OMM 1991

Evaluate the sum of all positive irreducible fractions less than $1$ and having the denominator $1991$ .

A company of $n$ soldiers is such that (i) $n$ is a palindrome number (read equally in both directions); (ii) if the soldiers arrange in rows of $3, 4$ or $5$ soldiers, then the last row contains $2, 3$ and $5$ soldiers, respectively. Find the smallest $n$ satisfying these conditions and prove that there are infinitely many such numbers $n$ .

Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them?

The diagonals $AC$ and $BD$ of a convex quarilateral $ABCD$ are orthogonal. Let $M,N,R,S$ be the midpoints of the sides $AB,BC,CD$ and $DA$ respectively, and let $W,X,Y,Z$ be the projections of the points $M,N,R$ and $S$ on the lines $CD,DA,AB$ and $BC$ , respectively. Prove that the points $M,N,R,S,W,X,Y$ and $Z$ lie on a circle.

The sum of squares of two consecutive integers can be a square, as in $3^2+4^2 =5^2$ . Prove that the sum of squares of $m$ consecutive integers cannot be a square for $m = 3$ or $6$ and find an example of $11$ consecutive integers the sum of whose squares is a square.

Given an $n$ -gon ( $n\ge 4$ ), consider a set $T$ of triangles formed by vertices of the polygon having the following property: Every two triangles in T have either two common vertices, or none. Prove that $T$ contains at most $n$ triangles.