Asian Pacific Mathematics Olympiad · 1998

Problems

Problem 1Let $\mathcal S$ be the set of all $n$ -tuples $(A_1,A_2,\ldots,A_n)$ in which each $A_i$ is a subset of $\{1,2,\ldots,1998\}$ . For $K\in\mathcal S$ , let $f(K)=|A_1\cup A_2\cup\cdots\cup A_n|$ . Find $\sum_{K\in\mathcal S}f(K)$ .Solutions: 1
Problem 2Show that $(36a+b)(a+36b)$ cannot be a power of $2$ for any positive integers $a$ and $b$ .Solutions: 1
Problem 3Let $x,y,z$ be positive real numbers and let $w=\sqrt[3]{xyz}$ . Prove that $(1+x/y)(1+y/z)(1+z/x)\ge2+2(x+y+z)/w$ .Solutions: 1
Problem 4Let ABC be a triangle and let D be the foot of the altitude from A. Let E and F be points on a line through D such that AE is perpendicular to BE, AF is perpendicular to CF, and E and F are different from D. Let M and N be the midpoints of BC and EF, respectively. Prove that AN is perpendicular to NM.Solutions: 1
Problem 5Determine the largest integer n with the property that n is divisible by every positive integer less than the cube root of n.Solutions: 1