Let for $j = 0, \dots, n$, $a_j = a_0 + jd$, where $a_0$, $d$ are fixed real numbers. Put $$A = \begin{pmatrix} a_0 & a_1 & a_2 & \dots & a_n \\ a_1 & a_0 & a_1 & \dots & a_{n-1} \\ a_2 & a_1 & a_0 & \dots & a_{n-2} \\ \dots & \dots & \dots & \dots & \dots \\ a_n & a_{n-1} & a_{n-2} & \dots & a_0 \end{pmatrix}.$$ Calculate $\det(A)$, where $\det(A)$ denotes the determinant of $A$.

Evaluate the definite integral $$\int_{-\pi}^{\pi} \frac{\sin nx}{(1 + 2^x) \sin x}\,dx,$$ where $n$ is a natural number.

The linear operator $A$ on the vector space $V$ is called an involution if $A^2 = E$ where $E$ is the identity operator on $V$. Let $\dim V = n < \infty$.

(i) Prove that for every involution $A$ on $V$ there exists a basis of $V$ consisting of eigenvectors of $A$.

(ii) Find the maximal number of distinct pairwise commuting involutions on $V$.

Let $a_1 = 1$, $a_n = \dfrac{1}{n} \sum\limits_{k=1}^{n-1} a_k a_{n-k}$ for $n \ge 2$. Show that

(i) $\limsup\limits_{n \to \infty} |a_n|^{1/n} < 2^{-1/2}$;

(ii) $\limsup\limits_{n \to \infty} |a_n|^{1/n} \ge 2/3$.

(i) Let $a$, $b$ be real numbers such that $b \le 0$ and $1 + ax + bx^2 \ge 0$ for every $x$ in $[0,1]$. Prove that $$\lim_{n \to +\infty} n \int_0^1 (1 + ax + bx^2)^n dx = \begin{cases} -\dfrac{1}{a} & \text{if } a < 0, \\ +\infty & \text{if } a \ge 0. \end{cases}$$

(ii) Let $f : [0,1] \to [0,\infty)$ be a function with a continuous second derivative and let $f''(x) \le 0$ for every $x$ in $[0,1]$. Suppose that $L = \lim\limits_{n \to \infty} n \int_0^1 (f(x))^n dx$ exists and $0 < L < +\infty$. Prove that $f'$ has a constant sign and $\min\limits_{x \in [0,1]} |f'(x)| = L^{-1}$.

Upper content of a subset $E$ of the plane $\mathbb{R}^2$ is defined as $$\mathcal{C}(E) = \inf \left\{ \sum_{i=1}^{n} \operatorname{diam}(E_i) \right\}$$ where $\inf$ is taken over all finite families of sets $E_1, \dots, E_n$, $n \in \mathbb{N}$, in $\mathbb{R}^2$ such that $E \subset \bigcup\limits_{i=1}^{n} E_i$.

Lower content of $E$ is defined as $$\begin{align*} \mathcal{K}(E) = \sup \{ \operatorname{lenght}(L) :{}& \text{$L$ is a closed line segment} \\ & \text{onto which $E$ can be contracted} \}.$$

\end{align*}K(E)=sup{lenght(L):​L is a closed line segmentonto which E can be contracted}.​ Show that

(a) $\mathcal{C}(L) = \operatorname{lenght}(L)$ if $L$ is a closed line segment;

(b) $\mathcal{C}(E) \ge \mathcal{K}(E)$;

(c) the equality in (b) needs not hold even if $E$ is compact.

Hint. If $E = T \cup T'$ where $T$ is the triangle with vertices $(-2,2)$, $(2,2)$ and $(0,4)$, and $T'$ is its reflexion about the $x$-axis, then $\mathcal{C}(E) = 8 > \mathcal{K}(E)$.

Remarks: All distances used in this problem are Euclidian. Diameter of a set $E$ is $\operatorname{diam}(E) = \sup \{ \operatorname{dist}(x,y) : x, y \in E \}$. Contraction of a set $E$ to a set $F$ is a mapping $f : E \mapsto F$ such that $\operatorname{dist}(f(x), f(y)) \le \operatorname{dist}(x,y)$ for all $x, y \in E$. A set $E$ can be contracted onto a set $F$ if there is a contraction $f$ of $E$ to $F$ which is onto, i.e., such that $f(E) = F$. Triangle is defined as the union of the three segments joining its vertices, i.e., it does not contain the interior.

Prove that if $f : [0,1] \to [0,1]$ is a continuous function, then the sequence of iterates $x_{n+1} = f(x_n)$ converges if and only if $$\lim_{n \to \infty} (x_{n+1} - x_n) = 0.$$

Let $\theta$ be a positive real number and let $\cosh t = \dfrac{e^t + e^{-t}}{2}$ denote the hyperbolic cosine. Show that if $k \in \mathbb{N}$ and both $\cosh k\theta$ and $\cosh(k+1)\theta$ are rational, then so is $\cosh \theta$.

Let $G$ be the subgroup of $GL_2(\mathbb{R})$, generated by $A$ and $B$, where $$A = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}.$$ Let $H$ consist of those matrices $\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ in $G$ for which $a_{11} = a_{22} = 1$.

(a) Show that $H$ is an abelian subgroup of $G$.

(b) Show that $H$ is not finitely generated.

Remarks. $GL_2(\mathbb{R})$ denotes, as usual, the group (under matrix multiplication) of all $2 \times 2$ invertible matrices with real entries (elements). Abelian means commutative. A group is finitely generated if there are a finite number of elements of the group such that every other element of the group can be obtained from these elements using the group operation.

Let $B$ be a bounded closed convex symmetric (with respect to the origin) set in $\mathbb{R}^2$ with boundary the curve $\Gamma$. Let $B$ have the property that the ellipse of maximal area contained in $B$ is the disc $D$ of radius 1 centered at the origin with boundary the circle $C$. Prove that $A \cap \Gamma \ne \emptyset$ for any arc $A$ of $C$ of length $l(A) \ge \dfrac{\pi}{2}$.

(i) Prove that $$\lim_{x \to +\infty} \sum_{n=1}^{\infty} \frac{nx}{(n^2 + x)^2} = \frac{1}{2}.$$

(ii) Prove that there is a positive constant $c$ such that for every $x \in [1, \infty)$ we have $$\left| \sum_{n=1}^{\infty} \frac{nx}{(n^2 + x)^2} - \frac{1}{2} \right| \le \frac{c}{x}.$$

(Carleman's inequality)

(i) Prove that for every sequence $\{a_n\}_{n=1}^{\infty}$, such that $a_n > 0$, $n = 1, 2, \dots$ and $\sum\limits_{n=1}^{\infty} a_n < \infty$, we have $$\sum_{n=1}^{\infty} (a_1 a_2 \cdots a_n)^{1/n} < e \sum_{n=1}^{\infty} a_n,$$ where $e$ is the natural log base.

(ii) Prove that for every $\varepsilon > 0$ there exists a sequence $\{a_n\}_{n=1}^{\infty}$, such that $a_n > 0$, $n = 1, 2, \dots$, $\sum\limits_{n=1}^{\infty} a_n < \infty$ and $$\sum_{n=1}^{\infty} (a_1 a_2 \cdots a_n)^{1/n} > (e - \varepsilon) \sum_{n=1}^{\infty} a_n.$$