Let $f : \mathbb{C} \longrightarrow \mathbb{C}$ be a holomorphic function. Choose the correct statement(s) from below:
(A) $f(\bar{z})$ is holomorphic;
(B) Suppose that $f(\mathbb{R}) \subseteq \mathbb{R}$. Then $f(\mathbb{R})$ is open in $\mathbb{R}$;
(C) the map $z \mapsto e^{f(z)}$ is holomorphic;
(D) Suppose that $f(\mathbb{C}) \subset \mathbb{R}$. Then $f(A)$ is closed in $\mathbb{C}$ for every closed subset $A$ of $\mathbb{C}$.

Let $f : \mathbb{R} \longrightarrow \mathbb{R}$ be a twice-differentiable function such that $f\left(\frac{1}{n}\right) = 0$ for every positive integer $n$. Choose the correct statement(s) from below:
(A) $f(0) = 0$;
(B) $f'(0) = 0$;
(C) $f''(0) = 0$;
(D) $f$ is a nonzero polynomial.

Let $A$ be a non-zero $4 \times 4$ complex matrix such that $A^2 = 0$. What is the largest possible rank of $A$?

A subspace $Y$ of $\mathbb{R}$ is said to be a retract of $\mathbb{R}$ if there exists a continuous map $r : \mathbb{R} \longrightarrow Y$ such that $r(y) = y$ for every $y \in Y$.
(A) Show that $[0,1]$ is a retract of $\mathbb{R}$.
(B) Determine (with appropriate justification) whether every closed subset of $\mathbb{R}$ is a retract of $\mathbb{R}$.
(C) Show that $(0,1)$ is not a retract of $\mathbb{R}$.

Let $N$ be a positive integer and $a_n$ be a complex number for every $-N \leq n \leq N$. Consider the holomorphic function on $\{z \in \mathbb{C} \mid z \neq 0\}$ given by $$F(z) = \sum_{n=-N}^{n=N} a_n z^n$$ Consider the function $f$ defined on the open unit disc $\{z \in \mathbb{C} : |z| < 1\}$ by $$f(z) = \frac{1}{2\pi i} \int_{\Gamma} \frac{F(\xi)}{\xi - z}\,d\xi$$ where $\Gamma$ is the boundary of the disc, oriented counterclockwise. Write down an expression for $f$ in terms of the coefficients $a_n$ of $F$.

Let $\phi : [0,1] \longrightarrow \mathbb{R}$ be a continuous function such that $$\int_0^1 \phi(t) e^{-at}\,\mathrm{d}t = 0$$ for every $a \in \mathbb{R}_+$. Show that for every non-negative integer $n$, $$\int_0^1 \phi(t) t^n\,\mathrm{d}t = 0$$

Let $U$ be a non-empty open subset of $\mathbb{R}$. Suppose that there exists a uniformly continuous homeomorphism $h : U \longrightarrow \mathbb{R}$. Show that $U = \mathbb{R}$.

A subgroup $H$ of a group $G$ is said to be a characteristic subgroup if $\sigma(H) = H$ for every group isomorphism $\sigma : G \longrightarrow G$ of $G$.
(A) Determine all the characteristic subgroups of $(\mathbb{Q}, +)$ (the additive group).
(B) Show that every characteristic subgroup of $G$ is normal in $G$. Determine whether the converse is true.

Write $V$ for the space of $3 \times 3$ skew-symmetric real matrices.
(A) Show that for $A \in SO_3(\mathbb{R})$ and $M \in V$, $AMA^t \in V$. Write $A \cdot M$ for this action.
(B) Let $\Phi : \mathbb{R}^3 \longrightarrow V$ be the map $$\begin{bmatrix} u \\ v \\ w \end{bmatrix} \mapsto \begin{bmatrix} 0 & w & -v \\ -w & 0 & u \\ v & -u & 0 \end{bmatrix}$$ With the usual action of $SO_3(\mathbb{R})$ on $\mathbb{R}^3$ and the above action on $V$, show that $\Phi(Av) = A \cdot \Phi(v)$ for every $A \in SO_3(\mathbb{R})$ and $v \in \mathbb{R}^3$.
(C) Show that there does not exist $M \in V$, $M \neq 0$ such that for every $A \in SO_3(\mathbb{R})$, $A \cdot M$ belongs to the span of $M$.

