We say a positive integer $n$ is $k$-special if none of its figures is zero and, for any permutation the figures of $n$, the resulting number is multiple of $k$. Let $m\ge 2$ be a positive integer. Find the number of $4$-special numbers with $m$ figures. Find the number of $3$-special numbers with $m$ figures.

Marc has an $n\times n$ board, where $n\ge 3$ is an integer, and an unlimited supply of green and red apples. Marc wants to place some apples on the board, so that the following conditions hold. Every cell of the board has exactly one apple, be it red or green. All rows and columns of the board have at least one red apple. No two rows or columns have the same apple color sequence. Note that rows are read from left to right, and columns are read from top to bottom. Also note that we do not allow a row and a column to have the same color sequence. Find, in terms of $n$, the minimal number of red apples that Marc needs in order to fill the board in this way.

Let $ABC$ be a triangle with circumcircle $\Omega$, and let $P$ be a point on the arc $BC$ of $\Omega$ not containing $A$. Let $\omega_B$ and $\omega_C$ be circles respectively passing through $B$ and $C$ and such that both of them are tangent to line $AP$ at point $P$. Let $R$, $R_B$, $R_C$ be the radii of $\Omega$, $\omega_B$, and $\omega_C$, respectively. Prove that if $h$ is the distance from $A$ to line $BC$, then \[ \frac{R_B+R_C}{R} \le \frac{BC}{h}. \]

Let $ABC$ be a triangle, and let $D$ and $E$ be two points on side $BC$ such that $BD = EC$. Let $X$ be a point on segment $AD$ such that $CX$ is parallel to the bisector of $\angle ADB$. Similarly, let $Y$ be a point on segment $AD$ such that $BY$ is parallel to the bisector of $\angle ADC$. Prove that $DE = XY$.

Prove that there do not exist positive integers $a,b,c$ such that the polynomial \[ P(x) = x^3 - 2^ax^2 + 3^bx - 6^c \]has three integer roots.

Given a positive integer $n\ge 3$, Arándano and Banana play a game. Initially, numbers $1,2,3,\dots,n$ are written on the blackboard. Alternatingly and starting with Arándano, the players erase numbers from the board one at a time, until exactly three numbers remain on the board. Banana wins the game if the last three numbers on the board are the sides of a nondegenerate triangle, and Arándano wins otherwise. Determine, in terms of $n$, who has a winning strategy.