IMO 1994

Problem 1. Let $m$ and $n$ be positive integers. Let $a_1,a_2,\dots ,a_m$ be distinct elements of $\{1,2,\dots ,n\}$ such that whenever $a_i+a_j\le n$ for some $i,j$, $1\le i\le j\le m$, there exists $k$, $1\le k\le m$, with $a_i+a_j=a_k$. Prove that

$\frac{a_1+a_2+\cdots +a_m}{m}\ge \frac{n+1}{2}.$

Problem 2. $ABC$ is an isosceles triangle with $AB=AC$. Suppose that

• $M$ is the midpoint of $BC$ and $O$ is the point on the line $AM$ such that $OB$ is perpendicular to $AB$;

• $Q$ is an arbitrary point on the segment $BC$ different from $B$ and $C$;

• $E$ lies on the line $AB$ and $F$ lies on the line $AC$ such that $E$, $Q$, $F$ are distinct and collinear.

Prove that $OQ$ is perpendicular to $EF$ if and only if $QE=QF$.

Problem 3. For any positive integer $k$, let $f(k)$ be the number of elements in the set $\{k+1,k+2,\dots ,2k\}$ whose base 2 representation has precisely three 1s.

• (a) Prove that, for each positive integer $m$, there exists at least one positive integer $k$ such that $f(k)=m$.

• (b) Determine all positive integers $m$ for which there exists exactly one $k$ with $f(k)=m$.

Problem 4. Determine all ordered pairs $(m,n)$ of positive integers such that

$\frac{n^3+1}{mn-1}$

is an integer.

Problem 5. Let $S$ be the set of real numbers strictly greater than $-1$. Find all functions $f:S\to S$ satisfying the two conditions:

• $f(x+f(y)+xf(y))=y+f(x)+yf(x)$ for all $x$ and $y$ in $S$;

• $\frac{f(x)}{x}$ is strictly increasing on each of the intervals $-1<x<0$ and $0<x$.

Problem 6. Show 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 $k\ge 2$.