Asian Pacific Mathematics Olympiad · 1993

Problems

Problem 1Let $ABCD$ be a quadrilateral with all sides equal and $\angle ABC=60^\circ$ . Let $\ell$ be a line through $D$ that does not meet the quadrilateral except at $D$ . Let $E,F$ be the intersections of $\ell$ with $AB,BC$ , respectively, and let $M=CE\cap AF$ . Prove that $CA^2=CM\cdot CE$ .Solutions: 1
Problem 2Find the total number of different integer values taken by $f(x)=\lfloor x\rfloor+\lfloor2x\rfloor+\left\lfloor\frac{5x}{3}\right\rfloor+\lfloor3x\rfloor+\lfloor4x\rfloor$ for real $x$ with $0\le x\le100$ .Solutions: 1
Problem 3Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ and $g(x)=c_{n+1}x^{n+1}+c_nx^n+\cdots+c_0$ be non-zero real polynomials such that $g(x)=(x+r)f(x)$ for some real $r$ . If $a=\max(|a_n|,\ldots,|a_0|)$ and $c=\max(|c_{n+1}|,\ldots,|c_0|)$ , prove that $\frac{a}{c}\le n+1$ .Solutions: 1
Problem 4Determine all positive integers $n$ for which $x^n+(2+x)^n+(2-x)^n=0$ has an integer solution.Solutions: 1
Problem 5Let $P_1,P_2,\ldots,P_{1993}=P_0$ be distinct points in the $xy$ -plane with integer coordinates. Assume that no point other than $P_i$ and $P_{i+1}$ on the segment $P_iP_{i+1}$ has both coordinates integers, for $i=0,1,\ldots,1992$ . Prove that for some $i$ , $0\le i\le1992$ , there is a point $Q=(q_x,q_y)$ on $P_iP_{i+1}$ such that both $2q_x$ and $2q_y$ are odd integers.Solutions: 1