Let $M$ be the set of the real numbers except for finitely many elements. Prove that for every positive integer $n$ there exists a polynomial $f(x)$ with $\deg f = n$, such that all the coefficients and the $n$ real roots of $f$ are all in $M$.

Given an integer $n\ge\ 3$, find the least positive integer $k$, such that there exists a set $A$ with $k$ elements, and $n$ distinct reals $x_{1},x_{2},\ldots,x_{n}$ such that $x_{1}+x_{2}, x_{2}+x_{3},\ldots, x_{n-1}+x_{n}, x_{n}+x_{1}$ all belong to $A$.

Let $H$ be the orthocenter of acute triangle $ABC$ and $D$ the midpoint of $BC$. A line through $H$ intersects $AB,AC$ at $F,E$ respectively, such that $AE=AF$. The ray $DH$ intersects the circumcircle of $\triangle ABC$ at $P$. Prove that $P,A,E,F$ are concyclic.

Prove that for every given positive integer $k$, there exist infinitely many $n$, such that $2^{n}+3^{n}-1, 2^{n}+3^{n}-2,\ldots, 2^{n}+3^{n}-k$ are all composite numbers.

Define a sequence $(x_{n})_{n\geq 1}$ by taking $x_{1}\in\left\{5,7\right\}$; when $k\ge 1$, $x_{k+1}\in\left\{5^{x_{k}},7^{x_{k}}\right\}$. Determine all possible last two digits of $x_{2009}$.

Given an acute triangle $ABC$, $D$ is a point on $BC$. A circle with diameter $BD$ intersects line $AB,AD$ at $X,P$ respectively (different from $B,D$).The circle with diameter $CD$ intersects $AC,AD$ at $Y,Q$ respectively (different from $C,D$). Draw two lines through $A$ perpendicular to $PX,QY$, the feet are $M,N$ respectively.Prove that $\triangle AMN$ is similar to $\triangle ABC$ if and only if $AD$ passes through the circumcenter of $\triangle ABC$.

A total of $n$ people compete in a mathematical match which contains $15$ problems where $n>12$. For each problem, $1$ point is given for a right answer and $0$ is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of $12$ people get is no less than $36$, then there are at least $3$ people that got the right answer of a certain problem, among the $n$ people. Find the least possible $n$.

The real numbers $a_{1},a_{2},\ldots ,a_{n}$ where $n\ge 3$ are such that $\sum_{i=1}^{n}a_{i}=0$ and $2a_{k}\le\ a_{k-1}+a_{k+1}$ for all $k=2,3,\ldots ,n-1$. Find the least $f(n)$ such that, for all $k\in\left\{1,2,\ldots ,n\right\}$, we have $|a_{k}|\le f(n)\max\left\{|a_{1}|,|a_{n}|\right\}$.