International Mathematical Olympiad  – IMO 1998  Problems

Problem 1

A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.

Problem 2

In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that\[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]

Apparently, I have the same proof as the one on Kalva..

We count the cases when two judges agree on the result of a certain contestant. Let this number be $N$. On the one hand, we have $N\le k\cdot \binom n2=k\cdot\frac{n(n-1)}2$. On the other hand, if we show that $m\cdot \left(\frac{n-1}2\right)^2\le N$, we’re done.

Consider the whole thing a $0,1$-matrix having the judges as rows and the contestants as columns, and having $a_{ij}=0$ is judge $j$ fails contestant $i$ and $a_{ij}=1$ otherwise. It suffices to show now that for each of the $m$ columns there are at least $\left(\frac{n-1}2\right)^2$ pairs of equal numbers.

Assume on a column there are $a$ $0$s and $b$ $1$s with $a+b=n=2k+1$. In this case the number of matching pairs on this column is $\frac{a(a-1)}2+\frac{b(b-1)}2=\frac{a^2+b^2}2-\frac n2$, which, knowing that $a+b$ is constant, is minimal when $|a-b|$ is minimal (i.e. when $a,b$ are as close to each other as possible), and this happens when $\{a,b\}=\{k,k+1\}$. If we compute the number of matching pairs on this column in this particular case, we get precisely $k^2=\left(\frac{n-1}2\right)^2$, so this is the minimal value. That is, on every column the number of matching pairs is $\geq \left(\frac{n-1}2\right)^2$. Thus, $N\geq m\cdot \left(\frac{n-1}2\right)^2$, Q.E.D.

Problem 3

For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.

Problem 4

Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.

Problem 5

Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.

Problem 6

Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$,

\[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]