Third Millennium International Mathematical Olympiad 2008
Grade 9

1. Two players take turns placing counters on a table. On each turn, a player puts on the table either 1 counter or as many counters as their are on the table. The player who makes the last move wins. At the beginning there are no counters on the table. Assuming both players found an optimal strategy, who will win the game if the initial number of counters is 9.

2. Vasya measured the angles between the rays on which lie the sides of a polygon. What is the minimum number of sides for which the angles could add up to 2008°?

3. Is it possible to draw 9 straight line segments in the plane so that all but one intersects with all the rest of the segments?

4. Find a quadratic equation each of whose coefficients has the absolute value of either 1 or 2008, with the cubes of its roots adding up the maximum possible value.

5. Circle of radii 3, 4, 5 pass through a common point, while the remaining pairwise intersections are collinear. Consider straight line segments joining the points of intersection. Find the number of pairs of perpendicular segments.

6. Is there a polygon whose perimeter and area equal 2008²⁰⁰⁸?

   

   

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