International Mathematical Olympiad · 1996

Problems

Problem 1We are given a positive integer $r$ and a rectangular board divided into $20\times12$ unit squares. A move from one square to another is permitted only when the distance between their centers is $\sqrt r$ . Find a sequence of moves leading between two adjacent corners of the board that lie on the long side. (a) Show that this is impossible if $r$ is divisible by $2$ or $3$ . (b) Prove that it is possible for $r=73$ . (c) Can it be done for $r=97$ ?Solutions: 1
Problem 2Let $P$ be a point inside triangle $ABC$ such that $\angle APB-\angle ACB=\angle APC-\angle ABC$ . Let $D$ and $E$ be the incenters of triangles $APB$ and $APC$ , respectively. Show that $AP$ , $BD$ , and $CE$ meet at a point.Solutions: 1
Problem 3Let $S$ be the set of non-negative integers. Find all functions $f:S\to S$ such that $f(m+f(n))=f(f(m))+f(n)$ for all $m,n\in S$ .Solutions: 1
Problem 4The positive integers $a$ and $b$ are such that $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value of the smaller of these two squares?Solutions: 1
Problem 5Let $ABCDEF$ be a convex hexagon such that $AB\parallel DE$ , $BC\parallel EF$ , and $CD\parallel FA$ . Let $R_A,R_C,R_E$ be the circumradii of triangles $FAB$ , $BCD$ , $DEF$ , respectively, and let $p$ be the perimeter of the hexagon. Prove that $R_A+R_C+R_E\ge p/2$ .Solutions: 1
Problem 6Let $p,q,n$ be positive integers with $p+q<n$ . Let $x_0,x_1,\ldots,x_n$ be integers with $x_0=x_n=0$ , and for each $1\le i\le n$ let $x_i-x_{i-1}=p$ or $-q$ . Show that there exist $i<j$ , with $(i,j)\ne(0,n)$ , such that $x_i=x_j$ .Solutions: 1