

1. p is a prime. Find the largest integer d such that p^{d} divides p^{4}!


2. There is a point inside an equilateral triangle side d whose distance from the vertices is 3, 4, 5. Find d.


3. Show that the only integral solution to xy + yz + zx = 3n^{2}  1, x + y + z = 3n with x ≥ y ≥ z is x = n+1, y = n, z = n1.


4. Show that if cos x/cos y + sin x/sin y = 1, then cos^{3}y/cos x + sin^{3}y/sin x = 1.


5. The numbers 1, 2, 3, ... , 64 are written in the cells of an 8 x 8 board (in some order, one per cell). Show that at least four 2 x 2 squares have sum > 100.


6. Show that there are positive reals a, b, c such that a^{2} + b^{2} + c^{2} > 2, a^{3} + b^{3} + c^{3} < 2, and a^{4} + b^{4} + c^{4} > 2.

