TR-MO 2000

A circle with center $O$ and a point $A$ in this circle are given. Let $P_{B}$ is the intersection point of $[AB]$ and the internal bisector of $\angle AOB$ where $B$ is a point on the circle such that $B$ doesn't lie on the line $OA$ , Find the locus of $P_{B}$ as $B$ varies.

Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$ . Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that $$P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).$$

Let $f(x,y)$ and $g(x,y)$ be real valued functions defined for every $x,y \in \{1,2,..,2000\}$ . If there exist $X,Y \subset \{1,2,..,2000\}$ such that $s(X)=s(Y)=1000$ and $x\notin X$ and $y\notin Y$ implies that $f(x,y)=g(x,y)$ than, what is the maximum number of $(x,y)$ couples where $f(x,y)\neq g(x,y)$ .

Let $p$ be a prime number. $T(x)$ is a polynomial with integer coefficients and degree from the set $\{0,1,...,p-1\}$ and such that $T(n) \equiv T(m) (mod p)$ for some integers m and n implies that $m \equiv n (mod p)$ . Determine the maximum possible value of degree of $T(x)$

A positive real number $a$ and two rays wich intersect at point $A$ are given. Show that all the circles which pass through $A$ and intersect these rays at points $B$ and $C$ where $|AB|+|AC|=a$ have a common point other than $A$ .

Find all continuous functions $f:[0,1]\to [0,1]$ for which there exists a positive integer $n$ such that $f^{n}(x)=x$ for $x \in [0,1]$ where $f^{0} (x)=x$ and $f^{k+1}=f(f^{k}(x))$ for every positive integer $k$ .