Asian Pacific Mathematics Olympiad · 1989

Problems

Problem 1Let $x_1,x_2,\ldots,x_n$ be positive real numbers and let $S=x_1+x_2+\cdots+x_n$ . Prove that $(1+x_1)(1+x_2)\cdots(1+x_n)\le 1+S+\dfrac{S^2}{2!}+\dfrac{S^3}{3!}+\cdots+\dfrac{S^n}{n!}$ .Solutions: 1
Problem 2Prove that the equation $6(6a^2+3b^2+c^2)=5n^2$ has no integer solutions except $a=b=c=n=0$ .Solutions: 1
Problem 3Let $A_1,A_2,A_3$ be three points in the plane, with $A_4=A_1$ and $A_5=A_2$ . For $n=1,2,3$ , let $B_n$ be the midpoint of $A_nA_{n+1}$ and $C_n$ the midpoint of $A_nB_n$ . Let $D_n=A_nC_{n+1}\cap B_nA_{n+2}$ and $E_n=A_nB_{n+1}\cap C_nA_{n+2}$ . Calculate the ratio of the area of $D_1D_2D_3$ to the area of $E_1E_2E_3$ .Solutions: 1
Problem 4Let $S$ be a set consisting of $m$ pairs $(a,b)$ of positive integers with $1\le a<b\le n$ . Show that there are at least $\dfrac{m(4m-n^2)}{3n}$ triples $(a,b,c)$ such that $(a,b),(a,c)$ , and $(b,c)$ belong to $S$ .Solutions: 1
Problem 5Determine all functions $f:\mathbb R\to\mathbb R$ such that (1) $f$ is strictly increasing, and (2) $f(x)+g(x)=2x$ for every real $x$ , where $g$ is the composition inverse of $f$ .Solutions: 1