

1. A graph G has n + k points. A is a subset of n points and B is the subset of the other k points. Each point of A is joined to at least k  m points of B where nm < k. Show that there is a point in B which is joined every point in A.


2. Find all real x such that 0 < x < π and 8/(3 sin x  sin 3x) + 3 sin^{2}x ≤ 5.


3. The real numbers x_{1}, x_{4}, y_{1}, y_{2} are positive and the real numbers x_{2}, x_{3}, y_{3}, y_{4} are negative. We have (x_{i}  a)^{2} + (y_{i}  b)^{2} ≤ c^{2} for i = 1, 2, 3, 4. Show that a^{2} + b^{2} ≤ c^{2}. State the result in geometric language.


4. Two circles centers O and O', radii R and R', meet at two points. A variable line L meets the circles at A, C, B, D in that order and AC/AD = CB/BD. The perpendiculars from O and O' to L have feet H and H'. Find the locus of H and H'. If OO'^{2} < R^{2} + R'^{2}, find a point P on L such that PO + PO' has the smallest possible value. Show that this value does not depend on the position of L. Comment on the case OO'^{2} > R^{2} + R'^{2}.
