IMO 1994

Let $ABC$ be an isosceles triangle with $AB = AC$ . $M$ 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 $BC$ 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 $QE = QF$ .

For any positive integer $k$ , let $f_k$ be the number of elements in the set $\{ k + 1, k + 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s.

(a) Prove that for any 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 $f(k) = m$ .

Find all ordered pairs $(m,n)$ where $m$ and $n$ are positive integers such that $\frac {n^3 + 1}{mn - 1}$ is an integer.

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

(a) $f(x + f(y) + xf(y)) = y + f(x) + yf(x)$ for all $x, y$ in $S$ ;

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

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 $n$ not in $A$ , each of which is a product of $k$ distinct elements of $S$ for some $k \geq 2$ .