International Mathematical Olympiad  – IMO 1972  Problems

Problem 1

Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum.

Problem 2

Prove that if $n \geq 4$, every quadrilateral that can be inscribed in a circle can be dissected into $n$ quadrilaterals each of which is inscribable in a circle.

Problem 3

Let $m$ and $n$ be arbitrary non-negative integers. Prove that\[\frac{(2m)!(2n)!}{m!n!(m+n)!}\]is an integer. ($0! = 1$.)

Problem 4

Find all solutions $(x_1, x_2, x_3, x_4, x_5)$ of the system of inequalities\[(x_1^2 – x_3x_5)(x_2^2 – x_3x_5) \leq 0 \\ ,(x_2^2 – x_4x_1)(x_3^2 – x_4x_1) \leq 0 \\ ,(x_3^2 – x_5x_2)(x_4^2 – x_5x_2) \leq 0 \\ ,(x_4^2 – x_1x_3)(x_5^2 – x_1x_3) \leq 0 \\ ,(x_5^2 – x_2x_4)(x_1^2 – x_2x_4) \leq 0\]where $x_1, x_2, x_3, x_4, x_5$ are positive real numbers.

Problem 5

Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y$, and satisfying the equation\[f(x + y) + f(x – y) = 2f(x)g(y)\]for all $x, y$. Prove that if $f(x)$ is not identically zero, and if $|f(x)| \leq 1$ for all $x$, then $|g(y)| \leq 1$ for all $y$.

Problem 6

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.