# [Solutions] Serbian Mathematical Olympiad 2008

1. Find all nonegative integers $x,y,z$ such that $12^x+y^4=2008^z$
2. Triangle $\triangle ABC$ is given. Points $D$ i $E$ are on line $AB$ such that $D - A - B - E, AD = AC$ and $BE = BC$. Bisector of internal angles at $A$ and $B$ intersect $BC,AC$ at $P$ and $Q$, and circumcircle of $ABC$ at $M$ and $N$. Line which connects $A$ with center of circumcircle of $BME$ and line which connects $B$ and center of circumcircle of $AND$ intersect at $X$. Prove that $CX \perp PQ$.
3. Let $a$, $b$, $c$ be positive real numbers such that $a + b + c = 1$. Prove inequality: $\frac{1}{bc + a + \frac{1}{a}} + \frac{1}{ac + b + \frac{1}{b}} + \frac{1}{ab + c + \frac{1}{c}} \leqslant \frac{27}{31}.$
4. Each point of a plane is painted in one of three colors. Show that there exists a triangle such that:
(i) all three vertices of the triangle are of the same color;
(ii) the radius of the circumcircle of the triangle is $2008$;
(iii) one angle of the triangle is either two or three times greater than one of the other two angles.
5. The sequence $(a_n)_{n\ge 1}$ is defined by $a_1 = 3$, $a_2 = 11$ and $a_n = 4a_{n-1}-a_{n-2}$, for $n \ge 3$. Prove that each term of this sequence is of the form $a^2 + 2b^2$ for some natural numbers $a$ and $b$.
6. In a convex pentagon $ABCDE$, let $\angle EAB = \angle ABC = 120^{\circ}$, $\angle ADB = 30^{\circ}$ and $\angle CDE = 60^{\circ}$. Let $AB = 1$. Prove that the area of the pentagon is less than $\sqrt {3}$.