OMCC 2019

Let $N=\overline{abcd}$ be a positive integer with four digits. We name plátano power to the smallest positive integer $p(N)=\overline{\alpha_1\alpha_2\ldots\alpha_k}$ that can be inserted between the numbers $\overline{ab}$ and $\overline{cd}$ in such a way the new number $\overline{ab\alpha_1\alpha_2\ldots\alpha_kcd}$ is divisible by $N$ . Determine the value of $p(2025)$ .

We have a regular polygon $P$ with 2019 vertices, and in each vertex there is a coin. Two players Azul and Rojo take turns alternately, beginning with Azul, in the following way: first, Azul chooses a triangle with vertices in $P$ and colors its interior with blue, then Rojo selects a triangle with vertices in $P$ and colors its interior with red, so that the triangles formed in each move don't intersect internally the previous colored triangles. They continue playing until it's not possible to choose another triangle to be colored. Then, a player wins the coin of a vertex if he colored the greater quantity of triangles incident to that vertex (if the quantities of triangles colored with blue or red incident to the vertex are the same, then no one wins that coin and the coin is deleted). The player with the greater quantity of coins wins the game. Find a winning strategy for one of the players.

Note. Two triangles can share vertices or sides.

Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Let $D$ be the foot of the altitude from $A$ to the side $BC$ , $M$ and $N$ the midpoints of $AB$ and $AC$ , and $Q$ the point on $\Gamma$ diametrically opposite to $A$ . Let $E$ be the midpoint of $DQ$ . Show that the lines perpendicular to $EM$ and $EN$ passing through $M$ and $N$ , respectively, meet on $AD$ .

Let $ABC$ be a triangle, $\Gamma$ its circumcircle and $l$ the tangent to $\Gamma$ through $A$ . The altitudes from $B$ and $C$ are extended and meet $l$ at $D$ and $E$ , respectively. The lines $DC$ and $EB$ meet $\Gamma$ again at $P$ and $Q$ , respectively. Show that the triangle $APQ$ is isosceles.

Let $a,\ b$ and $c$ be positive real numbers so that $a+b+c=1$ . Show that $$a\sqrt{a^2+6bc}+b\sqrt{b^2+6ac}+c\sqrt{c^2+6ab}\leq\frac{3\sqrt{2}}{4}$$

A triminó is a rectangular tile of $1\times 3$ . Is it possible to cover a $8\times8$ chessboard using $21$ triminós, in such a way there remains exactly one $1\times 1$ square without covering? In case the answer is in the affirmative, determine all the possible locations of such a unit square in the chessboard.