In the triangle $ABC$, $\angle B>90^\circ$, the incircle touches the sides $BC$ and $CA$ at $D$ and $E$, respectively. The lines $ED$ and $AB$ intersect at $P$. The incircle of the triangle $AEP$ touches the sides $PE$ and $AP$ at $D_1$ and $E_1$, respectively. The lines $E_1D_1$ and $AE$ intersect at $P_1$. Suppose $P,C,E,B$ are concyclic. Prove that $BE$ is parallel to $PP_1$.
Consider the polynomial $P(x)=x(x-1)\cdots(x-2024)$. Show that $P^{(n)}(x)=2025$ has at least $2024^n$ distinct real roots, where $n$ is any positive integer and $P^{(n)}(x)=P(P(\cdots P(x)\cdots))$ with $P$ iterated $n$ times.
For any 4-digit positive integer $n$, define $f(n)=(a+b)^2$, where $a,b$ are the numbers formed by the first two and last two digits of $n$, respectively (leading zeroes are allowed). Find all 4-digit positive integers $n$ such that $f(n)=n$.
Find the smallest positive integer $n$, or show no such $n$ exists, such that one can find infinitely many distinct $n$-tuplets of positive rationals $(a_1,a_2,\ldots,a_n)$ satisfying
$$\sum_{i=1}^n a_i\in\mathbb Z,\qquad\sum_{i=1}^n\frac1{a_i}\in\mathbb Z.$$
Alice starts with some coins in a box. On the $n$th minute (for $n=1,2,3,\ldots$), she can choose to put an additional coin into the box or do nothing, and then she writes the number $2c^2-n^2$ on the board, where $c$ is the number of coins in the box. Show that for any positive integer $m$, there will eventually be two numbers on the board that sum to a multiple of $m$.