CN-LEAGUE 2012

Suppose two regular pyramids with the same base $ABC$ : $P-ABC$ and $Q-ABC$ are circumscribed by the same sphere. If the angle formed by one of the lateral face and the base of pyramid $P-ABC$ is $\frac{\pi}{4}$ , find the tangent value of the angle formed by one of the lateral face and the base of the pyramid $Q-ABC$ .

Let $f(x)$ be an odd function on $\mathbb{R}$ , such that $f(x)=x^2$ when $x\ge 0$ . Knowing that for all $x\in [a,a+2]$ , the inequality $f(x+a)\ge 2f(x)$ holds, find the range of real number $a$ .

Find the sum of all integers $n$ satisfying the following inequality: $$\frac{1}{4}<\sin\frac{\pi}{n}<\frac{1}{3}.$$

There are $4$ distinct codes used in an intelligence station, one of them applied in each week. No two codes used in two adjacent weeks are the same code. Knowing that code $A$ is used in the first week, find the probability that code $A$ is used in the seventh week.

Given a function $f(x)=a\sin x-\frac{1}{2}\cos 2x+a-\frac{3}{a}+\frac{1}{2}$ , where $a\in\mathbb{R}, a\ne 0$ . (1) If for any $x\in\mathbb{R}$ , inequality $f(x)\le 0$ holds, find all possible value of $a$ .(2) If $a\ge 2$ , and there exists $x\in\mathbb{R}$ , such that $f(x)\le 0$ . Find all possible value of $a$ .

Given a sequence $\{a_n\}$ whose terms are non-zero real numbers. For any positive integer $n$ , the equality $$(\sum_{i=1}^{n}a_i)^2=\sum_{i=1}^{n}a_i^3$$ holds.(1) If $n=3$ , find all possible sequence $a_1,a_2,a_3$ ;(2) Does there exist such a sequence $\{a_n\}$ such that $a_{2011}=-2012$ ?

In the Cartesian plane $XOY$ , there is a rhombus $ABCD$ whose side lengths are all $4$ and $|OB|=|OD|=6$ , where $O$ is the origin.(1) Prove that $|OA|\cdot |OB|$ is a constant.(2) Find the locus of $C$ if $A$ is a point on the semicircle $$(x-2)^2+y^2=4 (2\le x\le 4).$$