International Mathematical Olympiad · 1994

Problems

Problem 1Let $a_1,\ldots,a_m$ be distinct elements of $\{1,\ldots,n\}$ such that whenever $a_i+a_j\le n$ (with $i\le j$ ), the sum is also one of the $a_k$ . Prove $\frac{a_1+\cdots+a_m}{m}\ge\frac{n+1}{2}$ .Solutions: 1
Problem 2In isosceles triangle $ABC$ with $AB=AC$ , let $M$ be the midpoint of $BC$ and let $O\in AM$ satisfy $OB\perp AB$ . For $Q\in BC$ , let $E\in AB$ and $F\in AC$ be distinct collinear points with $Q$ . Prove $OQ\perp EF$ if and only if $QE=QF$ .Solutions: 1
Problem 3For positive integer $k$ , let $f(k)$ be the number of integers in $\{k+1,\ldots,2k\}$ whose binary expansion has exactly three $1$ s. Prove every positive integer $m$ occurs as $f(k)$ , and determine all $m$ for which exactly one $k$ has $f(k)=m$ .Solutions: 1
Problem 4Find all ordered pairs of positive integers $(m,n)$ for which $\frac{n^3+1}{mn-1}$ is an integer.Solutions: 1
Problem 5Let $S$ be the set of all real numbers greater than $-1$ . Find all functions $f:S\to S$ such that $f(x+f(y)+xf(y))=y+f(x)+yf(x)$ for all $x,y\in S$ , and $\frac{f(x)}{x}$ is strictly increasing on each of the intervals $-1<x<0$ and $0<x$ .Solutions: 1
Problem 6Show that there exists a set $A$ of positive integers with the following property: for any infinite set $S$ of primes, there exist two positive integers $m\in A$ and $n\notin A$ , each of which is a product of $k$ distinct elements of $S$ for some $k\ge2$ .Solutions: 1