

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(AC) + 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.

