$AB$ is the diameter of circle $O$, $CD$ is a non-diameter chord that is perpendicular to $AB$. Let $E$ be the midpoint of $OC$, connect $AE$ and extend it to meet the circle at point $P$. Let $DP$ and $BC$ meet at $F$. Prove that $F$ is the midpoint of $BC$.

$p$ is a prime number that is greater than $2$. Let $\{ a_{n}\}$ be a sequence such that $ na_{n+1}= (n+1) a_{n}-\left( \frac{p}{2}\right)^{4}$. Show that if $a_{1}=5$, the $16 \mid a_{81}$.

$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$. Alternative formulation. Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.

Given a function $f(x)=x^{2}+ax+b$ with $a,b \in R$, if there exists a real number $m$ such that $\left| f(m) \right| \leq \frac{1}{4}$ and $\left| f(m+1) \right| \leq \frac{1}{4}$, then find the maximum and minimum of the value of $\Delta=a^{2}-4b$.

$a,b,c$ are positive numbers such that $a+b+c=3$, show that: \[\frac{a^{2}+9}{2a^{2}+(b+c)^{2}}+\frac{b^{2}+9}{2b^{2}+(a+c)^{2}}+\frac{c^{2}+9}{2c^{2}+(a+b)^{2}}\leq 5\]

Can we put positive integers $1,2,3, \cdots 64$ into $8 \times 8$ grids such that the sum of the numbers in any $4$ grids that have the form like $T$ ( $3$ on top and $1$ under the middle one on the top, this can be rotate to any direction) can be divided by $5$?

Given a sequence $\{ a_{n}\}$ such that $a_{n+1}=a_{n}+\frac{1}{2006}a_{n}^{2}$ , $n \in N$, $a_{0}=\frac{1}{2}$. Prove that $1-\frac{1}{2008}< a_{2006}< 1$.