Third Millennium International Mathematical Olympiad 2008
Grade 6

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 6.

2. A regiment has less than 1000 soldiers. The colonel plans to arrange the soldiers in the form of a rectangle. When he places 17 soldiers in a row, one row comes 1 soldier short. When he places 13 soldiers in a row one soldier remains. If the regiment consists of three battalions, each with the same number of soldiers, how big is a battalion?

3. An Architect plans a city with 2008 straight streets and a circular beltway around the city. The streets and the road in the city are required to meet at T-intersection. What may be the number of the intersections?

4. A water melon that weighs 3 lbs is sold for 11 cents, the one at 4 lbs for 13 cents, and the biggest that weighs 5 lbs for 17 cents. How should one select the water melons to maximize the total weight at the cost of 100 cents.

5. Find the last two digits of 2008²⁰⁰⁸.

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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