17th Swedish 1977


1. p is a prime. Find the largest integer d such that p^d divides p⁴!
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 = 3n² - 1, x + y + z = 3n with x ≥ y ≥ z is x = n+1, y = n, z = n-1.
4. Show that if cos x/cos y + sin x/sin y = -1, then cos³y/cos x + sin³y/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 a² + b² + c² > 2, a³ + b³ + c³ < 2, and a⁴ + b⁴ + c⁴ > 2.

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