Herzog Competition - 1993

Problem 1

Compute the length of an edge of a regular pentagon inscribed in a circle of unit radius. (Solve in radicals.)

Problem 2

Are there any angles $\theta$ for which $\sqrt{\sin \theta }$ and $\sqrt{\cos \theta }$ are both non-zero rational numbers?

Problem 3

Find all positive integers m and n for which

Problem 4

In a given tetrahedron the sum of the angles at each of its vertices is 180°. Prove that all the faces of the tetrahedron are congruent.

Problem 5

Show that the equation

has exactly one positive real solution.

Problem 6

Assume that $2\le x\le y$. Prove or disprove that $y^{x+1}\le x{y}^y$.