Let $n$ be a positive integer. Consider an $n \times n$ matrix with entries $1, 2, \dots, n^2$ written in order starting top left and moving along each row in turn left–to–right. We choose $n$ entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the sum of the selected entries?

Let $r, s, t$ be positive integers which are pairwise relatively prime. If $a$ and $b$ are elements of a commutative multiplicative group with unity element $e$, and $a^r = b^s = (ab)^t = e$, prove that $a = b = e$.

Does the same conclusion hold if $a$ and $b$ are elements of an arbitrary non-commutative group?

Find $\lim\limits_{t \nearrow 1} (1-t) \sum\limits_{n=1}^{\infty} \dfrac{t^n}{1 + t^n}$, where $t \nearrow 1$ means that $t$ approaches 1 from below.

Let $k$ be a positive integer. Let $p(x)$ be a polynomial of degree $n$ each of whose coefficients is $-1$, $1$ or $0$, and which is divisible by $(x-1)^k$. Let $q$ be a prime such that $\dfrac{q}{\ln q} < \dfrac{k}{\ln(n+1)}$. Prove that the complex $q$th roots of unity are roots of the polynomial $p(x)$.

Let $A$ be an $n \times n$ complex matrix such that $A \ne \lambda I$ for all $\lambda \in \mathbb{C}$. Prove that $A$ is similar to a matrix having at most one non-zero entry on the main diagonal.

Suppose that the differentiable functions $a, b, f, g : \mathbb{R} \to \mathbb{R}$ satisfy $$f(x) \ge 0,\ f'(x) \ge 0,\ g(x) > 0,\ g'(x) > 0 \quad \text{for all } x \in \mathbb{R},$$ $$\lim_{x \to \infty} a(x) = A > 0, \quad \lim_{x \to \infty} b(x) = B > 0, \quad \lim_{x \to \infty} f(x) = \lim_{x \to \infty} g(x) = \infty,$$ and $$\frac{f'(x)}{g'(x)} + a(x) \frac{f(x)}{g(x)} = b(x).$$ Prove that $$\lim_{x \to \infty} \frac{f(x)}{g(x)} = \frac{B}{A + 1}.$$

Let $r, s \ge 1$ be integers and $a_0, a_1, \dots, a_{r-1}, b_0, b_1, \dots, b_{s-1}$ be real non-negative numbers such that $$\begin{align*} (a_0 + a_1 x + a_2 x^2 + \dots + a_{r-1} x^{r-1} + x^r) &(b_0 + b_1 x + b_2 x^2 + \dots + b_{s-1} x^{s-1} + x^s) = \\ &= 1 + x + x^2 + \dots + x^{r+s-1} + x^{r+s}. \end{align*}$$ Prove that each $a_i$ and each $b_j$ equals either 0 or 1.

Let $a_0 = \sqrt{2}$, $b_0 = 2$, $a_{n+1} = \sqrt{2 - \sqrt{4 - a_n^2}}$, $b_{n+1} = \dfrac{2 b_n}{2 + \sqrt{4 + b_n^2}}$.

a) Prove that the sequences $(a_n)$, $(b_n)$ are decreasing and converge to 0.

b) Prove that the sequence $(2^n a_n)$ is increasing, the sequence $(2^n b_n)$ is decreasing and that these two sequences converge to the same limit.

c) Prove that there is a positive constant $C$ such that for all $n$ the following inequality holds: $0 < b_n - a_n < \dfrac{C}{8^n}$.

Find the maximum number of points on a sphere of radius 1 in $\mathbb{R}^n$ such that the distance between any two of these points is strictly greater than $\sqrt{2}$.

Let $A = (a_{k,\ell})_{k,\ell = 1,\dots,n}$ be an $n \times n$ complex matrix such that for each $m \in \{1, \dots, n\}$ and $1 \le j_1 < \dots < j_m \le n$ the determinant of the matrix $(a_{j_k, j_\ell})_{k,\ell = 1,\dots,m}$ is zero. Prove that $A^n = 0$ and that there exists a permutation $\sigma \in S_n$ such that the matrix $$(a_{\sigma(k), \sigma(\ell)})_{k,\ell = 1,\dots,n}$$ has all of its nonzero elements above the diagonal.

Let $\mathbb{R}$ be the set of real numbers. Prove that there is no function $f : \mathbb{R} \to \mathbb{R}$ with $f(0) > 0$, and such that $$f(x + y) \ge f(x) + y f(f(x)) \quad \text{for all } x, y \in \mathbb{R}.$$

For each positive integer $n$, let $f_n(\vartheta) = \sin\vartheta \cdot \sin(2\vartheta) \cdot \sin(4\vartheta) \cdots \sin(2^n \vartheta)$. For all real $\vartheta$ and all $n$, prove that $$|f_n(\vartheta)| \le \frac{2}{\sqrt{3}} |f_n(\pi/3)|.$$