .Solutions: 1