Let $A$ be the $n \times n$ matrix, whose $(i,j)$th entry is $i + j$ for all $i, j = 1, 2, \dots, n$. What is the rank of $A$?

For an integer $n \ge 3$ consider the sets $$S_n = \{ (x_1, x_2, \dots, x_n) : \forall i\ x_i \in \{0, 1, 2\} \}$$ $$A_n = \{ (x_1, x_2, \dots, x_n) \in S_n : \forall i \le n-2\ |\{x_i, x_{i+1}, x_{i+2}\}| \ne 1 \}$$ and $$B_n = \{ (x_1, x_2, \dots, x_n) \in S_n : \forall i \le n-1\ (x_i = x_{i+1} \Rightarrow x_i \ne 0) \}.$$ Prove that $|A_{n+1}| = 3 \cdot |B_n|$.

($|A|$ denotes the number of elements of the set $A$.)

Let $f : \mathbb{R} \to [0, \infty)$ be a continuously differentiable function. Prove that $$\left| \int_0^1 f^3(x)\,dx - f^2(0) \int_0^1 f(x)\,dx \right| \le \max_{0 \le x \le 1} |f'(x)| \left( \int_0^1 f(x)\,dx \right)^2.$$

Find all polynomials $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ ($a_n \ne 0$) satisfying the following two conditions:

(i) $(a_0, a_1, \dots, a_n)$ is a permutation of the numbers $(0, 1, \dots, n)$

and

(ii) all roots of $P(x)$ are rational numbers.

Let $f : (0, \infty) \to \mathbb{R}$ be a twice continuously differentiable function such that $$|f''(x) + 2x f'(x) + (x^2 + 1) f(x)| \le 1$$ for all $x$. Prove that $\lim\limits_{x \to \infty} f(x) = 0$.

Given a group $G$, denote by $G(m)$ the subgroup generated by the $m$th powers of elements of $G$. If $G(m)$ and $G(n)$ are commutative, prove that $G(\gcd(m, n))$ is also commutative. ($\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$.)

Let $f(x) = x^2 + bx + c$, where $b$ and $c$ are real numbers, and let $$M = \{ x \in \mathbb{R} : |f(x)| < 1 \}.$$ Clearly the set $M$ is either empty or consists of disjoint open intervals. Denote the sum of their lengths by $|M|$. Prove that $$|M| \le 2\sqrt{2}.$$

Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $(f(x))^n$ is a polynomial for every $n = 2, 3, \dots$. Does it follow that $f$ is a polynomial?

In the linear space of all real $n \times n$ matrices, find the maximum possible dimension of a linear subspace $V$ such that $$\forall X, Y \in V \quad \operatorname{trace}(XY) = 0.$$ (The trace of a matrix is the sum of the diagonal entries.)

Prove that if $f : \mathbb{R} \to \mathbb{R}$ is three times differentiable, then there exists a real number $\xi \in (-1, 1)$ such that $$\frac{f'''(\xi)}{6} = \frac{f(1) - f(-1)}{2} - f'(0).$$

Find all $r > 0$ such that whenever $f : \mathbb{R}^2 \to \mathbb{R}$ is a differentiable function such that $|\operatorname{grad} f(0,0)| = 1$ and $|\operatorname{grad} f(u) - \operatorname{grad} f(v)| \le |u - v|$ for all $u, v \in \mathbb{R}^2$, then the maximum of $f$ on the disk $\{ u \in \mathbb{R}^2 : |u| \le r \}$ is attained at exactly one point. ($\operatorname{grad} f(u) = (\partial_1 f(u), \partial_2 f(u))$ is the gradient vector of $f$ at the point $u$. For a vector $u = (a, b)$, $|u| = \sqrt{a^2 + b^2}$.)

Prove that if $p$ and $q$ are rational numbers and $r = p + q\sqrt{7}$, then there exists a matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \ne \pm \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ with integer entries and with $ad - bc = 1$ such that $$\frac{ar + b}{cr + d} = r.$$