$a_1,a_2,\cdots,a_n$ are positive real numbers, $a_1+a_2+\cdots,a_n=1$. Prove that $$\sum_{m=1}^n\frac{a_m}{\prod\limits_{k=1}^m(1+a_k)}\leq1-\frac{1}{2^n}.$$

In isosceles triangle $ABC$, $\angle CAB=\angle CBA=\alpha$, points $P,Q$ are on different sides of line $AB$, and $\angle CAP=\angle ABQ=\beta,\angle CBP=\angle BAQ=\gamma$. Prove that $P,C,Q$ are colinear.

Prove: (a) There are infinitely many positive intengers $n$, satisfying: $$\gcd(n,[\sqrt2n])=1.$$(b) There are infinitely many positive intengers $n$, satisfying: $$\gcd(n,[\sqrt2n])>1.$$

Can we put intengers $1,2,\cdots,12$ on a circle, number them $a_1,a_2,\cdots,a_{12}$ in order. For any $1\leq i<j\leq12$, $|a_i-a_j|\neq|i-j|$?

Let $\theta_{i}\in(0,\frac{\pi}{2})(i=1,2,\cdots,n)$. Prove: $$(\sum_{i=1}^n\tan\theta_{i})(\sum_{i=1}^n\cot\theta_{i})\geq(\sum_{i=1}^n\sin\theta_{i})^2+(\sum_{i=1}^n\cos\theta_{i})^2.$$

Four points $B,E,A,F$ lie on line $AB$ in order, four points $C,G,D,H$ lie on line $CD$ in order, satisfying: $$\frac{AE}{EB}=\frac{AF}{FB}=\frac{DG}{GC}=\frac{DH}{HC}=\frac{AD}{BC}.$$Prove that $FH\perp EG$.

Define sequence $(a_n):a_n=2^n+3^n+6^n+1(n\in\mathbb{Z}_+)$. Are there intenger $k\geq2$, satisfying that $\gcd(k,a_i)=1$ for all $k\in\mathbb{Z}_+$? If yes, find the smallest $k$. If not, prove this.

Set $A=\{1,2,\cdots,n\}$. If there exists nonempty sets $B,C$, such that $B\cap C=\varnothing,B\cup C=A$. Sum of Squares of all elements in $B$ is $M$, Sum of Squares of all elements in $C$ is $N$, $M-N=2016$. Find the minimum value of $n$.

Inscribed Triangle $ABC$ on circle $\odot O$. Bisector of $\angle ABC$ intersects $\odot O$ at $D$. Two lines $PB$ and $PC$ that are tangent to $\odot O$ intersect at $P$. $PD$ intersects $AC$ at $E$, $\odot O$ at $F$. $M$ is the midpoint of $BC$. Prove that $M,F,C,E$ are concyclic.

$m(m>1)$ is an intenger, define $(a_n)$: $a_0=m,a_{n}=\varphi(a_{n-1})$ for all positive intenger $n$. If for all nonnegative intenger $k$, $a_{k+1}\mid a_k$, find all $m$ that is not larger than $2016$. Note: $\varphi(n)$ means Euler Function.

$a_1=2,a_{n+1}=\frac{2^{n+1}a_n}{(n+\frac{1}{2})a_n+2^n}(n\in\mathbb{Z}_+)$ (a) Find $a_n$. (b) Let $b_n=\frac{n^3+2n^2+2n+2}{n(n+1)(n^2+1)a_n}$. Find $S_n=\sum_{i=1}^nb_i$.

Given a set $I=\{(x_1,x_2,x_3,x_4)|x_i\in\{1,2,\cdots,11\}\}$. $A\subseteq I$, satisfying that for any $(x_1,x_2,x_3,x_4),(y_1,y_2,y_3,y_4)\in A$, there exists $i,j(1\leq i<j\leq4)$, $(x_i-x_j)(y_i-y_j)<0$. Find the maximum value of $|A|$.