29th USAMO 2000


A1. Show that there is no real-valued 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. C₁ is a circle through A and B. We can find circle C₂ through B and C touching C₁, circle C₃ through C and A touching C₂, circle C₄ through A and B touching C₃ and so on. Show that C₇ is the same as C₁.
B3. x₁, x₂, ... , x_n, and y₁, y₂, ... , y_n are non-negative reals. Show that ∑ min(x_ix_j, y_iy_j) ≤ ∑ min(x_iy_j, x_jy_i), where each sum is taken over all n² pairs (i, j).

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.