International Competitions 1972

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.

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.

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

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.

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 $y$.

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