VMO 2000

Define a sequence of positive reals $x_0$, $x_1$, $x_2$, ... by $x_0$ = b, $x_{n+1}$ = √(c - √(c + $x_n$)). Find all values of c such that for all b in the interval (0, c), such a sequence exists and converges to a finite limit as n tends to infinity.

C and C' are circles centers O and O' respectively. X and X' are points on C and C' respectively such that the lines OX and O'X' intersect. M and M' are variable points on C and C' respectively, such that ∠XOM = ∠X'O'M' (both measured clockwise). Find the locus of the midpoint of MM'. Let OM and O'M' meet at Q. Show that the circumcircle of QMM' passes through a fixed point.

Let p(x) = $x^3$ + $153x^2$ - 111x + 38. Show that p(n) is divisible by $3^{2000}$ for at least nine positive integers n less than $3^{2000}$. For how many such n is it divisible?

Given an angle α such that 0 < α < π, show that there is a unique real monic quadratic $x^2$ + ax + b which is a factor of $p_n$(x) = sin α $x^n$ - sin(nα) x + sin(nα-α) for all n > 2. Show that there is no linear polynomial x + c which divides $p_n$(x) for all n > 2.

Find all n > 3 such that we can find n points in space, no three collinear and no four on the same circle, such that the circles through any three points all have the same radius.

p(x) is a polynomial with real coefficients such that p($x^2$ - 1) = p(x) p(-x). What is the largest number of real roots that p(x) can have?