Asian Pacific Mathematics Olympiad · 2025

Problems

Problem 1Let $ABC$ be an acute triangle inscribed in a circle $\Gamma$ . Let $A_1$ be the orthogonal projection of $A$ onto $BC$ , so that $AA_1$ is an altitude. Let $B_1$ and $C_1$ be the orthogonal projections of $A_1$ onto $AB$ and $AC$ , respectively. Point $P$ is such that quadrilateral $AB_1PC_1$ is convex and has the same area as triangle $ABC$ . Is it possible that $P$ lies strictly in the interior of circle $\Gamma$ ? Justify your answer.Solutions: 1
Problem 2Let $\alpha$ and $\beta$ be positive real numbers. Emerald makes a trip in the coordinate plane, starting from the origin $(0,0)$ . Each minute she moves one unit up or one unit to the right, restricting herself to the region $|x-y|<2025$ in the coordinate plane. By the time she visits a point $(x,y)$ , she writes down the integer $\lfloor x\alpha+y\beta\rfloor$ on it. It turns out that Emerald wrote each non-negative integer exactly once. Find all possible pairs $(\alpha,\beta)$ for which such a trip is possible.Solutions: 1
Problem 3Let $P(x)$ be a non-constant polynomial with integer coefficients such that $P(0)\neq0$ . Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of integers such that $P(i-j)$ divides $a_i-a_j$ for all distinct positive integers $i,j$ . Prove that the sequence $a_1,a_2,a_3,\ldots$ must be constant, that is, $a_n$ equals a constant $c$ for every positive integer $n$ .Solutions: 1
Problem 4Let $n\ge 3$ be an integer. There are $n$ cells on a circle, each assigned either 0 or 1, and a rooster occupies one cell. Repeatedly, if the rooster is on a cell assigned 0, it changes that number to 1 and moves to the next cell counterclockwise; if it is on a cell assigned 1, it changes that number to 0 and moves to the cell after next counterclockwise. Prove that after sufficiently many operations, whenever the rooster is on a cell $C$ , it goes around the circle exactly three times and stops again at $C$ , while every cell has the same number as it had immediately before those three laps.Solutions: 1
Problem 5Consider an infinite sequence $a_1,a_2,\ldots$ of positive integers such that $100!(a_m+a_{m+1}+\cdots+a_n)$ is a multiple of $a_{n-m+1}a_{n+m}$ for all positive integers $m,n$ with $m\le n$ . Prove that the sequence is either bounded or linear. A sequence is bounded if there is a constant $N$ such that $a_n<N$ for every positive integer $n$ , and it is linear if $a_n=n\cdot a_1$ for every positive integer $n$ .Solutions: 1