Let $D$ be an arbitrary point on the side $BC$ of triangle $ABC$. Points $E$ and $F$ are on $CA$ and $BA$ are such that $CD=CE$ and $BD=BF$. Lines $BE$ and $CF$ intersect at point $P$. Prove that when point $D$ varies along the line $BC$, $PD$ passes through a fixed point.

Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.

Given that $a_1, a_2, \ldots,a_{2020}$ are integers, find the maximal number of subsequences $a_i,a_{i+1}, ..., a_j$ ($0<i\leq j<2021$) with with sum $2021$

The following operation is allowed on the positive integers: if a number is even, we can divide it by $2$, otherwise we can multiply it by a power of $3$ (different from $3^0$) and add $1$. Prove that we can reach $1$ from any starting positive integer $n$.

Let $S=\{1,2, \ldots ,10^{10}\}$. Find all functions $f:S \rightarrow S$, such that $$f(x+1)=f(f(x))+1 \pmod {10^{10}}$$for each $x \in S$ (assume $f(10^{10}+1)=f(1)$).