Let $f : \mathbb{R} \longrightarrow \mathbb{R}$ be defined as $$f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x^2}\right), & \text{if } x \neq 0 \\ 0, & \text{otherwise} \end{cases}$$ Choose the correct statement(s) from below:
(A) $f$ is continuous;
(B) $f$ is discontinuous at 0;
(C) $f$ is differentiable;
(D) $f$ is continuously differentiable.

Let $A \in M_{m \times n}(\mathbb{R})$ be of rank $m$. Choose the correct statement(s) from below:
(A) The map $\mathbb{R}^n \longrightarrow \mathbb{R}^m$ given by $v \mapsto Av$ is injective;
(B) There exist matrices $B \in M_m(\mathbb{R})$ and $C \in M_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$;
(C) There exist matrices $B \in \mathrm{GL}_m(\mathbb{R})$ and $C \in \mathrm{GL}_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$;
(D) For every $(B, C) \in M_m(\mathbb{R}) \times M_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$, $C$ is uniquely determined by $B$.

A step starting at a point $P$ in the $XY$-plane consists of moving by one unit from $P$ in one of three directions: directly to the right or in the direction of one of the two rays that make the angle of $\pm 120^{\circ}$ with positive $X$-axis. (An opposite move, i.e. to the left/southeast/northeast, is not allowed.) A path consists of a number of such steps, each new step starting where the previous step ended. Points and steps in a path may repeat.
Find the number of paths starting at $(1,0)$ and ending at $(2,0)$ that consist of
(i) exactly 6 steps
(ii) exactly 7 steps.

Let $f : \mathbb{C} \longrightarrow \mathbb{C}$ be an entire function such that $f(z+1) = f(z+\imath) = f(z)$ for every $z \in \mathbb{C}$. Choose the correct statement(s) from below:
(A) $f$ is constant;
(B) $f(z) = 0$ for every $z \in \mathbb{C}$;
(C) There exist complex numbers $a, b$ such that for every $x, y \in \mathbb{R}$, $f(x + \imath y) = a\sin(x) + \imath b\cos(y)$;
(D) $f$ is not necessarily constant but $|f(z)|$ is constant.

What is the cardinality of the centre of $O_2(\mathbb{R})$? (Definition: The centre of a group $G$ is $\{g \in G \mid gh = hg \text{ for every } h \in G\}$. Hint: Reflection matrices and permutation matrices are orthogonal.)
(A) 1;
(B) 2;
(C) The cardinality of $\mathbb{N}$;
(D) The cardinality of $\mathbb{R}$.

A function $f(x)$ is defined by the following formulas
$$f(x) = \begin{cases} x^{2} + 1 & \text{when } x \text{ is irrational} \\ \tan(x) & \text{when } x \text{ is rational} \end{cases}$$
At how many $x$ in the interval $[0, 4\pi]$ is $f(x)$ continuous?

Let $U \subseteq \mathbb{R}$ be a non-empty open subset. Choose the correct statement(s) from below:
(A) $U$ is uncountable;
(B) $U$ contains a closed interval as a proper subset;
(C) $U$ is a countable union of disjoint open intervals;
(D) $U$ contains a convergent sequence of real numbers.

Let $R$ be a commutative ring. The characteristic of $R$ is the smallest positive integer $n$ such that $a + a + \cdots + a$ ($n$ times) is zero for every $a \in R$, if such an integer exists, and zero, otherwise. Choose the correct statement(s) from below:
(A) For every $n \in \mathbb{N}$, there exists a commutative ring whose characteristic is $n$;
(B) There exists an integral domain with characteristic 57;
(C) The characteristic of a field is either 0 or a prime number;
(D) For every prime number $p$, every commutative ring of characteristic $p$ contains $\mathbb{F}_p$ as a subring.

Consider the $\mathbb{Q}$-vector-space $$\{ f : \mathbb{R} \longrightarrow \mathbb{R} \mid f \text{ is continuous and } \mathrm{Image}(f) \subseteq \mathbb{Q} \}$$ What is its dimension?

Let $p$ be a prime number and $F$ a field of $p^{23}$ elements. Let $\phi : F \longrightarrow F$ be the field automorphism of $F$ sending $a$ to $a^p$. Let $K := \{ a \in F \mid \phi(a) = a \}$. What is the value of $[K : \mathbb{F}_p]$?

You are given a triangle ABC, a point D on segment AC, a point E on segment AB and a point F on segment BC. Let BD and CE intersect in point P. Join P with F. Suppose that $\angle\mathrm{EPB} = \angle\mathrm{BPF} = \angle\mathrm{FPC} = \angle\mathrm{CPD}$ and $\mathrm{PD} = \mathrm{PE} = \mathrm{PF}$.
For each statement below, state if it is true or false.
(i) AP must bisect $\angle\mathrm{BAC}$.
(ii) $\triangle\mathrm{ABC}$ must be isosceles.
(iii) $\mathrm{A}$, $\mathrm{P}$, $\mathrm{F}$ must be collinear.
(iv) $\angle\mathrm{BAC}$ must be $60^{\circ}$.

Let $U = \left\{ (x, y) \in \mathbb{R}^2 \mid 1 < x^2 + y^2 < 4 \right\}$. Let $p, q \in U$. Show that there is a continuous map $\gamma : [0,1] \longrightarrow U$ such that $\gamma(0) = p$ and $\gamma(1) = q$ and such that $\gamma$ is differentiable on $(0,1)$.

