Let $\{\varepsilon_n\}_{n=1}^{\infty}$ be a sequence of positive real numbers, such that $\lim\limits_{n \to \infty} \varepsilon_n = 0$. Find $$\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \ln \left( \frac{k}{n} + \varepsilon_n \right),$$ where $\ln$ denotes the natural logarithm.

Suppose $\sum\limits_{n=1}^{\infty} a_n$ converges. Do the following sums have to converge as well?

a) $a_1 + a_2 + a_4 + a_3 + a_8 + a_7 + a_6 + a_5 + a_{16} + a_{15} + \cdots + a_9 + a_{32} + \cdots$

b) $a_1 + a_2 + a_3 + a_4 + a_5 + a_7 + a_6 + a_8 + a_9 + a_{11} + a_{13} + a_{15} + a_{10} + a_{12} + a_{14} + a_{16} + a_{17} + a_{19} + \cdots$

Justify your answers.

Let $A$ and $B$ be real $n \times n$ matrices such that $A^2 + B^2 = AB$. Prove that if $BA - AB$ is an invertible matrix then $n$ is divisible by 3.

Let $\alpha$ be a real number, $1 < \alpha < 2$.

a) Show that $\alpha$ has a unique representation as an infinite product $$\alpha = \left( 1 + \frac{1}{n_1} \right) \left( 1 + \frac{1}{n_2} \right) \dots$$ where each $n_i$ is a positive integer satisfying $$n_i^2 \le n_{i+1}.$$

b) Show that $\alpha$ is rational if and only if its infinite product has the following property:

For some $m$ and all $k \ge m$, $$n_{k+1} = n_k^2.$$

For a natural $n$ consider the hyperplane $$\mathbb{R}^n_0 = \left\{ x = (x_1, x_2, \dots, x_n) \in \mathbb{R}^n : \sum_{i=1}^{n} x_i = 0 \right\}$$ and the lattice $\mathbb{Z}^n_0 = \{ y \in \mathbb{R}^n_0 : \text{all } y_i \text{ are integers} \}$. Define the (quasi–)norm in $\mathbb{R}^n$ by $\|x\|_p = \left( \sum\limits_{i=1}^{n} |x_i|^p \right)^{1/p}$ if $0 < p < \infty$, and $\|x\|_\infty = \max\limits_i |x_i|$.

a) Let $x \in \mathbb{R}^n_0$ be such that $$\max_i x_i - \min_i x_i \le 1.$$ For every $p \in [1, \infty]$ and for every $y \in \mathbb{Z}^n_0$ prove that $$\|x\|_p \le \|x + y\|_p.$$

b) For every $p \in (0,1)$, show that there is an $n$ and an $x \in \mathbb{R}^n_0$ with $\max\limits_i x_i - \min\limits_i x_i \le 1$ and an $y \in \mathbb{Z}^n_0$ such that $$\|x\|_p > \|x + y\|_p.$$

Suppose that $F$ is a family of finite subsets of $\mathbb{N}$ and for any two sets $A, B \in F$ we have $A \cap B \ne \emptyset$.

a) Is it true that there is a finite subset $Y$ of $\mathbb{N}$ such that for any $A, B \in F$ we have $A \cap B \cap Y \ne \emptyset$?

b) Is the statement a) true if we suppose in addition that all of the members of $F$ have the same size?

Justify your answers.

Let $f$ be a $C^3(\mathbb{R})$ non-negative function, $f(0) = f'(0) = 0$, $0 < f''(0)$. Let $$g(x) = \left( \frac{\sqrt{f(x)}}{f'(x)} \right)'$$ for $x \ne 0$ and $g(0) = 0$. Show that $g$ is bounded in some neighbourhood of 0. Does the theorem hold for $f \in C^2(\mathbb{R})$?

Let $M$ be an invertible matrix of dimension $2n \times 2n$, represented in block form as $$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \quad \text{and} \quad M^{-1} = \begin{bmatrix} E & F \\ G & H \end{bmatrix}.$$ Show that $\det M \cdot \det H = \det A$.

Show that $\sum\limits_{n=1}^{\infty} \dfrac{(-1)^{n-1} \sin{(\log n)}}{n^\alpha}$ converges if and only if $\alpha > 0$.

a) Let the mapping $f : M_n \to \mathbb{R}$ from the space $M_n = \mathbb{R}^{n^2}$ of $n \times n$ matrices with real entries to reals be linear, i.e.: $$\tag{1} f(A + B) = f(A) + f(B), \quad f(cA) = c f(A)$$ for any $A, B \in M_n$, $c \in \mathbb{R}$. Prove that there exists a unique matrix $C \in M_n$ such that $f(A) = \operatorname{tr}(AC)$ for any $A \in M_n$. (If $A = \{a_{ij}\}_{i,j=1}^{n}$ then $\operatorname{tr}(A) = \sum\limits_{i=1}^{n} a_{ii}$).

b) Suppose in addition to (1) that $$\tag{2} f(A \cdot B) = f(B \cdot A)$$ for any $A, B \in M_n$. Prove that there exists $\lambda \in \mathbb{R}$ such that $f(A) = \lambda \cdot \operatorname{tr}(A)$.

Let $X$ be an arbitrary set, let $f$ be an one-to-one function mapping $X$ onto itself. Prove that there exist mappings $g_1, g_2 : X \to X$ such that $f = g_1 \circ g_2$ and $g_1 \circ g_1 = id = g_2 \circ g_2$, where $id$ denotes the identity mapping on $X$.

Let $f : [0,1] \to \mathbb{R}$ be a continuous function. Say that $f$ “crosses the axis” at $x$ if $f(x) = 0$ but in any neighbourhood of $x$ there are $y$, $z$ with $f(y) < 0$ and $f(z) > 0$.

a) Give an example of a continuous function that “crosses the axis” infiniteley often.

b) Can a continuous function “cross the axis” uncountably often?

Justify your answer.