Let $S$ be an infinite set of real numbers such that $|s_1 + s_2 + \dots + s_k| < 1$ for every finite subset $\{s_1, s_2, \dots, s_k\} \subset S$. Show that $S$ is countable.

Let $P(x) = x^2 - 1$. How many distinct real solutions does the following equation have: $$\underbrace{P(P(\dots(P}_{2004}(x))\dots)) = 0\ ?$$

Let $S_n$ be the set of all sums $\sum\limits_{k=1}^{n} x_k$, where $n \ge 2$, $0 \le x_1, x_2, \dots, x_n \le \frac{\pi}{2}$ and $$\sum_{k=1}^{n} \sin x_k = 1.$$

a) Show that $S_n$ is an interval. [10 points]

b) Let $l_n$ be the length of $S_n$. Find $\lim\limits_{n \to \infty} l_n$. [10 points]

Suppose $n \ge 4$ and let $M$ be a finite set of $n$ points in $\mathbb{R}^3$, no four of which lie in a plane. Assume that the points can be coloured black or white so that any sphere which intersects $M$ in at least four points has the property that exactly half of the points in the intersection of $M$ and the sphere are white. Prove that all of the points in $M$ lie on one sphere.

Let $X$ be a set of $\binom{2k-4}{k-2} + 1$ real numbers, $k \ge 2$. Prove that there exists a monotone sequence $\{x_i\}_{i=1}^{k} \subseteq X$ such that $$|x_{i+1} - x_1| \ge 2 |x_i - x_1|$$ for all $i = 2, \dots, k-1$.

For every complex number $z \notin \{0, 1\}$ define $$f(z) := \sum (\log z)^{-4},$$ where the sum is over all branches of the complex logarithm.

a) Show that there are two polynomials $P$ and $Q$ such that $f(z) = P(z)/Q(z)$ for all $z \in \mathbb{C} \setminus \{0,1\}$. [10 points]

b) Show that for all $z \in \mathbb{C} \setminus \{0,1\}$ $$f(z) = z\, \frac{z^2 + 4z + 1}{6 (z-1)^4}.$$ [10 points]

Let $A$ be a real $4 \times 2$ matrix and $B$ be a real $2 \times 4$ matrix such that $$AB = \begin{pmatrix} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \end{pmatrix}.$$ Find $BA$.

Let $f, g : [a, b] \to [0, \infty)$ be continuous and non-decreasing functions such that for each $x \in [a, b]$ we have $$\int_a^x \sqrt{f(t)}\,dt \le \int_a^x \sqrt{g(t)}\,dt$$ and $\int_a^b \sqrt{f(t)}\,dt = \int_a^b \sqrt{g(t)}\,dt$.

Prove that $\int_a^b \sqrt{1 + f(t)}\,dt \ge \int_a^b \sqrt{1 + g(t)}\,dt$.

Let $D$ be the closed unit disk in the plane, and let $p_1, p_2, \dots, p_n$ be fixed points in $D$. Show that there exists a point $p$ in $D$ such that the sum of the distances of $p$ to each of $p_1, p_2, \dots, p_n$ is greater than or equal to 1.

For $n \ge 1$ let $M$ be an $n \times n$ complex matrix with distinct eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_k$, with multiplicities $m_1, m_2, \dots, m_k$, respectively. Consider the linear operator $L_M$ defined by $L_M(X) = MX + XM^T$, for any complex $n \times n$ matrix $X$. Find its eigenvalues and their multiplicities. ($M^T$ denotes the transpose of $M$; that is, if $M = (m_{k,l})$, then $M^T = (m_{l,k})$.)

Prove that $$\int_0^1 \int_0^1 \frac{dx\,dy}{x^{-1} + |\ln y| - 1} \le 1.$$

For $n \ge 0$ define matrices $A_n$ and $B_n$ as follows: $A_0 = B_0 = (1)$ and for every $n > 0$ $$A_n = \begin{pmatrix} A_{n-1} & A_{n-1} \\ A_{n-1} & B_{n-1} \end{pmatrix} \quad \text{and} \quad B_n = \begin{pmatrix} A_{n-1} & A_{n-1} \\ A_{n-1} & 0 \end{pmatrix}.$$ Denote the sum of all elements of a matrix $M$ by $S(M)$. Prove that $S(A_n^{k-1}) = S(A_k^{n-1})$ for every $n, k \ge 1$.