OME 1984

At a position $O$ of an airport in a plateau there is a gun which can rotate arbitrarily. Two tanks moving along two given segments $AB$ and $CD$ attack the airport. Determine, by a ruler and a compass, the reach of the gun, knowing that the total length of the parts of the trajectories of the two tanks reachable by the gun is equal to a given length $\ell$ .

Find the number of five-digit numbers whose square ends in the same five digits in the same order.

If $p$ and $q$ are positive numbers with $p+q = 1$ , knowing that any real numbers $x,y$ satisfy $(x-y)^2 \ge 0$ , show that $\frac{x+y}{2} \ge \sqrt{xy}$ , $\frac{x^2+y^2}{2} \ge \big(\frac{x+y}{2}\big)^2$ ,

$\big(p+\frac{1}{p}\big)^2+\big(q+\frac{1}{q}\big)^2 \ge \frac{25}{2}$

Evaluate $\lim_{n\to \infty} cos\frac{x}{2}cos\frac{x}{2^2} cos\frac{x}{2^3}...cos\frac{x}{2^n}$

Let $A$ and $A'$ be fixed points on two equal circles in the plane and let $AB$ and $A' B'$ be arcs of these circles of the same length $x$ . Find the locus of the midpoint of segment $BB'$ when $x$ varies: (a) if the arcs have the same direction, (b) if the arcs have opposite directions.

Consider the circle $\gamma$ with center at point $(0,3)$ and radius $3$ , and a line $r$ parallel to the axis $Ox$ at a distance $3$ from the origin. A variable line through the origin meets $\gamma$ at point $M$ and $r$ at point $P$ . Find the locus of the intersection point of the lines through $M$ and $P$ parallel to $Ox$ and $Oy$ respectively.