International Competitions 1994

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

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

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

1. $M$ is the midpoint of $BC$ and $O$ is the point on the line $AM$ such that $OB$ is perpendicular to $AB$;
2. $Q$ is an arbitrary point on the segment $BC$ different from $B$ and $C$;
3. $E$ 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 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 $f(k) = m$.

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

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

is an integer.

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

1. $f(x + f(y) + xf(y)) = y + f(x) + yf(x)$ for all $x$ and $y$ in $S$;
2. $\frac{f(x)}{x}$ is strictly increasing on each of the 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 $n \notin A$ each of which is a product of $k$ distinct elements of $S$ for some $k \geq 2$.