IMO 1972

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

Given $n>4$ , prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$ .

Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \end{eqnarray*}

$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$ . $f$ is not identically zero and $|f(x)|\le1$ for all $x$ . Prove that $|g(x)|\le1$ for all $x$ .

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