Let $m > 1$ be an integer and consider the following equivalence relation on $\mathbb{C} \setminus \{0\}$: $z_1 \sim z_2$ if $z_1 = z_2 e^{\frac{2\pi \imath a}{m}}$ for some $a \in \mathbb{Z}$. Write $X$ for the set of equivalence classes and $\pi : \mathbb{C} \setminus \{0\} \longrightarrow X$ for the map that takes $z$ to its equivalence class. Define a topology on $X$ by setting $U \subseteq X$ to be open if and only if $\pi^{-1}(U)$ is open in the euclidean topology of $\mathbb{C} \setminus \{0\}$. Determine (with appropriate justification) whether $X$ is compact.

Let $\mathbb{k}$ be a field, $n$ a positive integer and $G$ a finite subgroup of $\mathrm{GL}_n(\mathbb{k})$ such that $|G| > 1$. Further assume that every $g \in G$ is upper-triangular and all the diagonal entries of $g$ are 1.
(A) Show that $\operatorname{char}\,\mathbb{k} > 0$. (Hint: consider the minimal polynomials of elements of $G$.)
(B) Show that the order of $g$ is a power of $\operatorname{char}\,\mathbb{k}$, for every $g \in G$.
(C) Show that the centre of $G$ has at least two elements.

Consider $f : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined as follows: $$f(a,b) := \lim_{n \rightarrow \infty} \frac{1}{n} \log_{e}\left[e^{na} + e^{nb}\right]$$
For each statement, state if it is true or false.
(a) $f$ is not onto i.e. the range of $f$ is not all of $\mathbb{R}$.
(b) For every $a$ the function $x \mapsto f(a,x)$ is continuous everywhere.
(c) For every $b$ the function $x \mapsto f(x,b)$ is differentiable everywhere.
(d) We have $f(0,x) = x$ for all $x \geq 0$.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$. For each statement, state if it is true or false.
(a) There is no continuous function $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx < \frac{1}{4}$.
(b) There is only one continuous function $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx = \frac{1}{4}$.
(c) There are infinitely many continuous functions $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx = \frac{1}{4}$.

For a natural number $n$ denote by $\operatorname{Map}(n)$ the set of all functions $f : \{1,2,\ldots,n\} \rightarrow \{1,2,\ldots,n\}$. For $f, g \in \operatorname{Map}(n)$, $f \circ g$ denotes the function in $\operatorname{Map}(n)$ that sends $x$ to $f(g(x))$.
(a) Let $f \in \operatorname{Map}(n)$. If for all $x \in \{1,\ldots,n\}$ $f(x) \neq x$, show that $f \circ f \neq f$.
(b) Count the number of functions $f \in \operatorname{Map}(n)$ such that $f \circ f = f$.

(a) Count the number of roots $w$ of the equation $z^{2019} - 1 = 0$ over complex numbers that satisfy $|w + 1| \geq \sqrt{2 + \sqrt{2}}$.
(b) Find all real numbers $x$ that satisfy the following equation: $$\frac{8^{x} + 27^{x}}{12^{x} + 18^{x}} = \frac{7}{6}$$

Let $ABCD$ be a parallelogram. Let $O$ be a point in its interior such that $\angle AOB + \angle DOC = 180^{\circ}$. Show that $\angle ODC = \angle OBC$.

Three positive real numbers $x, y, z$ satisfy $$\begin{aligned} x^{2} + y^{2} &= 3^{2} \\ y^{2} + yz + z^{2} &= 4^{2} \\ x^{2} + \sqrt{3}\,xz + z^{2} &= 5^{2} \end{aligned}$$ Find the value of $2xy + xz + \sqrt{3}\,yz$.

