Problem 1
Evaluate $$ \int_0^{\pi/2} \dfrac{\cos^4 x + \sin x \cos^3 x + \sin^2 x \cos^2 x + \sin^3 x \cos x} {\sin^4 x + \cos^4 x + 2 \sin x \cos^3 x + 2 \sin^2 x \cos^2 x + 2 \sin^3 x \cos x} dx. $$
Problem 2
Solve in real numbers the equation $3x - x^3 = \sqrt{x+2}$.
Problem 3
Find nonzero complex numbers $a, b, c, d, e$ such that
$$ \begin{align*} a + b + c + d + e & = -1 \newline a^2 + b^2 + c^2 + d^2 + e^2 & = 15 \newline \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e} & = -1 \newline \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} + \frac{1}{d^2} + \frac{1}{e^2} & = 15 \newline abcde & = -1 \end{align*} $$
Problem 4
Define $f (n)$ for $n$ a positive integer by $f (1) = 3$ and $f (n + 1) = 3^{f (n)}$. What are the last two digits of $f (2012)$?
Problem 5
Determine whether the series $\displaystyle \sum_{n=2}^\infty \frac{1}{\ln n} - \left(\frac{1}{\ln n}\right)^{(n+1)/n}$ is convergent.
Problem 6
Define a sequence $(a_n)$ for $n$ a positive integer inductively by $a_1 = 1$ and $a_n = \dfrac{n}{\underset{1 \le d < n}{\prod_{d|n} a_d}}$. Thus $a_2 = 2$, $a_3 = 3$, $a_4 = 2$, etc. Find $a_{999000}$.
Problem 7
Let $A_1, A_2, A_3$ be $2 \times 2$ matrices with entries in $\mathbb{C}$ (the complex numbers). Let $\text{tr}$ denote the trace of a matrix (so $\text{tr} \begin{pmatrix} a & b \newline c & d \end{pmatrix} = a + d$). Suppose ${A_1, A_2, A_3}$ is closed under matrix multiplication (i.e., given $i, j$, there exists $k$ such that $A_i A_j = A_k$), and $\text{tr}(A_1 + A_2 + A_3) \neq 3$. Prove that there exists $i$ such that $A_i A_j = A_j A_i$ for all $j$ (here $i, j$ are 1, 2 or 3).