OME 1996

The natural numbers $a$ and $b$ are such that $\frac{a+1}{b}+ \frac{b+1}{a}$ is an integer. Show that the greatest common divisor of a and b is not greater than $\sqrt{a+b}$ .

Let $G$ be the centroid of a triangle $ABC$ . Prove that if $AB+GC = AC+GB$ , then the triangle is isosceles

For each real value of $p$ , find all real solutions of the equation $\sqrt{x^2 - p}+2\sqrt{x^2-1} = x$ .

At Port Aventura there are $16$ secret agents, each of whom is watching one or more other agents. It is known that if agent $A$ is watching agent $B$ , then $B$ is not watching $A$ . Moreover, any $10$ agents can be ordered so that the first is watching the second, the second is watching the third, etc, the last is watching the first. Show that any $11$ agents can also be so ordered.

A regular pentagon is constructed externally on each side of a regular pentagon of side $1$ . The figure is then folded and the two edges of the external pentagons meeting at each vertex of the original pentagon are glued together. Find the volume of water that can be poured into the obtained container.