1994 IMO Problems

Problems of the 1994 IMO.

Day I

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 \[\frac{a_1+a_2+...+a_m}{m} \ge \frac{n+1}{2}\].


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 $\{k + 1, k + 2,\dots, 2k\}$ whose base 2 representation has precisely three $1$s.

  • (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 $f(k) = m$.


Day II

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 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 $0<x$.


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 $k \ge 2$.


1994 IMO (Problems) • Resources
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1993 IMO
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1995 IMO
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