25th Annual
Virginia Tech Regional Mathematics Contest
From 8:30a.m. to 11:00a.m., November 1, 2003

Fill out the individual registration form

1. An investor buys stock worth $10,000 and holds it for n business days. Each day he has an equal chance of either gaining 20% or losing 10%. However in the case he gains every day (i.e. n gains of 20%), he is deemed to have lost all his money, because he must have been involved with insider trading. Find a (simple) formula, with proof, of the amount of money he will have on average at the end of the n days.

2. Find S_{n = 1}^$\infty$xⁿ/(n(n + 1)) = x/(1*2) + x²/(2*3) + x³/(3*4) + ... for | x| < 1.

3. Determine all invertible 2 by 2 matrices A with complex numbers as entries satisfying A = A^-1 = A', where A' denotes the transpose of A.

4. It is known that 2cos³p/7 - cos²p/7 - cosp/7 is a rational number. Write this rational number in the form p/q, where p and q are integers with q positive.

5. In the diagram below, X is the midpoint of BC, Y is the midpoint of AC, and Z is the midpoint of AB. Also / ABC + / PQC = / ACB + / PRB = 90^o. Prove that / PXC = 90^o.

   
   
   

6. Let f : [0, 1] - > [0, 1] be a continuous function such that f (f (f (x))) = x for all x e [0, 1]. Prove that f (x) = x for all x e [0, 1]. Here [0, 1] denotes the closed interval of all real numbers between 0 and 1, including 0 and 1.

7. Let T be a solid tetrahedron whose edges all have length 1. Determine the volume of the region consisting of points which are at distance at most 1 from some point in T (your answer should involve $\sqrt{2}$,$\sqrt{3}$,p).