Let $a_1,a_2,\ldots,a_n$ are real numbers, prove that; \[\sum_{i=1}^na_i^2-\sum_{i=1}^n a_i a_{i+1} \le \left\lfloor \frac{n}{2}\right\rfloor(M-m)^2.\] where $a_{n+1}=a_1,M=\max_{1\le i\le n} a_i,m=\min_{1\le i\le n} a_i$.

On the circumcircle of the acute triangle $ABC$, $D$ is the midpoint of $ \stackrel{\frown}{BC}$. Let $X$ be a point on $ \stackrel{\frown}{BD}$, $E$ the midpoint of $ \stackrel{\frown}{AX}$, and let $S$ lie on $ \stackrel{\frown}{AC}$. The lines $SD$ and $BC$ have intersection $R$, and the lines $SE$ and $AX$ have intersection $T$. If $RT \parallel DE$, prove that the incenter of the triangle $ABC$ is on the line $RT.$

Let $A$ be a set consist of finite real numbers,$A_1,A_2,\cdots,A_n$ be nonempty sets of $A$, such that (a) The sum of the elements of $A$ is $0,$ (b) For all $x_i \in A_i(i=1,2,\cdots,n)$,we have $x_1+x_2+\cdots+x_n>0$. Prove that there exist $1\le k\le n,$ and $1\le i_1<i_2<\cdots<i_k\le n$, such that \[|A_{i_1}\bigcup A_{i_2} \bigcup \cdots \bigcup A_{i_k}|<\frac{k}{n}|A|.\] Where $|X|$ denote the numbers of the elements in set $X$.

Let $n$ be an given positive integer, the set $S=\{1,2,\cdots,n\}$.For any nonempty set $A$ and $B$, find the minimum of $|A\Delta S|+|B\Delta S|+|C\Delta S|,$ where $C=\{a+b|a\in A,b\in B\}, X\Delta Y=X\cup Y-X\cap Y.$

Let $a_i,b_i,i=1,\cdots,n$ are nonnegitive numbers,and $n\ge 4$,such that $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0$. Find the maximum of $\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}$

Let $m,n$ be positive integer numbers. Prove that there exist infinite many couples of positive integer nubmers $(a,b)$ such that \[a+b| am^a+bn^b , \quad\gcd(a,b)=1.\]