A1. 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 .
A2. 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 .
A3. 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 ?
B1. 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.
B2. 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.
B3. 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 ? |