KURSCHAK 1969

Show that if $2 + 2\sqrt{28n^2 + 1}$ is an integer, then it is a square (for $n$ an integer).

A triangle has side lengths $a, b, c$ and angles $A, B, C$ as usual (with $b$ opposite $B$ etc). Show that if $$a(1 - 2 \cos A) + b(1 - 2 \cos B) + c(1 - 2 \cos C) = 0$$ then the triangle is equilateral.

We are given $64$ cubes, each with five white faces and one black face. One cube is placed on each square of a chessboard, with its edges parallel to the sides of the board. We are allowed to rotate a complete row of cubes about the axis of symmetry running through the cubes or to rotate a complete column of cubes about the axis of symmetry running through the cubes. Show that by a sequence of such rotations we can always arrange that each cube has its black face uppermost