

1. A, B are points inside a circle C. Show that there is a circle through A and B which lies entirely inside C.


2. Each point in a 3 x 7 array is colored yellow or blue. Show that it is always possible to find 4 points of the same color forming a rectangle with sides parallel to the side of the array.


3. Show that ( (a+1)/(b+1) )^{b+1} ≥ (a/b)^{b} for positive reals a, b.


4. Find all positive integers m, n such that all roots of (x^{2}  mx + n)(x^{2}  nx + m) are positive integers.


5. Find all positive integer solutions to a^{3}  b^{3}  c^{3} = 3abc, a^{2} = 2(a + b + c).


6. The positive integers a_{1}, a_{2}, ... , a_{14} satisfy ∑ 3^{ai} = 6558. Show that they must be two copies of each of 1, 2, ... , 7.

