Third Millennium International Mathematical Olympiad 2008
Grade 5

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 5.

2. How many squares can be found in a 5×5 grid?

3. Ann intends to place into each box the same number of toys. If she puts 12 toys into each box, 5 toys remain over. If she tries to put 15 toys into each box, the last of the boxes gets only 2 toys. However, after adding one more box, Ann has managed to achieve her goal. How many toys went into each box?

4. Find is the last digit of 2008²⁰⁰⁸.

5. Is it possible to draw 5 straight line segments in the plane so all but one intersects all the remaining segments.

6. A series of bus tickets includes all 6-digit number from 000000 through 999999. A girl collects the tickets whose numbers are divisible by 78 and a boy collects the tickets whose numbers are divisible by 77, but not by 78. Which kind of tickets in the series is more numerous: those of intersect to the girl or to the boy?

   

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