Asian Pacific Mathematics Olympiad · 2020

Problems

Problem 1Let $\Gamma$ be the circumcircle of triangle $ABC$ . Let $D$ be a point on side $BC$ . The tangent to $\Gamma$ at $A$ intersects the line through $D$ parallel to $BA$ at point $E$ . The segment $CE$ intersects $\Gamma$ again at $F$ . Suppose $B$ , $D$ , $F$ , $E$ are concyclic. Prove that $AC$ , $BF$ , $DE$ are concurrent.Solutions: 1
Problem 2Show that $r=2$ is the largest real number $r$ that satisfies the following condition: if a sequence $a_1,a_2,\ldots$ of positive integers fulfills the inequalities $a_n\le a_{n+2}\le\sqrt{a_n^2+r\,a_{n+1}}$ for every positive integer $n$ , then there exists a positive integer $M$ such that $a_{n+2}=a_n$ for every $n\ge M$ .Solutions: 1
Problem 3Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n>m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways.Solutions: 1
Problem 4Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property: for any infinite sequence $a_1,a_2,\ldots$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i<j$ and an integer $k$ such that $a_i+a_{i+1}+\cdots+a_j=P(k)$ .Solutions: 1
Problem 5Let $n\ge3$ be a fixed integer. The number $1$ is written $n$ times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers $a$ and $b$ , replacing them with the numbers $1$ and $a+b$ , then adding one stone to the first bucket and $\gcd(a,b)$ stones to the second bucket. After some finite number of moves, there are $s$ stones in the first bucket and $t$ stones in the second bucket, where $s$ and $t$ are positive integers. Find all possible values of the ratio $\dfrac{t}{s}$ .Solutions: 1