Third Millennium International Mathematical Olympiad 2009
Grade 10

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.

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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.

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3. Find all natural numbers M for which (7^M - M²)^M - M^2M = 2009.

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4. By definition, the shortest pass between two points on the surface of a cube is the shortest broken line on the surface of the cube joining the two points. (Which is a straight line segment whenever the given points lie on the same face.) Three shortest passes joining three points split the surface of a cube into two regions. The region with the smallest area is called a triangle. Find the largest area of a triangle on the surface of a unit cube.

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

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6. In a Cartesian system of coordinates, a circle of radius r with center at (p, q) meets the parabola with the equation y = ax² + bx + c at four distinct points. Prove that there is another parabola passing through the same four points and find its equation.

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