CN-LEAGUE 2008

Given a convex quadrilateral with $\angle B+\angle D<180$ .Let $P$ be an arbitrary point on the plane,define $f(P)=PA*BC+PD*CA+PC*AB$ . (1)Prove that $P,A,B,C$ are concyclic when $f(P)$ attains its minimum. (2)Suppose that $E$ is a point on the minor arc $AB$ of the circumcircle $O$ of $ABC$ ,such that $AE=\frac{\sqrt 3}{2}AB,BC=(\sqrt 3-1)EC,\angle ECA=2\angle ECB$ .Knowing that $DA,DC$ are tangent to circle $O$ , $AC=\sqrt 2$ ,find the minimum of $f(P)$ .

Let $f(x)$ be a periodic function with periods $T$ and $1$ ( $0<T<1$ ).Prove that: (1)If $T$ is rational,then there exists a prime $p$ such that $\frac{1}{p}$ is also a period of $f$ ; (2)If $T$ is irrational,then there exists a strictly decreasing infinite sequence { $a_n$ },with $1>a_n>0$ for all positive integer $n$ ,such that all $a_n$ are periods of $f$ .

For all $k=1,2,\ldots,2008$ , $a_k>0$ .Prove that iff $\sum_{k=1}^{2008}a_k>1$ ,there exists a function $f:N\rightarrow R$ satisfying (1) $0=f(0)<f(1)<f(2)<\ldots$ ; (2) $f(n)$ has a finite limit when $n$ approaches infinity; (3) $f(n)-f({n-1})=\sum_{k=1}^{2008}a_kf({n+k})-\sum_{k=0}^{2007}a_{k+1}f({n+k})$ ,for all $n=1,2,3,\ldots$ .