IMO 1972

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 $n\ge 4$, 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

$\frac{(2m)!(2n)!}{m\prime n!(m+n)!}$

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

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 $\left|f(x)\right|$ $\le 1$ for all $x$, then $\left|g(y)\right|\le 1$ for all $y$.

Problem 6. Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.