Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfy the following property for all real numbers $x$ and all polynomials $P$ with real coefficients: If $P(f(x)) = 0$, then $f(P(x)) = 0$.

For which positive integers $n$ does there exist a positive integer $m$ such that among the numbers $m + n, 2m + (n - 1), \dots, nm + 1$, there are no two that share a common factor greater than $1$?

Let $P$ and $Q$ be points in the interior of a triangle $ABC$ such that $\angle APC = \angle AQB = 90^{\circ}$, $\angle ACP = \angle PBC$, and $\angle ABQ = \angle QCB$. Suppose that lines $BP$ and $CQ$ meet at a point $R$. Show that $AR$ is perpendicular to $PQ$.

Let $ABC$ be an obtuse triangle with $AB = AC$, and let $\Gamma$ be the circle that is tangent to $AB$ at $B$ and to $AC$ at $C$. Let $D$ be the point on $\Gamma$ furthest from $A$ such that $AD$ is perpendicular to $BC$. Point $E$ is the intersection of $AB$ and $DC$, and point $F$ lies on line $AB$ such that $BC = BF$ and $B$ lies on segment $AF$. Finally, let $P$ be the intersection of lines $AC$ and $DB$. Show that $PE = PF$.

Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that: No two numbers are equal. If a number $m$ is placed on square $C$, then at least $k$ of the squares orthogonally adjacent to $C$ have a multiple of $m$ written on them. What is the greatest value of $k$ for which this is possible?

Let $a$ and $b$ be fixed positive integers. We say that a prime $p$ is fun if there exists a positive integer $n$ satisfying the following conditions: $p$ divides $a^{n!} + b$. $p$ divides $a^{(n + 1)!} + b$. $p < 2n^2 + 1$. Show that there are finitely many fun primes.