CAPS 2003

Given an integer $n \ge 2$ , solve in real numbers the system of equations \begin{align*} \max{1, x_1} &= x_2 \max{2, x_2} &= 2x_3 &\cdots \max{n, x_n} &= nx_1. \end{align*}

In an acute-angled triangle $ABC$ the angle at $B$ is greater than $45^\circ$ . Points $D,E, F$ are the feet of the altitudes from $A,B,C$ respectively, and $K$ is the point on segment $AF$ such that $\angle DKF = \angle KEF$ . (a) Show that such a point $K$ always exists. (b) Prove that $KD^2 = FD^2 + AF \cdot BF$ .

Numbers $p,q,r$ lies in the interval $(\frac{2}{5},\frac{5}{2})$ nad satisfy $pqr=1$ . Prove that there exist two triangles of the same area, one with the sides $a,b,c$ and the other with the sides $pa,qb,rc$ .

Point $P$ lies on the median from vertex $C$ of a triangle $ABC$ . Line $AP$ meets $BC$ at $X$ , and line $BP$ meets $AC$ at $Y$ . Prove that if quadrilateral $ABXY$ is cyclic, then triangle $ABC$ is isosceles.

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the condition $$f(f(x) + y) = 2x + f(f(y) - x) \text{ for all } x, y \in\mathbb{R}.$$