Asian Pacific Mathematics Olympiad · 1995

Problems

Problem 1Determine all sequences of real numbers $(a_1,a_2,\ldots,a_{1995})$ satisfying $2\sqrt{a_n-(n-1)}\ge a_{n+1}-(n-1)$ for $n=1,2,\ldots,1994$ , and $2\sqrt{a_{1995}-1994}\ge a_1+1$ .Solutions: 1
Problem 3Let $PQRS$ be a cyclic quadrilateral such that the lines $PQ$ and $RS$ are not parallel. Let $\mathcal A$ be the set of points of tangency of a circle through $P,Q$ and a circle through $R,S$ . Determine $\mathcal A$ .Solutions: 1
Problem 4Let $C$ be a circle of radius $R$ and center $O$ , and let $S$ be a fixed interior point. Let $AA'$ and $BB'$ be perpendicular chords through $S$ . The rectangles $SAMB$ , $SB'N'A'$ , $SA'M'B'$ , and $SBNA$ are formed. Determine the locus of the four points $M,N',M',N$ as $A$ moves around $C$ .Solutions: 1
Problem 5Find the minimum positive integer $k$ for which there exists a function $f:\mathbb Z\to\{1,2,\dots,k\}$ such that $f(x)\ne f(y)$ whenever $|x-y|\in\{5,7,12\}$ .Solutions: 1