International Mathematical Olympiad · 1982

Problems

Problem 1The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0$ , $f(3)>0$ , $f(9999)=3333$ , and for all $m,n$ : $f(m+n)-f(m)-f(n)=0 \text{ or } 1.$ Determine $f(1982)$ .Solutions: 1
Problem 2A non-isosceles triangle $A_1A_2A_3$ has sides $a_1$ , $a_2$ , $a_3$ with the side $a_i$ lying opposite to the vertex $A_i$ . Let $M_i$ be the midpoint of the side $a_i$ , and let $T_i$ be the point where the inscribed circle of triangle $A_1A_2A_3$ touches the side $a_i$ . Denote by $S_i$ the reflection of the point $T_i$ in the interior angle bisector of the angle $A_i$ . Prove that the lines $M_1S_1$ , $M_2S_2$ and $M_3S_3$ are concurrent.Solutions: 1
Problem 3Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\cdots$ . (a) Prove that for every such sequence there is an $n\ge1$ such that $S_n=\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots+\frac{x_{n-1}^2}{x_n}\ge3.999$ . (b) Find such a sequence for which $S_n<4$ for all $n$ .Solutions: 1
Problem 4Prove that if $n$ is a positive integer such that $x^3-3xy^2+y^3=n$ has an integer solution $(x,y)$ , then it has at least three such solutions. Show that the equation has no integer solutions for $n=2891$ .Solutions: 1
Problem 5The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by interior points $M$ and $N$ respectively, so that $\frac{AM}{AC}=\frac{CN}{CE}=r$ . Determine $r$ if $B,M,N$ are collinear.Solutions: 1
Problem 6Let $S$ be a square with side length $100$ . Let $L$ be a simple closed polygonal path inside $S$ , composed of segments $A_0A_1,A_1A_2,\ldots,A_{n-1}A_n$ with $A_0=A_n$ . Suppose that every point $P$ on the boundary of $S$ is at distance at most $1/2$ from some point of $L$ . Prove that there are points $X,Y$ of $L$ whose distance is at most $1$ and for which the length of the part of $L$ between $X$ and $Y$ is at least $198$ .Solutions: 1