3^rd Annual Eastern Oregon University Mathematics Competition Exam – 2006

1. The circle above has a radius of length 1. The segment AB is a diameter of the circle, angle BAC measures 30 degrees, and angle BAD measures 45 degrees. Determine the area of the shaded region.

2. In how many zeros does the standard decimal representation of the quantity (100!) end?

3. Let [x] denote the largest integer less than or equal to x (i.e. the “floor” function).

So, for instance, [13.791] = 13, [p] = 3, and [– 2.7] = –3.

₁

Compute the value of ò [log₂(x)] dx.

⁰

4. Two integers are chosen at random (independently, uniformly, and with repetition allowed) from the set {1, 2, 3, …, N}. Show that the probability that the sum of the two integers is even is never less than the probability that the sum is odd, regardless of the value of N.

5. For n any positive integer define the function f₁(n) as follows:

f1(n) =

{

(n/2) if n is even (3n + 5) if n is odd

Then recursively define f_N(n) = f₁( f _N–1(n) ) for N ³ 2.

Determine the value of f_1,000,000(7).

6. Suppose x and y are integers such that (7x + 5y) is a multiple of 13. Prove that (3x + 4y) must then also be a multiple of 13.

7. Is it possible for a function f : R à R to satisfy the following condition:

f(x + y) = f(x) + y³ for all real numbers x and y?

If so, find such a function. If not, prove it.