International Mathematical Olympiad · 1981

Problems

Problem 1Let $P$ be a point inside a given triangle $ABC$ , and let $D$ , $E$ , $F$ be the feet of the perpendiculars from $P$ to the lines $BC$ , $CA$ , $AB$ respectively. Find all positions of $P$ for which $\dfrac{BC}{PD} + \dfrac{CA}{PE} + \dfrac{AB}{PF}$ is least.Solutions: 1
Problem 2Let $n$ and $r$ be integers with $1 \le r \le n$ , and consider all $\binom{n}{r}$ subsets of $r$ elements of the set $\{1,2,\dots,n\}$ . Each such subset has a smallest element. Let $F(n,r)$ denote the arithmetic mean of these smallest elements. Prove that $F(n,r) = \frac{n+1}{r+1}.$ Solutions: 2
Problem 3Determine the maximum value of $m^2+n^2$ , where $m$ and $n$ are integers satisfying $m, n \in \{1, 2, \ldots, 1981\}$ and $(n^2-mn-m^2)^2 = 1$ .Solutions: 1
Problem 4(a) For which integers $n>2$ does there exist a set of $n$ consecutive positive integers such that the largest number in the set divides the least common multiple of the remaining $n-1$ numbers? (b) For which integers $n>2$ is there exactly one such set?Solutions: 2
Problem 5Three congruent circles have a common point $O$ and lie inside a given triangle. Each circle is tangent to two of the triangle's sides. Prove that the incenter and the circumcenter of the triangle, together with the point $O$ , are collinear.Solutions: 1
Problem 6The function $f(x,y)$ satisfies $f(0,y) = y+1,$ $f(x+1,0) = f(x,1),$ $f(x+1,y+1) = f\big(x, f(x+1,y)\big)$ for all non-negative integers $x,y$ . Determine $f(4,1981)$ .Solutions: 1