OLYMPIC-REVENGE 2021

Let $a$ , $b$ , $c$ , $k$ be positive reals such that $ab+bc+ca \leq 1$ and $0 < k \leq \frac{9}{2}$ . Prove that:

$$\sqrt[3]{ \frac{k}{a} + (9-3k)b} + \sqrt[3]{\frac{k}{b} + (9-3k)c} + \sqrt[3]{\frac{k}{c} + (9-3k)a } \leq \frac{1}{abc}.$$

Proposed by Zhang Yanzong and Song Qing

Evan is a $n$ -dimensional being that lives in a house formed by the points of $\mathbb{Z}_{\geq 0}^n$ . His room is the set of points in which coordinates are all less than or equal to $2021$ . Evan's room has been infested with bees, so he decides to flush them out through $\textit{captures}$ . A $\textit{capture}$ can be performed by eliminating a bee from point $(a_1, a_2, \ldots, a_n)$ and replacing it with $n$ bees, one in each of the points: $$(a_1 + 1, a_2 , \ldots, a_n), (a_1, a_2 + 1, \ldots, a_n), \ldots, (a_1, a_2, \ldots, a_n + 1)$$ However, two bees can never occupy the same point in the house. Determine, for every $n$ , the greatest value $A (n)$ of bees such that, for some initial arrangement of these bees in Evan's room, he is able to accomplish his goal with a finite amount of $\textit{captures}$ .

Let $I, C, \omega$ and $\Omega$ be the incenter, circumcenter, incircle and circumcircle, respectively, of the scalene triangle $XYZ$ with $XZ > YZ > XY$ . The incircle $\omega$ is tangent to the sides $YZ, XZ$ and $XY$ at the points $D, E$ and $F$ . Let $S$ be the point on $\Omega$ such that $XS, CI$ and $YZ$ are concurrent. Let $(XEF) \cap \Omega = R$ , $(RSD) \cap (XEF) = U$ , $SU \cap CI = N$ , $EF \cap YZ = A$ , $EF \cap CI = T$ and $XU \cap YZ = O$ .

Prove that $NARUTO$ is cyclic.

On a chessboard, Po controls a white queen and plays, in alternate turns, against an invisible black king (there are only those two pieces on the board). The king cannot move to a square where he would be in check, neither capture the queen. Every time the king makes a move, Po receives a message from beyond that tells which direction the king has moved (up, right, up-right, etc). His goal is to make the king unable to make a movement.

Can Po reach his goal with at most $150$ moves, regardless the starting position of the pieces?