Let $X$ be a nonsingular matrix with columns $X_1, X_2, \dots, X_n$. Let $Y$ be a matrix with columns $X_2, X_3, \dots, X_n, 0$. Show that the matrices $A = Y X^{-1}$ and $B = X^{-1} Y$ have rank $n-1$ and have only $0$'s for eigenvalues.

Let $f$ be a continuous function on $[0,1]$ such that for every $x \in [0,1]$ we have $\int\limits_x^1 f(t)\,dt \ge \dfrac{1-x^2}{2}$. Show that $\int\limits_0^1 f^2(t)\,dt \ge \dfrac{1}{3}$.

Let $f$ be twice continuously differentiable on $(0, +\infty)$ such that $\lim\limits_{x \to 0+} f'(x) = -\infty$ and $\lim\limits_{x \to 0+} f''(x) = +\infty$. Show that $$\lim_{x \to 0+} \frac{f(x)}{f'(x)} = 0.$$

Let $F : (1, \infty) \to \mathbb{R}$ be the function defined by $$F(x) := \int_x^{x^2} \frac{dt}{\ln t}.$$ Show that $F$ is one-to-one (i.e. injective) and find the range (i.e. set of values) of $F$.

Let $A$ and $B$ be real $n \times n$ matrices. Assume that there exist $n+1$ different real numbers $t_1, t_2, \dots, t_{n+1}$ such that the matrices $$C_i = A + t_i B, \quad i = 1, 2, \dots, n+1,$$ are nilpotent (i.e. $C_i^n = 0$).

Show that both $A$ and $B$ are nilpotent.

Let $p > 1$. Show that there exists a constant $K_p > 0$ such that for every $x, y \in \mathbb{R}$ satisfying $|x|^p + |y|^p = 2$, we have $$(x - y)^2 \le K_p \left( 4 - (x+y)^2 \right).$$

Let $A$ be $3 \times 3$ real matrix such that the vectors $Au$ and $u$ are orthogonal for each column vector $u \in \mathbb{R}^3$. Prove that:

a) $A^\top = -A$, where $A^\top$ denotes the transpose of the matrix $A$;

b) there exists a vector $v \in \mathbb{R}^3$ such that $Au = v \times u$ for every $u \in \mathbb{R}^3$, where $v \times u$ denotes the vector product in $\mathbb{R}^3$.

Let $\{b_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $b_0 = 1$, $b_n = 2 + \sqrt{b_{n-1}} - 2 \sqrt{1 + \sqrt{b_{n-1}}}$. Calculate $$\sum_{n=1}^{\infty} b_n 2^n.$$

Let all roots of an $n$-th degree polynomial $P(z)$ with complex coefficients lie on the unit circle in the complex plane. Prove that all roots of the polynomial $$2 z P'(z) - n P(z)$$ lie on the same circle.

a) Prove that for every $\varepsilon > 0$ there is a positive integer $n$ and real numbers $\lambda_1, \dots, \lambda_n$ such that $$\max_{x \in [-1,1]} \left| x - \sum_{k=1}^{n} \lambda_k x^{2k+1} \right| < \varepsilon.$$

b) Prove that for every odd continuous function $f$ on $[-1,1]$ and for every $\varepsilon > 0$ there is a positive integer $n$ and real numbers $\mu_1, \dots, \mu_n$ such that $$\max_{x \in [-1,1]} \left| f(x) - \sum_{k=1}^{n} \mu_k x^{2k+1} \right| < \varepsilon.$$ Recall that $f$ is odd means that $f(x) = -f(-x)$ for all $x \in [-1,1]$.

a) Prove that every function of the form $$f(x) = \frac{a_0}{2} + \cos x + \sum_{n=2}^{N} a_n \cos(nx)$$ with $|a_0| < 1$, has positive as well as negative values in the period $[0, 2\pi)$.

b) Prove that the function $$F(x) = \sum_{n=1}^{100} \cos(n^{\frac{3}{2}} x)$$ has at least 40 zeros in the interval $(0, 1000)$.

Suppose that $\{f_n\}_{n=1}^{\infty}$ is a sequence of continuous functions on the interval $[0,1]$ such that $$\int_0^1 f_m(x) f_n(x)\,dx = \begin{cases} 1 & \text{if } n = m \\ 0 & \text{if } n \ne m \end{cases}$$ and $$\sup \{ |f_n(x)| : x \in [0,1] \text{ and } n = 1,2,\dots \} < +\infty.$$ Show that there exists no subsequence $\{f_{n_k}\}$ of $\{f_n\}$ such that $\lim\limits_{k \to \infty} f_{n_k}(x)$ exists for all $x \in [0,1]$.