For a field $F$, $F^{\times}$ denotes the multiplicative group ($F \backslash \{0\}, \times$). Choose the correct statement(s) from below:
(A) Every finite subgroup of $\mathbb{R}^{\times}$ is cyclic;
(B) The order of every non-trivial finite subgroup of $\mathbb{R}^{\times}$ is a prime number;
(C) There are infinitely many non-isomorphic non-trivial finite subgroups of $\mathbb{R}^{\times}$;
(D) The order of every non-trivial finite subgroup of $\mathbb{C}^{\times}$ is a prime number.

Let $R$ be a commutative ring with 1 and $I$ and $J$ ideals of $R$. Choose the correct statement(s) from below:
(A) If $I$ or $J$ is maximal then $IJ = I \cap J$;
(B) If $IJ = I \cap J$, then $I$ or $J$ is maximal;
(C) If $IJ = I \cap J$, then $1 \in I + J$;
(D) If $1 \in I + J$ then $IJ = I \cap J$.

Let $(X, d)$ and $(Y, \rho)$ be metric spaces and $f : X \longrightarrow Y$ a homeomorphism. Choose the correct statement(s) from below:
(A) If $B \subseteq Y$ is compact, then $f^{-1}(B)$ is compact;
(B) If $B \subseteq Y$ is bounded, then $f^{-1}(B)$ is bounded;
(C) If $B \subseteq Y$ is connected, then $f^{-1}(B)$ is connected;
(D) If $\{y_n\}$ is Cauchy in $Y$, then $\{f^{-1}(y_n)\}$ is Cauchy in $X$.

Let $a, b \in \mathbb{R}$, and consider the $\mathbb{R}$-linear map $f : \mathbb{C} \longrightarrow \mathbb{C},\ z \mapsto az + b\bar{z}$. Choose the correct statement(s) from below:
(A) $f$ is onto (i.e., surjective) if $ab \neq 0$;
(B) $f$ is one-one (i.e., injective) if $ab \neq 0$;
(C) $f$ is onto if $a^2 \neq b^2$;
(D) if $a^2 = b^2$, $f$ is not one-one.

Let $$f(x,y) = \begin{cases} \frac{x^3 y^3}{x^2 + y^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}$$ Choose the correct statement(s) from below:
(A) $f$ is continuous on $\mathbb{R}^2$;
(B) $f$ is continuous at every point of $\mathbb{R}^2 \backslash \{(0,0)\}$;
(C) $f$ is differentiable at every point of $\mathbb{R}^2 \backslash \{(0,0)\}$;
(D) $f$ is not differentiable at $(0,0)$.

Let $K$ be the smallest subfield of $\mathbb{C}$ containing all the roots of unity. Choose the correct statement(s) from below:
(A) $\mathbb{C}$ is algebraic over $K$;
(B) $K$ has countably many elements;
(C) Irreducible polynomials in $K[X]$ do not have multiple roots;
(D) The characteristic of $K$ is zero.

Let $f : \mathbb{R} \longrightarrow \mathbb{R}$ be twice continuously differentiable. Suppose further that $f''(x) \geq 0$ for every $x \in \mathbb{R}$. Choose the correct statement(s) from below:
(A) $f$ is bounded;
(B) $f$ is constant;
(C) If $f$ is bounded, then it is infinitely differentiable;
(D) $\int_0^x f(t)\,\mathrm{d}t$ is infinitely differentiable with respect to $x$.

Let $f(z)$ be a power-series (with complex coefficients) centred at $0 \in \mathbb{C}$ and with a radius of convergence 2. Suppose that $f(0) = 0$. Choose the correct statement(s) from below:
(A) $f^{-1}(0) = \{0\}$;
(B) If $f$ is a non-constant function on $\{|z| < 2\}$, then $f^{-1}(0) = \{0\}$;
(C) If $f$ is a non-constant function, then for all $\zeta \in \mathbb{C}$ with sufficiently small $|\zeta|$, the equation $f(z) = \zeta$ has a solution;
(D) $$\int_{\gamma} f^{(n)}(z)\,\mathrm{d}z = 0$$ for every $n \geq 1$, where $\gamma$ is a unit circle centred at 0, oriented clockwise, and $f^{(n)}$ is the $n$th derivative of $f(z)$.