Suppose that $f$ and $g$ are real-valued functions on the real line and $f(r) \le g(r)$ for every rational $r$. Does this imply that $f(x) \le g(x)$ for every real $x$ if

a) $f$ and $g$ are non-decreasing?

b) $f$ and $g$ are continuous?

Let $A$, $B$ and $C$ be real square matrices of the same size, and suppose that $A$ is invertible. Prove that if $(A - B) C = B A^{-1}$, then $C (A - B) = A^{-1} B$.

In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from 1 to $n$ and let $a_i$ be the number of friends of the $i$-th resident. Suppose that $\sum\limits_{i=1}^{n} a_i^2 = n^2 - n$. Let $k$ be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of $k$.

Let $p(z) = a_0 + a_1 z + a_2 z^2 + \dots + a_n z^n$ be a complex polynomial. Suppose that $1 = c_0 \ge c_1 \ge \dots \ge c_n \ge 0$ is a sequence of real numbers which is convex (i.e. $2 c_k \le c_{k-1} + c_{k+1}$ for every $k = 1, 2, \dots, n-1$), and consider the polynomial $$q(z) = c_0 a_0 + c_1 a_1 z + c_2 a_2 z^2 + \dots + c_n a_n z^n.$$ Prove that $$\max_{|z| \le 1} \bigl| q(z) \bigr| \le \max_{|z| \le 1} \bigl| p(z) \bigr|.$$

Let $n$ be a positive integer. An $n$-simplex in $\mathbb{R}^n$ is given by $n+1$ points $P_0, P_1, \dots, P_n$, called its vertices, which do not all belong to the same hyperplane. For every $n$-simplex $S$ we denote by $v(S)$ the volume of $S$, and we write $C(S)$ for the center of the unique sphere containing all the vertices of $S$.

Suppose that $P$ is a point inside an $n$-simplex $S$. Let $S_i$ be the $n$-simplex obtained from $S$ by replacing its $i$-th vertex by $P$. Prove that $$v(S_0) C(S_0) + v(S_1) C(S_1) + \dots + v(S_n) C(S_n) = v(S) C(S).$$

Let $\ell$ be a line and $P$ a point in $\mathbb{R}^3$. Let $S$ be the set of points $X$ such that the distance from $X$ to $\ell$ is greater than or equal to two times the distance between $X$ and $P$. If the distance from $P$ to $\ell$ is $d > 0$, find the volume of $S$.

Suppose $f : \mathbb{R} \to \mathbb{R}$ is a two times differentiable function satisfying $f(0) = 1$, $f'(0) = 0$, and for all $x \in [0, \infty)$, $$f''(x) - 5 f'(x) + 6 f(x) \ge 0.$$ Prove that for all $x \in [0, \infty)$, $$f(x) \ge 3 e^{2x} - 2 e^{3x}.$$

Let $A, B \in M_n(\mathbb{C})$ be two $n \times n$ matrices such that $$A^2 B + B A^2 = 2 A B A.$$ Prove that there exists a positive integer $k$ such that $(AB - BA)^k = 0$.

Let $p$ be a prime number and $\mathbb{F}_p$ be the field of residues modulo $p$. Let $W$ be the smallest set of polynomials with coefficients in $\mathbb{F}_p$ such that

• the polynomials $x + 1$ and $x^{p-2} + x^{p-3} + \dots + x^2 + 2x + 1$ are in $W$, and
• for any polynomials $h_1(x)$ and $h_2(x)$ in $W$ the polynomial $r(x)$, which is the remainder of $h_1(h_2(x))$ modulo $x^p - x$, is also in $W$.

Let $M$ be the vector space of $m \times p$ real matrices. For a vector subspace $S \subseteq M$, denote by $\delta(S)$ the dimension of the vector space generated by all columns of all matrices in $S$.

Say that a vector subspace $T \subseteq M$ is a covering matrix space if $$\bigcup_{A \in T, A \ne 0} \ker A = \mathbb{R}^p.$$ Such a $T$ is minimal if it does not contain a proper vector subspace $S \subset T$ which is also a covering matrix space.

(a) (8 points) Let $T$ be a minimal covering matrix space and let $n = \dim T$. Prove that $$\delta(T) \le \binom{n}{2}.$$

(b) (2 points) Prove that for every positive integer $n$ we can find $m$ and $p$, and a minimal covering matrix space $T$ as above such that $\dim T = n$ and $\delta(T) = \binom{n}{2}$.