Let $\triangle ABC$ satisfies $\cos A:\cos B:\cos C=1:1:2$, then $\sin A=\sqrt[s]{t}$($s\in\mathbb{N},t\in\mathbb{Q^+}$ and $t$ is an irreducible fraction). Find $s+t$.

A and B two people are throwing n fair coins.X and Y are the times they get heads. If throwing coins are mutually independent events, (1)When n=5, what is the possibility of X=Y? (2)When n=6, what is the possibility of X=Y+1?

Let $M$ is a four digit positive interger. Write $M$ backwards and get a new number $N$.(e.g $M=1234$ then $N=4321$) Let $C$ is the sum of every digit of $M$. If $M,N,C$ satisfies (i)$d=\gcd(M-C,N-C)$ and $d<10$ (ii)$\dfrac{M-C}{d}=\lfloor\dfrac{N}{2}+1\rfloor$ (1)Find $d$. (2)If there are "m(s)" $M$ satisfies (i) and (ii), and the largest $M$=$M_{max}$. Find $(m,M_{max})$

Let $(a,b)=(a_n,a_{n+1}),\forall n\in\mathbb{N}$ all be positive interger solutions that satisfies $$1\leq a\leq b$$and $$\dfrac{a^2+b^2+a+b+1}{ab}\in\mathbb{N}$$And the value of $a_n$ is only determined by the following recurrence relation:$ a_{n+2} = pa_{n+1} + qa_n + r$ Find $(p,q,r)$.

Let $S$ is the set of permutation of {1,2,3,4,5,6,7,8} (1)For all $\sigma=\sigma_1\sigma_2...\sigma_8\in S$ Evaluate the sum of S=$\sigma_1\sigma_2+\sigma_3\sigma_4+\sigma_5\sigma_6+\sigma_7\sigma_8$. Then for all elements in $S$,what is the arithmetic mean of S? (Notice $S$ and S are different.) (2)In $S$, how many permutations are there which satisfies "For all $k=1,2,...,7$,the digit after k is not (k+1)"?

Let $a,b,c$ be positive reals. Find the minimum value of $$\dfrac{13a+13b+2c}{2a+2b}+\dfrac{24a-b+13c}{2b+2c}+\dfrac{(-a+24b+13c)}{2c+2a}$$. (1)What is the minimum value? (2)If the minimum value occurs when $(a,b,c)=(a_0,b_0,c_0)$,then find $\frac{b_0}{a_0}+\frac{c_0}{b_0}$.

[$XYZ$] denotes the area of $\triangle XYZ$ We have a $\triangle ABC$,$BC=6,CA=7,AB=8$ (1)If $O$ is the circumcenter of $\triangle ABC$, find [$OBC$]:[$OCA$]:[$OAB$] (2)If $H$ is the orthocenter of $\triangle ABC$, find [$HBC$]:[$HCA$]:[$HAB$]