Math Mania 2007

In a certain motel, if two students share each room, then two students will be left without a room. If three students share each room, then two rooms will be left empty. How many rooms are in the motel?
A car went up a hill at an average speed of 30 mph and went down the hill at an average speed of 60 mph. What was the average speed for the entire trip?
What is the angle (in degrees) between the minute hand and hour hand of a clock that reads 6:20?
Find the next term in the sequence: 1, 4, 13, 31, 61, 106,...
What are the last two digits in the decimal expansion of 2007²⁰⁰⁷ ?
What is the smallest positive integer that is not a divisor of 90! = 90×89×...×3×2×1?

Tiebreaker — How many (decimal) digits are there in 2007! = 2007×2006×...×3×2×1? (after it is expanded out)? The team getting closest to the correct answer wins.

1. 8 rooms
2. 40 mph
3. 70 degrees
4. The next term is 169.
5. The last two digits are 43.
6. The smallest positive integer that is not a divisor is 97.
Tiebreaker — There are 5759 digits.

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