Third Millennium International Mathematical Olympiad 2008
Grade 10

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 10.

2. Draw the diagonals and the midlines of a parallelogram and label the endpoints and the intersections of these line segments. In how many ways may one choose three labels that do not correspond to collinear points.

3. Construct a triangle with integer sides and an angle of 60°. Prove that there are infinitely many of such triangles, no two similar.

4. Find all pairs (p, q) such that the distance between the root of the polynomial x² + px + q equals 2008.

5. Asteroid is in the form of a parallelopiped. There is a pair of wolves sitting at a pair of opposite vertices. Each of the wolves rules over the part of the surface of the asteroid it may reach faster than its opposite number. What is the relation between the dimensions of the parallelopiped if each wolf controls at least one whole of its faces?

6. How many digits are there in the decimal representation of 2008²⁰⁰⁸?

   

   

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