2. This 12-digit integer is not divisible by 11. What is the smallest positive integer x that must be subtracted to make the resulting number evenly divisible by 11?

1 1 0 0 1 1 0 0 0 0 6 3

Answer. x =

3. Given S = 1 + 1/2² + 1/3² + 1/4² + ... = p²/6. Let T = 1/2² + 1/4² + 1/6² + ... What is 2S/T?

4. Two concentric circles A and B are such that the inner circle A covers 1/16 the area of circle B. You want to paint circle A red, and for the remaining area between A and B, one half black and the other half white. If 2 gallons of red paint are needed to cover the area of A, how many gallons of black paint will be needed? Give your answer in terms of gallons.

5. Let a_n = a₁a₂...a_n. If P_N = (n²)(1/(n+1)²), what is the limiting value of P_N as N approaches infinity?

6. Given log₁₀ a + log₁₀ b = 1 and a, b are positive integers, find a²b².

Answer. a²b² =

7. A door hinged along an axis about O closes at a constant speed of 30° per second. The door initially is at 90° when a mouse at O starts to run along a straight line at an angle q=60° at a constant speed (see figure). If the door is 4 ft wide, what is the minimum speed the mouse must run to avoid getting hit by the door? Express your answer in terms of ft per second.

Answer. Minimum speed = ft per second

8. Tom puts $1 into a drawer on day 1. The next day he puts in $2, and $3 the following day, and so on. At the end of day N, he finds out that there are exactly $465 in the drawer. What is N?

9. Let a₁, a₂,..., a_N be a sequence of N numbers. Given the average of these numbers is A. A new number a_N+1=13 is added to the sequence. If the new average is still A, what is A?

10. A room is 10 ft wide and 20 ft long. You want to cover its floor with identical rectangular tiles of dimensions a ft x b ft (ab). What is the maximum number of tiles you need if the tiles must have integral dimensions, i.e. a and b must be integer?