38th Vietnam 2000 problems

A1.  Define a sequence of positive reals x0, x1, x2, ... by x0 = b, xn+1 = √(c - √(c + xn)). 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) = x3 + 153x2 - 111x + 38. Show that p(n) is divisible by 32000 for at least nine positive integers n less than 32000. 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 x2 + ax + b which is a factor of pn(x) = sin α xn - sin(nα) x + sin(nα-α) for all n > 2. Show that there is no linear polynomial x + c which divides pn(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(x2 - 1) = p(x) p(-x). What is the largest number of real roots that p(x) can have?

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
29 July 2002