JBMO 2026

Let $a, b, c$ be positive real numbers such that $$a^2 + b^2 + c^2 \geq 3.$$ Prove that $$\frac{a^4}{a^2 + 2b + 2c} + \frac{b^4}{b^2 + 2c + 2a} + \frac{c^4}{c^2 + 2a + 2b} \geq \frac{3}{5}.$$

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

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:

• green $\rightarrow$ red,
• red $\rightarrow$ blue,
• blue $\rightarrow$ 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.

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.