KURSCHAK 2004

Given is a triangle $ABC$ , its circumcircle $\omega$ , and a circle $k$ that touches $\omega$ from the outside, and also touches rays $AB$ and $AC$ in $P$ and $Q$ , respectively. Prove that the $A$ -excenter of $\triangle ABC$ is the midpoint of $\overline{PQ}$ .

Find the smallest positive integer $n\neq 2004$ for which there exists a polynomial $f\in\mathbb{Z}[x]$ such that the equation $f(x)=2004$ has at least one, and the equation $f(x)=n$ has at least $2004$ different integer solutions.

We have placed some red and blue points along a circle. The following operations are permitted:

(a) we may add a red point somewhere and switch the color of its neighbors,

(b) we may take off a red point from somewhere and switch the color of its neighbors (if there are at least $3$ points on the circle and there is a red one too).

Initially, there are two blue points on the circle. Using a number of these operations, can we reach a state with exactly two red point?