A1. Show that the triangle ABC is rightangled iff sin A + sin B + sin C = cos A + cos B + cos C + 1.  
A2. Find all integral values of m such that x^{3} + 2x + m divides x^{12}  x^{11} + 3x^{10} + 11x^{3}  x^{2} + 23x + 30.  
A3. Given two points A, B not in the plane p, find the point X in the plane such that XA/XB has the smallest possible value.  
B1. Find all real solutions to:
w^{2} + x^{2} + y^{2} + z^{2} = 50 w^{2}  x^{2} + y^{2}  z^{2} = 24 wx = yz w  x + y  z = 0. 

B2. x_{1}, x_{2}, x_{3}, ... , x_{n} are reals in the interval [a, b]. M = (x_{1} + x_{2} + ... + x_{n})/n, V = (x_{1}^{2} + x_{2}^{2} + ... + x_{n}^{2})/n. Show that M^{2} ≥ 4Vab/(a + b)^{2}.  
B3. Two circles touch externally at A. P is a point inside one of the circles, not on the line of centers. A variable line L through P meets one circle at B (and possibly another point) and the other circle at C (and possibly another point). Find L such that the circumcircle of ABC touches the line of centers at A. 
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
23 July 2002