41st Swedish 2001


1. Show that if we take any six numbers from the following array, one from each row and column, then the product is always the same: 4 6 10 14 22 26 6 9 15 21 33 39 10 15 25 35 55 65 16 24 40 56 88 104 18 27 45 63 99 117 20 30 50 70 110 130
2. Show that (√52 + 5)^1/3 - (√52 - 5)^1/3 is rational.
3. Show that if b = (a+c)/2 in the triangle ABC, then cos(A-C) + 4 cos B = 3.
4. ABC is a triangle. A circle through A touches the side BC at D and intersects the sides AB and AC again at E, F respectively. EF bisects ∠AFD and ∠ADC = 80^o. Find ∠ABC.
5. Find all polynomials p(x) such that p'(x)² = c p(x) p''(x) for some constant c.
6. A chessboard is covered with 32 dominos. Each domino covers two adjacent squares. Show that the number of horizontal dominos with a white square on the left equals the number with a white square on the right.

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