SE-SKOLORNAS 2001

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

Show that $\sqrt[3]{\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5}$ is rational.

Show that if $b = \frac{a+c}{2}$ in the triangle $ABC$ , then $\cos (A-C) + 4 \cos B = 3$ .

$ABC$ 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 $\angle AFD$ and $\angle ADC = 80^o$ . Find $\angle ABC$ .

Find all polynomials $p(x)$ such that $p'(x)^2 = c p(x) p''(x)$ for some constant $c$ .

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.