Asian Pacific Mathematics Olympiad · 2024

Problems

Problem 1Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of quadrilateral $BCED$ . Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$ , respectively, such that both $P$ and $Q$ lie between $B$ and $C$ . Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y\neq X$ . Prove that points $A$ , $X$ , and $Y$ are collinear.Solutions: 1
Problem 2Consider a $100\times100$ table, and identify the cell in row $a$ and column $b$ , $1\le a,b\le100$ , with the ordered pair $(a,b)$ . Let $k$ be an integer such that $51\le k\le99$ . A $k$ -knight is a piece that moves one cell vertically or horizontally and $k$ cells in the other direction; that is, it moves from $(a,b)$ to $(c,d)$ such that $(|a-c|,|b-d|)$ is either $(1,k)$ or $(k,1)$ . The $k$ -knight starts at cell $(1,1)$ and performs several moves. A sequence of moves is a sequence of cells $(x_0,y_0)=(1,1),(x_1,y_1),\ldots,(x_n,y_n)$ such that, for all $i=1,2,\ldots,n$ , $1\le x_i,y_i\le100$ and the $k$ -knight can move from $(x_{i-1},y_{i-1})$ to $(x_i,y_i)$ . In this case each cell $(x_i,y_i)$ is said to be reachable. For each $k$ , find $L(k)$ , the number of reachable cells.Solutions: 1
Problem 3Let $n$ be a positive integer and $a_1,a_2,\ldots,a_n$ be positive real numbers. Prove that $\sum_{i=1}^{n}\frac{1}{2^i}\left(\frac{2}{1+a_i}\right)^{2^i}\ge\frac{2}{1+a_1a_2\cdots a_n}-\frac{1}{2^n}.$ Solutions: 1
Problem 4Prove that for every positive integer $t$ there is a unique permutation $a_0,a_1,\ldots,a_{t-1}$ of $0,1,\ldots,t-1$ such that, for every $0\le i\le t-1$ , the binomial coefficient $\binom{t+i}{2a_i}$ is odd and $2a_i\neq t+i$ .Solutions: 1
Problem 5Line $\ell$ intersects sides $BC$ and $AD$ of cyclic quadrilateral $ABCD$ at its interior points $R$ and $S$ , respectively, and intersects ray $DC$ beyond point $C$ at $Q$ , and ray $BA$ beyond point $A$ at $P$ . The circumcircles of triangles $QCR$ and $QDS$ intersect at $N\neq Q$ , while the circumcircles of triangles $PAS$ and $PBR$ intersect at $M\neq P$ . Let lines $MP$ and $NQ$ meet at point $X$ , lines $AB$ and $CD$ meet at point $K$ , and lines $BC$ and $AD$ meet at point $L$ . Prove that point $X$ lies on line $KL$ .Solutions: 1