1972 IMO Problems

Problems of the 14th IMO 1972 in Poland.

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.

Solution

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.

Solution

Problem 3

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

Solution

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.

Solution

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 $y$ .