A sequence of positive real numbers $x_0,x_1,x_2,\ldots$ is defined by
$$x_0=b,\qquad x_{n+1}=\sqrt{c-\sqrt{c+x_n}}.$$
Find all values of $c$ such that, for every $b\in(0,c)$, this sequence exists and converges to a limit as $n$ tends to infinity.
Two circles $C,C'$ have centers $O,O'$, respectively, with intersecting reference lines $OX$ and $O'X'$. Points $M,M'$ move on $C,C'$ so that the counterclockwise directed angles $\angle XOM$ and $\angle X'O'M'$ are equal. Determine the locus of the midpoint of $MM'$. Suppose $OM$ and $O'M'$ intersect at $Q$. Prove that the circle through $Q,M,M'$ always passes through a fixed point.
Let $p(x)=x^3+153x^2-111x+38$. Prove that $p(n)$ is divisible by $3^{2000}$ for at least nine positive integers $n\lt 3^{2000}$. How many such integers are there?
Let $0\lt \alpha\lt \pi$. Prove that there is a unique quadratic polynomial $x^2+ax+b$ that is a factor of
$$p_n(x)=\sin\alpha\,x^n-\sin(n\alpha)x+\sin(n\alpha-\alpha)$$
for every integer $n\gt 2$. Prove that no linear polynomial $x+c$ divides $p_n(x)$ for every $n\gt 2$.
Find all integers $n\gt 3$ for which there exist $n$ points in space such that no three are collinear, no four are concyclic, and all circles through any three of the points have the same radius.