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

Solution

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 $QE = QF$ .

Solution

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 $f(k) = m$ .

Solution

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.

Solution

Problem 5

Problem 6

1994 IMO (Problems) • Resources
Preceded by 1993 IMO	1 • 2 • 3 • 4 • 5 • 6	Followed by 1995 IMO
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1994 IMO Problems

Contents

Day I

Problem 1

Problem 2

Problem 3

Day II

Problem 4

Problem 5

Problem 6