Third Millennium International Mathematical Olympiad 2009
Grade 9

1. A group of 49 numbers contains four 4's, five 5's, six 6's, seven 7's, eight 8's, nine 9's and ten 3's. Split the numbers into seven groups of seven numbers each so that the numbers in all seven groups have the same sum.

   Solution

2. Find the area of the ring between two concentric circles, if the chord of the bigger circle that is tangent to the smaller one has the length of 2009.

   Solution

3. Find all pairs of natural numbers A and B for which (A^B - B²)² - (2^B)² = 2009.

   Solution

4. None of the group of nine distinct natural numbers exceeds 16. Prove that, out of these nine, at least one divides at least one other.

   Solution

5. Find all natural numbers that equal the sum of squares of their digits.

   Solution

6. Through the vertices of a 9×9 grid square draw a broken line with nodes at the grid points that encloses the least possible area.

   Solution

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