Problem 1. Let \(a, b, c\) be positive real numbers such that
\[
a^2 + b^2 + c^2 \geq 3.
\]
Prove that
\[
\frac{a^4}{a^2 + 2b + 2c} + \frac{b^4}{b^2 + 2c + 2a} + \frac{c^4}{c^2 + 2a + 2b} \geq \frac{3}{5}.
\]

 

Problem 2. Determine all pairs \((a, b)\) of positive integers such that
\[(a + 1)\text{ divides } (b + 2) \quad \text{and} \quad b \text{ divides } 2a^2.\]

Problem 3. Let \(n \geq 3\) be an integer. There are \(n\) colored lamps arranged in a circle. Pressing the button of a lamp once changes its color as follows:
\[
\text{green} \rightarrow \text{red}, \quad \text{red} \rightarrow \text{blue}, \quad \text{blue} \rightarrow \text{green}.
\]
Initially, all lamps are colored red. Aladdin makes moves on these lamps. Each move consists of the following three steps:
• he chooses a lamp L, without pressing its button;
• he presses the button of the clockwise neighbor of L once;
• he presses the button of the counterclockwise neighbor of L twice.
For each \(n\), determine the maximum possible number of lamps that are colored green simultaneously, after finitely many moves.

Problem 4. Let \(ABC\) be a triangle with \(AB \neq AC\) and let \(I\) be its incenter. Let \(P\) and \(Q\) be points inside triangle \(ABC\) such that \(PB = PC > QC = QB\). Lines \(BP\) and \(CQ\) meet at \(X\). Suppose that line \(AI\) is tangent to the circumcircle of triangle \(IPQ\). Prove that the circumcircles of triangles \(ABQ, ACP\) and \(PQX\) have a common point.