Asian Pacific Mathematics Olympiad · 2015

Problems

Problem 1Let $ABC$ be a triangle, and let $D$ be a point on side $BC$ . A line through $D$ intersects side $AB$ at $X$ and ray $AC$ at $Y$ . The circumcircle of triangle $BXD$ intersects the circumcircle $\omega$ of triangle $ABC$ again at point $Z\ne B$ . The lines $ZD$ and $ZY$ intersect $\omega$ again at $V$ and $W$ , respectively. Prove that $AB=VW$ .Solutions: 1
Problem 2Let $S=\{2,3,4,\ldots\}$ denote the set of integers that are greater than or equal to $2$ . Does there exist a function $f:S\to S$ such that $f(a)f(b)=f(a^2b^2)$ for all $a,b\in S$ with $a\ne b$ ?Solutions: 1
Problem 3A sequence of real numbers $a_0,a_1,\ldots$ is said to be good if: (i) $a_0$ is a positive integer; (ii) for each non-negative integer $i$ , $a_{i+1}=2a_i+1$ or $a_{i+1}=\frac{a_i}{a_i+2}$ ; (iii) there exists a positive integer $k$ such that $a_k=2014$ . Find the smallest positive integer $n$ such that there exists a good sequence with $a_n=2014$ .Solutions: 1
Problem 4Let $n$ be a positive integer. Consider $2n$ distinct lines on the plane, no two of which are parallel. Of the $2n$ lines, $n$ are colored blue, the other $n$ are colored red. Let $B$ be the set of all points on the plane that lie on at least one blue line, and $R$ the set of all points on the plane that lie on at least one red line. Prove that there exists a circle that intersects $B$ in exactly $2n-1$ points, and also intersects $R$ in exactly $2n-1$ points.Solutions: 1
Problem 5Determine all sequences $a_0,a_1,a_2,\ldots$ of positive integers with $a_0\ge2015$ such that for all integers $n\ge1$ : (i) $a_{n+2}$ is divisible by $a_n$ ; (ii) $|s_{n+1}-(n+1)a_n|=1$ , where $s_{n+1}=a_{n+1}-a_n+a_{n-1}-\cdots+(-1)^{n+1}a_0$ .Solutions: 1