| 1. Show that if 2 + 2 √(28 n2 + 1) is an integer, then it is a square (for n an integer). | |
| 2. 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. | |
| 3. 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. |
The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.
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(C) John Scholes
jscholes@kalva.demon.co.uk
19 Apr 2003
Last corrected/updated 19 Apr 2003