International Mathematical Olympiad · 1989

Problems

Problem 1Prove that the set $\{1, 2, \ldots, 1989\}$ can be expressed as the union of $117$ pairwise disjoint subsets $A_1, A_2, \ldots, A_{117}$ , each containing $17$ elements, such that the sum of the elements is the same in every $A_i$ .Solutions: 1
Problem 2Let $ABC$ be an acute-angled triangle. The internal bisector of angle $A$ meets the circumcircle of $ABC$ again at $A_1$ ; points $B_1$ and $C_1$ are defined similarly. Let $A_0$ be the point where the line $AA_1$ meets the external bisectors of angles $B$ and $C$ ; points $B_0$ and $C_0$ are defined similarly. Prove that the area of triangle $A_0B_0C_0$ is twice the area of the hexagon $AC_1BA_1CB_1$ , and that this area is at least four times the area of triangle $ABC$ .Solutions: 1
Problem 3Let $n$ and $k$ be positive integers, and let $S$ be a set of $n$ points in the plane such that no three points of $S$ are collinear. Suppose that for every point $P\in S$ there are at least $k$ points of $S$ equidistant from $P$ . Prove that $k<\frac12+\sqrt{2n}$ .Solutions: 1
Problem 4Let $ABCD$ be a convex quadrilateral with $AB=AD+BC$ . Suppose that an interior point $P$ has distance $h$ from the line $CD$ and satisfies $AP=h+AD$ and $BP=h+BC$ . Prove that $\frac1{\sqrt h}\ge\frac1{\sqrt{AD}}+\frac1{\sqrt{BC}}$ .Solutions: 1
Problem 5Prove that for every positive integer $n$ there exist $n$ consecutive positive integers none of which is a prime or a power of a prime.Solutions: 1
Problem 6A permutation $(x_1,x_2,\ldots,x_{2n})$ of $\{1,2,\ldots,2n\}$ has property $P$ if $|x_i-x_{i+1}|=n$ for at least one $i\in\{1,\ldots,2n-1\}$ . Prove that for every positive integer $n$ there are more permutations with property $P$ than without it.Solutions: 1