

A1. Solve the following equations for x, y and z:
x + y + z = a; x^{2} + y^{2} + z^{2} = b^{2}; xy = z^{2} .
What conditions must a and b satisfy for x, y and z to be distinct positive numbers?


A2. Let a, b, c be the sides of a triangle and A its area. Prove that:
a^{2} + b^{2} + c^{2} ≥ 4√3 A
When do we have equality?


A3. Solve the equation cos^{n}x  sin^{n}x = 1, where n is a natural number.


B1. P is inside the triangle ABC. PA intersects BC in D, PB intersects AC in E, and PC intersects AB in F. Prove that at least one of AP/PD, BP/PE, CP/PF does not exceed 2, and at least one is not less than 2.


B2. Construct the triangle ABC, given the lengths AC=b, AB=c and the acute ∠AMB = α, where M is the midpoint of BC. Prove that the construction is possible if and only if
b tan(α/2) ≤ c < b.
When does equality hold?


B3. Given 3 noncollinear points A, B, C and a plane p not parallel to ABC and such that A, B, C are all on the same side of p. Take three arbitrary points A', B', C' in p. Let A'', B'', C'' be the midpoints of AA', BB', CC' respectively, and let O be the centroid of A'', B'', C''. What is the locus of O as A', B', C' vary?

