Problem 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.
Problem 2
Find $\displaystyle{\sum_{n=1}^{\infty} \frac{x^n}{n(n+1)}} = \frac{x}{1 \cdot 2} + \frac{x^2}{2 \cdot 3} + \frac{x^3}{3 \cdot 4} + \dots$ for $|x| < 1$.
Problem 3
Determine all invertible 2 by 2 matrices $A$ with complex numbers as entries satisfying $A = A^{-1} = A^\top$, where $A^\top$ denotes the transpose of $A$.
Problem 4
It is known that $2 \cos^3 \dfrac{\pi}{7} - \cos^2 \dfrac{\pi}{7} - \cos \dfrac{\pi}{7}$ is a rational number. Write this rational number in the form $p/q$, where $p$ and $q$ are integers with $q$ positive.
Problem 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 $\angle ABC + \angle PQC = \angle ACB + \angle PRB = 90^{\circ}$. Prove that $\angle PXC = 90^{\circ}$.
Problem 6
Let $f : [0, 1] \to [0, 1]$ be a continuous function such that $f(f(f(x))) = x$ for all $x \in [0, 1]$. Prove that $f(x) = x$ for all $x \in [0, 1]$. Here $[0, 1]$ denotes the closed interval of all real numbers between 0 and 1, including 0 and 1.
Problem 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 $\sqrt{2}, \sqrt{3}, \pi$).