Third Millennium International Mathematical Olympiad 2008
Grade 12

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 12.

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 120°. Prove that there are infinitely many of such triangles, no two similar.

4. Two polynomials have completely different values, except for two common roots. What least number of roots may have the derivative of their product.

5. What is the maximum number of faces of a dodecahedron that may be cut by a single plane.

6. Is there a parallelopiped whose volume, surface area and the sum of the edges are all equal to 2008?

   

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