Asian Pacific Mathematics Olympiad · 2017

Problems

Problem 1We call a $5$ -tuple of integers arrangeable if its elements can be labeled $a,b,c,d,e$ in some order so that $a-b+c-d+e=29$ . Determine all $2017$ -tuples of integers $n_1,n_2,\ldots,n_{2017}$ such that if we place them in a circle in clockwise order, then any $5$ -tuple of numbers in consecutive positions on the circle is arrangeable.Solutions: 1
Problem 2Let $ABC$ be a triangle with $AB<AC$ . Let $D$ be the intersection point of the internal bisector of $\angle BAC$ and the circumcircle of $ABC$ . Let $Z$ be the intersection point of the perpendicular bisector of $AC$ with the external bisector of $\angle BAC$ . Prove that the midpoint of segment $AB$ lies on the circumcircle of triangle $ADZ$ .Solutions: 1
Problem 3Let $A(n)$ denote the number of sequences $a_1\ge a_2\ge\cdots\ge a_k$ of positive integers for which $a_1+\cdots+a_k=n$ and each $a_i+1$ is a power of two ( $i=1,2,\ldots,k$ ). Let $B(n)$ denote the number of sequences $b_1\ge b_2\ge\cdots\ge b_m$ of positive integers for which $b_1+\cdots+b_m=n$ and each inequality $b_j\ge2b_{j+1}$ holds ( $j=1,2,\ldots,m-1$ ). Prove that $A(n)=B(n)$ for every positive integer $n$ .Solutions: 1
Problem 4Call a rational number $r$ powerful if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p,q$ and some integer $k>1$ . Let $a,b,c$ be positive rational numbers such that $abc=1$ . Suppose there exist positive integers $x,y,z$ such that $a^x+b^y+c^z$ is an integer. Prove that $a,b,c$ are all powerful.Solutions: 1
Problem 5Let $n$ be a positive integer. A pair of $n$ -tuples $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$ with integer entries is called an exquisite pair if $|a_1b_1+\cdots+a_nb_n|\le1$ . Determine the maximum number of distinct $n$ -tuples with integer entries such that any two of them form an exquisite pair.Solutions: 1