Question bank IMO 1972

A set of $10$ positive integers is given such that the decimal expansion of each of them has two digits. Prove that there are two disjoint subsets of the set with equal sums of their elements.

Prove that for each $n\ge4$ every cyclic quadrilateral can be decomposed into $n$ cyclic quadrilaterals.

Let $m$ and $n$ be nonnegative integers. Prove that $\dfrac{(2m)!(2n)!}{m!n!(m+n)!}$ is an integer $(0!=1)$.

Find all solutions in positive real numbers $x_i$ $(i=1,2,3,4,5)$ of the following system of inequalities:
$$\begin{aligned}(x_1^2-x_3x_5)(x_2^2-x_3x_5)&\le0&&\text{(i)}\\ (x_2^2-x_4x_1)(x_3^2-x_4x_1)&\le0&&\text{(ii)}\\ (x_3^2-x_5x_2)(x_4^2-x_5x_2)&\le0&&\text{(iii)}\\ (x_4^2-x_1x_3)(x_5^2-x_1x_3)&\le0&&\text{(iv)}\\ (x_5^2-x_2x_4)(x_1^2-x_2x_4)&\le0&&\text{(v)}.\end{aligned}$$

Let $f$ and $\varphi$ be real functions defined in the interval $(-\infty,\infty)$ satisfying the functional equation
$$f(x+y)+f(x-y)=2\varphi(y)f(x),$$
for arbitrary real $x,y$ (give examples of such functions). Prove that if $f(x)$ is not identically $0$ and $|f(x)|\le1$ for all $x$, then $|\varphi(x)|\le1$ for all $x$.

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