JBMO 2024

Let $a, b, c$ be positive real numbers such that

$$a^2 + b^2 + c^2 = \frac{1}{4}.$$

Prove that

$$\frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}} + \frac{1}{\sqrt{a^2 + b^2}} \leq \frac{\sqrt{2}}{(a + b)(b + c)(c + a)}.$$

Let $ABC$ be a triangle such that $AB < AC$. Let the excircle opposite to $A$ be tangent to the lines $AB, AC$ and $BC$ at points $D, E$ and $F$, respectively, and let $J$ be its centre. Let $P$ be a point on the side $BC$. The circumcircles of the triangles $BDP$ and $CEP$ intersect for the second time at $Q$. Let $R$ be the foot of the perpendicular from $A$ to the line $FJ$. Prove that the points $P, Q$ and $R$ are collinear.

(The excircle of a triangle $ABC$ opposite to $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)

Find all triples of positive integers $(x, y, z)$ that satisfy the equation

$$2020^x + 2^y = 2024^z.$$

Three friends Archie, Billie and Charlie play a game. At the beginning of the game, each of them has a pile of 2024 pebbles. Archie makes the first move, Billie makes the second, Charlie makes the third and they continue to make moves in the same order. In each move, the player making the move must choose a positive integer $n$ greater than any previously chosen number by any player, take $2n$ pebbles from his pile and distribute them equally to the other two players. If a player cannot make a move, the game ends and that player loses the game.

Determine all the players who have a strategy such that, regardless of how the other two players play, they will not lose the game.