Virginia Tech Regional Mathematics Contest

- 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. - Find
S
_{n = 1}^{}*x*^{n}/(*n*(*n*+ 1)) =*x*/(1*2) +*x*^{2}/(2*3) +*x*^{3}/(3*4) + ... for |*x*| < 1. - 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*. - It is known that
2cos
^{3}p/7 - cos^{2}p/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. - 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}.

- 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. - 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 ,,p).