(a) Let $a_1, a_2, \dots$ be a sequence of real numbers such that $a_1 = 1$ and $a_{n+1} > \frac{3}{2} a_n$ for all $n$. Prove that the sequence $$\frac{a_n}{\left( \frac{3}{2} \right)^{n-1}}$$ has a finite limit or tends to infinity. (10 points)

(b) Prove that for all $\alpha > 1$ there exists a sequence $a_1, a_2, \dots$ with the same properties such that $$\lim \frac{a_n}{\left( \frac{3}{2} \right)^{n-1}} = \alpha.$$ (10 points)

Let $a_1, a_2 \dots, a_{51}$ be non-zero elements of a field. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_1 \dots, b_{51}$. If this new sequence is a permutation of the original one, what can be the characteristic of the field? (The characteristic of a field is $p$, if $p$ is the smallest positive integer such that $\underbrace{x + x + \dots + x}_{p} = 0$ for any element $x$ of the field. If there exists no such $p$, the characteristic is 0.)

Let $A$ be an $n \times n$ real matrix such that $3 A^3 = A^2 + A + I$ ($I$ is the identity matrix). Show that the sequence $A^k$ converges to an idempotent matrix. (A matrix $B$ is called idempotent if $B^2 = B$.)

Determine the set of all pairs $(a, b)$ of positive integers for which the set of positive integers can be decomposed into two sets $A$ and $B$ such that $a \cdot A = b \cdot B$.

Let $g : [0,1] \to \mathbb{R}$ be a continuous function and let $f_n : [0,1] \to \mathbb{R}$ be a sequence of functions defined by $f_0(x) = g(x)$ and $$f_{n+1}(x) = \frac{1}{x} \int_0^x f_n(t)\,dt \quad (x \in (0,1],\ n = 0, 1, 2, \dots).$$ Determine $\lim\limits_{n \to \infty} f_n(x)$ for every $x \in (0,1]$.

Let $f(z) = a_n z^n + a_{n-1} z^{n-1} + \dots + a_1 z + a_0$ be a polynomial with real coefficients. Prove that if all roots of $f$ lie in the left half-plane $\{ z \in \mathbb{C} : \operatorname{Re} z < 0 \}$ then $$a_k a_{k+3} < a_{k+1} a_{k+2}$$ holds for every $k = 0, 1, \dots, n-3$.

Let $A$ and $B$ be $n \times n$ real matrices such that $AB + A + B = 0$. Prove that $AB = BA$.

Evaluate the limit $$\lim_{x \to 0+} \int_x^{2x} \frac{\sin^m t}{t^n}\,dt \quad (m, n \in \mathbb{N}).$$

Let $A$ be a closed subset of $\mathbb{R}^n$ and let $B$ be the set of all those points $b \in \mathbb{R}^n$ for which there exists exactly one point $a_0 \in A$ such that $$|a_0 - b| = \inf_{a \in A} |a - b|.$$ Prove that $B$ is dense in $\mathbb{R}^n$; that is, the closure of $B$ is $\mathbb{R}^n$.

Find all positive integers $n$ for which there exists a family $F$ of three-element subsets of $S = \{1, 2, \dots, n\}$ satisfying the following two conditions:

(i) for any two different elements $a, b \in S$, there exists exactly one $A \in F$ containing both $a, b$;

(ii) if $a, b, c, x, y, z$ are elements of $S$ such that if

$\{a,b,x\}, \{a,c,y\}, \{b,c,z\} \in F$, then $\{x,y,z\} \in F$.

(a) Show that for each function $f : \mathbb{Q} \times \mathbb{Q} \to \mathbb{R}$ there exists a function $g : \mathbb{Q} \to \mathbb{R}$ such that $f(x, y) \le g(x) + g(y)$ for all $x, y \in \mathbb{Q}$.

(b) Find a function $f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ for which there is no function $g : \mathbb{R} \to \mathbb{R}$ such that $f(x, y) \le g(x) + g(y)$ for all $x, y \in \mathbb{R}$.

Let $(a_n)_{n \in \mathbb{N}}$ be the sequence defined by $$a_0 = 1, \qquad a_{n+1} = \frac{1}{n+1} \sum_{k=0}^{n} \frac{a_k}{n - k + 2}.$$ Find the limit $$\lim_{n \to \infty} \sum_{k=0}^{n} \frac{a_k}{2^k},$$ if it exists.