

A1. Show that there is no realvalued function f on the reals such that ( f(x) + f(y) )/2 ≥ f( (x+y)/2 ) + x  y for all x, y.


A2. The incircle of the triangle ABC touches BC, CA, AB at D, E, F respectively. We have AF ≤ BD ≤ CE, the inradius is r and we have 2/AF + 5/BD + 5/CE = 6/r. Show that ABC is isosceles and find the lengths of its sides if r = 4.


A3. A player starts with A blue cards, B red cards and C white cards. He scores points as he plays each card. If he plays a blue card, his score is the number of white cards remaining in his hand. If he plays a red card it is three times the number of blue cards remaining in his hand. If he plays a white card, it is twice the number of red cards remaining in his hand. What is the lowest possible score as a function of A, B and C and how many different ways can it be achieved?


B1. How many squares of a 1000 x 1000 chessboard can be chosen, so that we cannot find three chosen squares with two in the same row and two in the same column?


B2. ABC is a triangle. C_{1} is a circle through A and B. We can find circle C_{2} through B and C touching C_{1}, circle C_{3} through C and A touching C_{2}, circle C_{4} through A and B touching C_{3} and so on. Show that C_{7} is the same as C_{1}.


B3. x_{1}, x_{2}, ... , x_{n}, and y_{1}, y_{2}, ... , y_{n} are nonnegative reals. Show that ∑ min(x_{i}x_{j}, y_{i}y_{j}) ≤ ∑ min(x_{i}y_{j}, x_{j}y_{i}), where each sum is taken over all n^{2} pairs (i, j).

