a) Let $A$ be a $n \times n$, $n \ge 2$, symmetric, invertible matrix with real positive elements. Show that $z_n \le n^2 - 2n$, where $z_n$ is the number of zero elements in $A^{-1}$.

b) How many zero elements are there in the inverse of the $n \times n$ matrix $$A = \begin{pmatrix} 1 & 1 & 1 & 1 & \dots & 1 \\ 1 & 2 & 2 & 2 & \dots & 2 \\ 1 & 2 & 1 & 1 & \dots & 1 \\ 1 & 2 & 1 & 2 & \dots & 2 \\ \dots & \dots & \dots & \dots & \dots & \dots \\ 1 & 2 & 1 & 2 & \dots & \dots \end{pmatrix}?$$

Let $f \in C^1(a,b)$, $\lim\limits_{x \to a+} f(x) = +\infty$, $\lim\limits_{x \to b-} f(x) = -\infty$ and $f'(x) + f^2(x) \ge -1$ for $x \in (a,b)$. Prove that $b - a \ge \pi$ and give an example where $b - a = \pi$.

Given a set $S$ of $2n-1$, $n \in \mathbb{N}$, different irrational numbers. Prove that there are $n$ different elements $x_1, x_2, \dots, x_n \in S$ such that for all non-negative rational numbers $a_1, a_2, \dots, a_n$ with $a_1 + a_2 + \cdots + a_n > 0$ we have that $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n$ is an irrational number.

Let $\alpha \in \mathbb{R} \setminus \{0\}$ and suppose that $F$ and $G$ are linear maps (operators) from $\mathbb{R}^n$ into $\mathbb{R}^n$ satisfying $F \circ G - G \circ F = \alpha F$.

a) Show that for all $k \in \mathbb{N}$ one has $F^k \circ G - G \circ F^k = \alpha k F^k$.

b) Show that there exists $k \ge 1$ such that $F^k = 0$.

a) Let $f \in C[0,b]$, $g \in C(\mathbb{R})$ and let $g$ be periodic with period $b$. Prove that $\int\limits_0^b f(x) g(nx)\,dx$ has a limit as $n \to \infty$ and $$\lim_{n \to \infty} \int_0^b f(x) g(nx)\,dx = \frac{1}{b} \int_0^b f(x)\,dx \cdot \int_0^b g(x)\,dx.$$

b) Find $$\lim_{n \to \infty} \int_0^{\pi} \frac{\sin x}{1 + 3\cos^2 nx}\,dx.$$

Let $f \in C^2[0,N]$ and $|f'(x)| < 1$, $f''(x) > 0$ for every $x \in [0,N]$. Let $0 \le m_0 < m_1 < \cdots < m_k \le N$ be integers such that $n_i = f(m_i)$ are also integers for $i = 0,1,\dots,k$. Denote $b_i = n_i - n_{i-1}$ and $a_i = m_i - m_{i-1}$ for $i = 1,2,\dots,k$.

a) Prove that $$-1 < \frac{b_1}{a_1} < \frac{b_2}{a_2} < \cdots < \frac{b_k}{a_k} < 1.$$

b) Prove that for every choice of $A > 1$ there are no more than $N/A$ indices $j$ such that $a_j > A$.

c) Prove that $k \le 3 N^{2/3}$ (i.e. there are no more than $3 N^{2/3}$ integer points on the curve $y = f(x)$, $x \in [0,N]$).

Let $f \in C^1[a,b]$, $f(a) = 0$ and suppose that $\lambda \in \mathbb{R}$, $\lambda > 0$, is such that $$|f'(x)| \le \lambda |f(x)|$$ for all $x \in [a,b]$. Is it true that $f(x) = 0$ for all $x \in [a,b]$?

Let $f : \mathbb{R}^2 \to \mathbb{R}$ be given by $f(x,y) = (x^2 - y^2) e^{-x^2 - y^2}$.

a) Prove that $f$ attains its minimum and its maximum.

b) Determine all points $(x,y)$ such that $\frac{\partial f}{\partial x}(x,y) = \frac{\partial f}{\partial y}(x,y) = 0$ and determine for which of them $f$ has global or local minimum or maximum.

Let $f$ be a real-valued function with $n+1$ derivatives at each point of $\mathbb{R}$. Show that for each pair of real numbers $a$, $b$, $a < b$, such that $$\ln \left( \frac{f(b) + f'(b) + \cdots + f^{(n)}(b)} {f(a) + f'(a) + \cdots + f^{(n)}(a)} \right) = b - a$$ there is a number $c$ in the open interval $(a,b)$ for which $$f^{(n+1)}(c) = f(c).$$ Note that $\ln$ denotes the natural logarithm.

Let $A$ be a $n \times n$ diagonal matrix with characteristic polynomial $$(x - c_1)^{d_1} (x - c_2)^{d_2} \dots (x - c_k)^{d_k},$$ where $c_1, c_2, \dots, c_k$ are distinct (which means that $c_1$ appears $d_1$ times on the diagonal, $c_2$ appears $d_2$ times on the diagonal, etc. and $d_1 + d_2 + \cdots + d_k = n$). Let $V$ be the space of all $n \times n$ matrices $B$ such that $AB = BA$. Prove that the dimension of $V$ is $$d_1^2 + d_2^2 + \cdots + d_k^2.$$

Let $x_1, x_2, \dots, x_k$ be vectors of $m$-dimensional Euclidian

space, such that $x_1 + x_2 + \cdots + x_k = 0$. Show that there exists a permutation $\pi$ of the integers $\{1,2,\dots,k\}$ such that $$\biggl\| \sum_{i=1}^{n} x_{\pi(i)} \biggr\| \le \left( \sum_{i=1}^{k} \| x_i \|^2 \right)^{1/2} \quad \text{for each } n = 1,2,\dots,k.$$ Note that $\|\cdot\|$ denotes the Euclidian norm.

Find $$\lim_{N \to \infty} \frac{\ln^2 N}{N} \sum_{k=2}^{N-2} \frac{1}{\ln k \cdot \ln (N-k)}.$$ Note that $\ln$ denotes the natural logarithm.