NL-BXMO-TST 2011

All positive integers are coloured either red or green, such that the following conditions are satisfied:

• There are equally many red as green integers.
• The sum of three (not necessarily distinct) red integers is red.
• The sum of three (not necessarily distinct) green integers is green. Find all colourings that satisfy these conditions.

In an acute triangle $ABC$ the angle $\angle C$ is greater than $\angle A$ . Let $E$ be such that $AE$ is a diameter of the circumscribed circle $\Gamma$ of \vartriangle ABC. Let $K$ be the intersection of $AC$ and the tangent line at $B$ to $\Gamma$ . Let $L$ be the orthogonal projection of $K$ on $AE$ and let $D$ be the intersection of $KL$ and $AB$ . Prove that $CE$ is the bisector of $\angle BCD$ .

Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$ .

Let $n \ge 2$ be an integer. Let $a$ be the greatest positive integer such that $2^a | 5^n - 3^n$ . Let $b$ be the greatest positive integer such that $2^b \le n$ . Prove that $a \le b + 3$ .

A trapezoid $ABCD$ is given with $BC // AD$ . Assume that the bisectors of the angles $BAD$ and $CDA$ intersect on the perpendicular bisector of the line segment $BC$ . Prove that $|AB|= |CD|$ or $|AB| +|CD| =|AD|$ .