DE-MO 2008

The triangle $\triangle SFA$ has a right angle at $F$ . The points $P$ and $Q$ lie on the line $SF$ such that the point $P$ lies between $S$ and $F$ and the point $F$ is the midpoint of the segment $[PQ]$ . The circle $k_1$ is th incircle of the triangle $\triangle SPA$ . The circle $k_2$ lies outside the triangle $\triangle SQA$ and touches the segment $[QA]$ and the lines $SQ$ and $SA$ . Prove that the sum of the radii of the circles $k_1$ and $k_2$ equals the length of $[FA]$ .

Find all functions $f$ defined on non-negative real numbers having the following properties: (i) For all non-negative $x$ it is $f(x) \geq 0$ . (ii) It is $f\left(1\right)=\frac 12$ . (iii) For all non-negative numbers $x,y$ it is $f\left( y \cdot f(x) \right) \cdot f(x) = f(x+y)$ .

Find the smallest constant $C$ such that for all real $x,y$ $$1+(x+y)^2 \leq C \cdot (1+x^2) \cdot (1+y^2)$$ holds.

Inside a square of sidelength $1$ there are finitely many disks that are allowed to overlap. The sum of all circumferences is $10$ . Show that there is a line intersecting or touching at least $4$ disks.

Find all real numbers $x$ such that $4x^5 - 7$ and $4x^{13} - 7$ are both perfect squares.