OME 1997

Compute the sum of the squares of the first $100$ terms of an arithmetic progression, given that their sum is $-1$ and that the sum of those among them having an even index is $1$ .

A square of side $5$ is divided into $25$ unit squares. Let $A$ be the set of the $16$ interior points of the initial square which are vertices of the unit squares. What is the largest number of points of $A$ no three of which form an isosceles right triangle?

For each parabola $y = x^2+ px+q$ intersecting the coordinate axes in three distinct points, consider the circle passing through these points. Prove that all these circles pass through a single point, and find this point.

Let $p$ be a prime number. Find all integers $k$ for which $\sqrt{k^2 -pk}$ is a positive integer.

Prove that in every convex quadrilateral of area $1$ , the sum of the lengths of the sides and diagonals is not smaller than $2(2+\sqrt2)$ .

The exact quantity of gas needed for a car to complete a single loop around a track is distributed among $n$ containers placed along the track. Prove that there exists a position starting at which the car, beginning with an empty tank of gas, can complete a loop around the track without running out of gas. The tank of gas is assumed to be large enough.