If $I, J$ are two maximal ideals in a PID that is not a field, then show that $IJ$ is never a prime ideal.

Let $f : \mathbb{C} \longrightarrow \mathbb{C}$ be an entire function. Suppose that $f(z) \in \mathbb{R}$ if $z$ is on the real axis or on the imaginary axis. Show that $f'(z) = 0$ at $z = 0$.

Let $A \subseteq \mathbb{R}^n$ be a closed proper subset. For $x, y \in \mathbb{R}^n$, denote the usual (Euclidean) distance between them by $d(x, y)$. Let $x \in \mathbb{R}^n \setminus A$; define $\delta := \inf\{ d(x, y) \mid y \in A \}$. Show that there exists $y \in A$ such that $\delta = d(x, y)$.

Let $F$ be a field and $V$ a finite-dimensional vector-space over $F$. Let $T : V \longrightarrow V$ be a linear transformation, such that for every $v \in V$, there exists $n \in \mathbb{N}$ such that $T^n(v) = v$.
(A) Show that if $F = \mathbb{C}$, then $T$ is diagonalizable.
(B) Show that if $\operatorname{char}(F) > 0$, then there exists a non-diagonalizable $T$ satisfying the above hypothesis.

Let $F = \mathbb{Q}(\omega, \sqrt[3]{2})$, where $\omega \in \mathbb{C}$ is a primitive cube-root of unity. Find a $\mathbb{Q}$-basis for $F$ (with proof). Let $\mu : F \longrightarrow F$ be the $\mathbb{Q}$-linear map given by $\mu(a) = \omega^2 a$. Find the matrix of $\mu$ with respect to the basis obtained above.

Let $G$ be a non-trivial subgroup of the group $(\mathbb{R}, +)$. Show that either $G$ is dense in $\mathbb{R}$ or that $G = \mathbb{Z} \cdot l$ where $l = \inf\{ x \in G \mid x > 0 \}$.

Let $S^1 = \{ z \in \mathbb{C} : |z| = 1 \}$. Consider the map $\mathrm{Sq} : S^1 \longrightarrow S^1$, $$\operatorname{Sq}(z) = z^2$$ Show that there does not exist a continuous map $\mathrm{Sqrt} : S^1 \longrightarrow S^1$ such that $\mathrm{Sq} \circ \mathrm{Sqrt} = Id_{S^1}$. (That is, $(\operatorname{Sqrt}(w))^2 = w$.) (Hint: If such a map existed, show that there would be a bijective continuous map $S^1 \times \{1, -1\} \longrightarrow S^1$.)

Consider the following construction in a circle. Choose points $A, B, C$ on the given circle such that $\angle ABC$ is $60^\circ$. Draw another circle that is tangential to the chords $AB$, $BC$ and to the original circle. Do the above construction in the unit circle to obtain a circle $S_1$. Repeat the process in $S_1$ to obtain another circle $S_2$. What is the radius of $S_2$?

Let $f$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ (where $\mathbb{R}$ is the set of all real numbers) that satisfies the following property: For every natural number $n$ $$f(n) = \text{the smallest prime factor of } n.$$ For example, $f(12) = 2$, $f(105) = 3$. Calculate the following.
(a) $\lim_{x \rightarrow \infty} f(x)$.
(b) The number of solutions to the equation $f(x) = 2016$.

Answer the following questions
(a) Evaluate $$\lim_{x \rightarrow 0^{+}} \left( x^{x^{x}} - x^{x} \right)$$ (b) Let $A = \frac{2\pi}{9}$, i.e., $A = 40$ degrees. Calculate the following $$1 + \cos A + \cos 2A + \cos 4A + \cos 5A + \cos 7A + \cos 8A$$ (c) Find the number of solutions to $e^{x} = \frac{x}{2017} + 1$.

Consider the ideal $I := (ux, uy, vx, uv)$ in the polynomial ring $\mathbb{Q}[u,v,x,y]$, where $u,v,x,y$ are indeterminates. Choose the correct statement(s) from below:
(A) Every prime ideal containing $I$ contains the ideal $(x,y)$;
(B) Every prime ideal containing $I$ contains the ideal $(x,y)$ or the ideal $(u,v)$;
(C) Every maximal ideal containing $I$ contains the ideal $(u,v)$;
(D) Every maximal ideal containing $I$ contains the ideal $(u,v,x,y)$.

Let $f$ be an irreducible cubic polynomial over $\mathbb{Q}$ with at most one real root and $\mathbb{k}$ the smallest subfield of $\mathbb{C}$ containing the roots of $f$. Choose the correct statement(s) from below:
(A) $\sigma(K) \subseteq K$ where $\sigma$ denotes complex conjugation;
(B) $[K : \mathbb{Q}]$ is an even number;
(C) $[(K \cap \mathbb{R}) : \mathbb{Q}]$ is an even number;
(D) $K$ is uncountable.

For a positive integer $n$, let $S_n$ denote the permutation group on $n$ symbols. Choose the correct statement(s) from below:
(A) For every positive integer $n$ and for every $m$ with $1 \leq m \leq n$, $S_n$ has a cyclic subgroup of order $m$;
(B) For every positive integer $n$ and for every $m$ with $n < m < n!$, $S_n$ has a cyclic subgroup of order $m$;
(C) There exist positive integers $n$ and $m$ with $n < m < n!$ such that $S_n$ has a cyclic subgroup of order $m$;
(D) For every positive integer $n$ and for every group $G$ of order $n$, $G$ is isomorphic to a subgroup of $S_n$.