Asian Pacific Mathematics Olympiad · 2014

Problems

Problem 1For a positive integer $m$ , denote by $S(m)$ and $P(m)$ the sum and product, respectively, of the digits of $m$ . Show that for each positive integer $n$ , there exist positive integers $a_1,a_2,\ldots,a_n$ satisfying $S(a_1)<S(a_2)<\cdots<S(a_n)$ and $S(a_i)=P(a_{i+1})$ for $i=1,2,\ldots,n$ , where $a_{n+1}=a_1$ .Solutions: 1
Problem 2Let $S=\{1,2,\ldots,2014\}$ . For each non-empty subset $T\subseteq S$ , one of its members is chosen as its representative. Find the number of ways to assign representatives to all non-empty subsets of $S$ so that if $D\subseteq S$ is a disjoint union of non-empty subsets $A,B,C\subseteq S$ , then the representative of $D$ is also the representative of at least one of $A,B,C$ .Solutions: 1
Problem 3Find all positive integers $n$ such that for any integer $k$ there exists an integer $a$ for which $a^3+a-k$ is divisible by $n$ .Solutions: 1
Problem 4Let $n$ and $b$ be positive integers. We say $n$ is $b$ -discerning if there exists a set consisting of $n$ different positive integers less than $b$ that has no two different subsets $U$ and $V$ such that the sum of all elements in $U$ equals the sum of all elements in $V$ . (a) Prove that $8$ is $100$ -discerning. (b) Prove that $9$ is not $100$ -discerning.Solutions: 1
Problem 5Circles $\omega$ and $\Omega$ meet at points $A$ and $B$ . Let $M$ be the midpoint of the arc $AB$ of circle $\omega$ ( $M$ lies inside $\Omega$ ). A chord $MP$ of circle $\omega$ intersects $\Omega$ at $Q$ ( $Q$ lies inside $\omega$ ). Let $\ell_P$ be the tangent line to $\omega$ at $P$ , and let $\ell_Q$ be the tangent line to $\Omega$ at $Q$ . Prove that the circumcircle of the triangle formed by the lines $\ell_P$ , $\ell_Q$ , and $AB$ is tangent to $\Omega$ .Solutions: 1