Is it true that if $f : [0,1] \to [0,1]$ is

a) monotone increasing

b) monotone decreasing

then there exists an $x \in [0,1]$ for which $f(x) = x$?

Let $p(x) = x^5 + x$ and $q(x) = x^5 + x^2$. Find all pairs $(w, z)$ of complex numbers with $w \ne z$ for which $p(w) = p(z)$ and $q(w) = q(z)$.

$A$ and $B$ are square complex matrices of the same size and $$\operatorname{rank}(AB - BA) = 1.$$ Show that $(AB - BA)^2 = 0$.

a) Show that if $(x_i)$ is a decreasing sequence of positive numbers then $$\left( \sum_{i=1}^{n} x_i^2 \right)^{1/2} \le \sum_{i=1}^{n} \frac{x_i}{\sqrt{i}}.$$

b) Show that there is a constant $C$ so that if $(x_i)$ is a decreasing sequence of positive numbers then $$\sum_{m=1}^{\infty} \frac{1}{\sqrt{m}} \left( \sum_{i=m}^{\infty} x_i^2 \right)^{1/2} \le C \sum_{i=1}^{\infty} x_i.$$

Let $R$ be a ring of characteristic zero (not necessarily commutative). Let $e$, $f$ and $g$ be idempotent elements of $R$ satisfying $e + f + g = 0$. Show that $e = f = g = 0$.

($R$ is of characteristic zero means that, if $a \in R$ and $n$ is a positive integer, then $na \ne 0$ unless $a = 0$. An idempotent $x$ is an element satisfying $x = x^2$.)

Let $f : \mathbb{R} \to (0, \infty)$ be an increasing differentiable function for which $\lim\limits_{x \to \infty} f(x) = \infty$ and $f'$ is bounded.

Let $F(x) = \int\limits_0^x f$. Define the sequence $(a_n)$ inductively by $$\tag{1} a_0 = 1, \qquad a_{n+1} = a_n + \frac{1}{f(a_n)},$$ and the sequence $(b_n)$ simply by $b_n = F^{-1}(n)$. Prove that $\lim\limits_{n \to \infty} (a_n - b_n) = 0$.

a) Show that the unit square can be partitioned into $n$ smaller squares if $n$ is large enough.

b) Let $d \ge 2$. Show that there is a constant $N(d)$ such that, whenever $n \ge N(d)$, a $d$-dimensional unit cube can be partitioned into $n$ smaller cubes.

Let $f$ be continuous and nowhere monotone on $[0,1]$. Show that the set of points on which $f$ attains local minima is dense in $[0,1]$.

(A function is nowhere monotone if there exists no interval where the function is monotone. A set is dense if each non-empty open interval contains at least one element of the set.)

Let $p(z)$ be a polynomial of degree $n$ with complex coefficients. Prove that there exist at least $n+1$ complex numbers $z$ for which $p(z)$ is 0 or 1.

Suppose the graph of a polynomial of degree 6 is tangent to a straight line at 3 points $A_1$, $A_2$, $A_3$, where $A_2$ lies between $A_1$ and $A_3$.

a) Prove that if the lengths of the segments $A_1 A_2$ and $A_2 A_3$ are equal, then the areas of the figures bounded by these segments and the graph of the polynomial are equal as well.

b) Let $k = \dfrac{A_2 A_3}{A_1 A_2}$, and let $K$ be the ratio of the areas of the appropriate figures. Prove that $$\frac{2}{7} k^5 < K < \frac{7}{2} k^5.$$

Let $\mathbb{R}^+$ be the set of positive real numbers. Find all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that for all $x, y \in \mathbb{R}^+$ $$f(x) f(y f(x)) = f(x + y).$$

For an $m \times m$ real matrix $A$, $e^A$ is defined as $\sum\limits_{n=0}^{\infty} \dfrac{1}{n!} A^n$. (The sum is convergent for all matrices.) Prove or disprove, that for all real polynomials $p$ and $m \times m$ real matrices $A$ and $B$, $p(e^{AB})$ is nilpotent if and only if $p(e^{BA})$ is nilpotent. (A matrix $A$ is nilpotent if $A^k = 0$ for some positive integer $k$.)