International Mathematical Olympiad · 1972

Problems

Problem 1Prove 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.Solutions: 1
Problem 2Prove that if $n\ge4$ , every quadrilateral that can be inscribed in a circle can be dissected into $n$ quadrilaterals each of which is inscribable in a circle.Solutions: 1
Problem 3Let $m$ and $n$ be arbitrary non-negative integers. Prove that $\dfrac{(2m)!(2n)!}{m!n!(m+n)!}$ is an integer. ( $0!=1$ ).Solutions: 1
Problem 4Find all positive real solutions $(x_1,x_2,x_3,x_4,x_5)$ of $(x_1^2-x_3x_5)(x_2^2-x_3x_5)\le0$ , $(x_2^2-x_4x_1)(x_3^2-x_4x_1)\le0$ , $(x_3^2-x_5x_2)(x_4^2-x_5x_2)\le0$ , $(x_4^2-x_1x_3)(x_5^2-x_1x_3)\le0$ , and $(x_5^2-x_2x_4)(x_1^2-x_2x_4)\le0$ .Solutions: 1
Problem 5Let $f$ and $g$ be real-valued functions defined for all real $x,y$ , satisfying $f(x+y)+f(x-y)=2f(x)g(y)$ for all $x,y$ . Prove that if $f$ is not identically zero and $|f(x)|\le1$ for all $x$ , then $|g(y)|\le1$ for all $y$ .Solutions: 1
Problem 6Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.Solutions: 1