CN-LEAGUE 2011

Let $P,Q$ be the midpoints of diagonals $AC,BD$ in cyclic quadrilateral $ABCD$ . If $\angle BPA=\angle DPA$ , prove that $\angle AQB=\angle CQB$ .

We want to arrange $7$ students to attend $5$ sports events, but students $A$ and $B$ can't take part in the same event, every event has its own participants, and every student can only attend one event. How many arrangements are there?

In a tetrahedral $ABCD$ , given that $\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}$ , $AD=BD=3$ , and $CD=2$ . Find the radius of the circumsphere of $ABCD$ .

The line $x-2y-1=0$ insects the parabola $y^2=4x$ at two different points $A, B$ . Let $C$ be a point on the parabola such that $\angle ACB=\frac{\pi}{2}$ . Find the coordinate of point $C$ .

Given that $a_{n}= \binom{200}{n} \cdot 6^{\frac{200-n}{3}} \cdot (\dfrac{1}{\sqrt{2}})^n$ ( $1 \leq n \leq 95$ ). How many integers are there in the sequence $\{a_n\}$ ?

Let $f(x)=|\log(x+1)|$ and let $a,b$ be two real numbers ( $a<b$ ) satisfying the equations $f(a)=f\left(-\frac{b+1}{a+1}\right)$ and $f\left(10a+6b+21\right)=4\log 2$ . Find $a,b$ .

A sequence $a_n$ satisfies $a_1 =2t-3$ ( $t \ne 1,-1$ ), and $a_{n+1}=\dfrac{(2t^{n+1}-3)a_n+2(t-1)t^n-1}{a_n+2t^n-1}$ . [list]i) Find $a_n$ ,ii) If $t>0$ , compare $a_{n+1}$ with $a_n$ .[/list]

A line $\ell$ with slope of $\frac{1}{3}$ insects the ellipse $C:\frac{x^2}{36}+\frac{y^2}{4}=1$ at points $A,B$ and the point $P\left( 3\sqrt{2} , \sqrt{2}\right)$ is above the line $\ell$ .

[list](1) Prove that the locus of the incenter of triangle $PAB$ is a segment,(2) If $\angle APB=\frac{\pi}{3}$ , then find the area of triangle $PAB$ .[/list]