

A1. Prove that there are infinitely many positive integers m, such that n^{4} + m is not prime for any positive integer n.


A2. Let f(x) = cos(a_{1} + x) + 1/2 cos(a_{2} + x) + 1/4 cos(a_{3} + x) + ... + 1/2^{n1} cos(a_{n} + x), where a_{i} are real constants and x is a real variable. If f(x_{1}) = f(x_{2}) = 0, prove that x_{1}  x_{2} is a multiple of π.


A3. For each of k = 1, 2, 3, 4, 5 find necessary and sufficient conditions on a > 0 such that there exists a tetrahedron with k edges length a and the remainder length 1.


B1. C is a point on the semicircle diameter AB, between A and B. D is the foot of the perpendicular from C to AB. The circle K_{1} is the incircle of ABC, the circle K_{2} touches CD, DA and the semicircle, the circle K_{3} touches CD, DB and the semicircle. Prove that K_{1}, K_{2} and K_{3} have another common tangent apart from AB.


B2. Given n > 4 points in the plane, no three collinear. Prove that there are at least (n3)(n4)/2 convex quadrilaterals with vertices amongst the n points.


B3. Given real numbers x_{1}, x_{2}, y_{1}, y_{2}, z_{1}, z_{2}, satisfying x_{1} > 0, x_{2} > 0, x_{1}y_{1} > z_{1}^{2}, and x_{2}y_{2} > z_{2}^{2}, prove that:
8/((x_{1} + x_{2})(y_{1} + y_{2})  (z_{1} + z_{2})^{2}) ≤ 1/(x_{1}y_{1}  z_{1}^{2}) + 1/(x_{2}y_{2}  z_{2}^{2}).
Give necessary and sufficient conditions for equality.

