International Mathematical Olympiad  – IMO 1994  Problems

Problem 1

Let $ m$ and $ n$ be two positive integers. Let $ a_1$, $ a_2$, $ \ldots$, $ a_m$ be $ m$ different numbers from the set $ \{1, 2,\ldots, n\}$ such that for any two indices $ i$ and $ j$ with $ 1\leq i \leq j \leq m$ and $ a_i + a_j \leq n$, there exists an index $ k$ such that $ a_i + a_j = a_k$. Show that
\[ \frac {a_1 + a_2 + … + a_m}{m} \geq \frac {n + 1}{2}.
\]

Problem 2

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$.

Problem 3

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$.

Problem 4

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

Problem 5

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$.

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