Which of the following topological spaces is/are connected?
(A) $\mathrm{GL}_{1}(\mathbb{R})$
(B) $\mathrm{GL}_{1}(\mathbb{C})$
(C) $\mathrm{GL}_{2}(\mathbb{R})$
(D) $\left\{\left[\begin{array}{cc}x & -y \\ y & x\end{array}\right] : x, y \in \mathbb{R}, x^{2}+y^{2}=1\right\}$

Consider $f : \{z \in \mathbb{C} : |z| > 1\} \longrightarrow \mathbb{C},\ f(z) = \frac{1}{z}$. Choose the correct statement(s):
(A) There are infinitely many entire functions $g$ such that $g(z) = f(z)$ for every $z \in \mathbb{C}$ with $|z| > 1$.
(B) There does not exist an entire function $g$ such that $g(z) = f(z)$ for every $z \in \mathbb{C}$ with $|z| > 1$.
(C) $g : \mathbb{C} \longrightarrow \mathbb{C}$ with $$g(z) = \begin{cases} 1 - \frac{1}{2}z^{2}, & |z| \leq 1 \\ \frac{1}{z}, & |z| > 1 \end{cases}$$ is an entire function such that $g(z) = f(z)$ for every $z \in \mathbb{C}$ with $|z| > 1$.

Consider all finite letter-strings formed by using only two letters A and B. We consider the usual dictionary order on these strings.
Formal rule: To compare two strings $w_1$ and $w_2$, read them from left to right. We say ``$w_1$ is smaller than $w_2$'' or ``$w_1 < w_2$'' if the first letter in which $w_1$ and $w_2$ differ is A in $w_1$ and B in $w_2$ (for example, $\mathrm{ABAA} < \mathrm{ABB}$ by looking at the third letters) or if $w_2$ is obtained by appending some letters at the end of $w_1$ (e.g. $\mathrm{AB} < \mathrm{ABAA}$).
For each of the statements below, state if it is true or false. Write your answers as a sequence of three letters (T for True and F for False) in correct order.
(a) Let $w$ be an arbitrary string. There exists a unique string $y$ satisfying both the following properties: (i) $w < y$ and (ii) there is no string $x$ with $w < x < y$.
(b) It is possible to give an infinite decreasing sequence of strings, i.e. a sequence $w_1, w_2, \ldots$, such that $w_{i+1} < w_i$ for each positive integer $i$.
(c) Fewer than 50 strings are smaller than ABBABABB.

Let $$G = \left\{\left(\begin{array}{cc}a & b \\ 0 & a^{-1}\end{array}\right) : a, b \in \mathbb{R}, a > 0\right\}, \quad N = \left\{\left(\begin{array}{cc}1 & b \\ 0 & 1\end{array}\right) : b \in \mathbb{R}\right\}.$$ Which of the following are true?
(A) $G/N$ is isomorphic to $\mathbb{R}$ under addition.
(B) $G/N$ is isomorphic to $\{a \in \mathbb{R} : a > 0\}$ under multiplication.
(C) There is a proper normal subgroup $N'$ of $G$ which properly contains $N$.
(D) $N$ is isomorphic to $\mathbb{R}$ under addition.

Choose the correct statement(s):
(A) There is a continuous surjective function from $[0,1)$ to $\mathbb{R}$;
(B) $\mathbb{R}$ and $[0,1)$ are homeomorphic to each other;
(C) There is a bijective function from $[0,1)$ to $\mathbb{R}$;
(D) Bounded subspaces of $\mathbb{R}$ cannot be homeomorphic to $\mathbb{R}$.

Which of the following complex numbers has/have a prime number as the degree of its minimal polynomial over $\mathbb{Q}$?
(A) $\zeta_{7}$, a primitive 7th root of unity;
(B) $\sqrt{2}+\sqrt{3}$;
(C) $\sqrt{-1}$;
(D) $\sqrt[3]{2}$.

Consider the polynomial $p(x) = (x + a_1)(x + a_2) \cdots (x + a_{10})$ where $a_i$ is a real number for each $i = 1, \ldots, 10$. Suppose all of the eleven coefficients of $p(x)$ are positive. For each of the following statements, decide if it is true or false. Write your answers as a sequence of four letters (T/F) in correct order.
(i) The polynomial $p(x)$ must have a global minimum.
(ii) Each $a_i$ must be positive.
(iii) All real roots of $p'(x)$ must be negative.
(iv) All roots of $p'(x)$ must be real.

Let $R$ be an integral domain such that every non-zero prime ideal of $R[X]$ (where $X$ is an indeterminate) is maximal. Choose the correct statement(s):
(A) $R$ is a field;
(B) $R$ contains $\mathbb{Z}$ as a subring;
(C) Every ideal in $R[X]$ is principal;
(D) $R$ contains $\mathbb{F}_{p}$ as a subring for some prime number $p$.

Let $f : \mathbb{R} \longrightarrow \mathbb{R}$ be such that $\int_{-\infty}^{\infty}|f(x)|\,\mathrm{d}x < \infty$. Define $F : \mathbb{R} \longrightarrow \mathbb{R}$ by $F(x) = \int_{-\infty}^{x} f(t)\,\mathrm{d}t$. Choose the correct statement(s):
(A) $f$ is continuous;
(B) $F$ is continuous;
(C) $F$ is uniformly continuous;
(D) There exists a positive real number $M$ such that $|f(x)| < M$ for all $x \in \mathbb{R}$.

Let $\omega \in \mathbb{C}$ be a primitive third root of unity. How many distinct possible images of $\omega$ are there under all the field homomorphisms $\mathbb{Q}(\omega) \longrightarrow \mathbb{C}$.

Let $C := \{z \in \mathbb{C} : |z| = 5\}$. What is value of $M$ such that $$2\pi\imath M = \int_{C} \frac{1}{z^{2}-5z+6}\,\mathrm{d}z\,?$$

Consider the set $\mathbb{R}[X]$ of polynomials in $X$ with real coefficients as a real vector space. Let $T$ be the $\mathbb{R}$-linear operator on $\mathbb{R}[X]$ given by $$T(f) = \frac{\mathrm{d}^{2}f}{\mathrm{d}X^{2}} - \frac{\mathrm{d}f}{\mathrm{d}X} + f$$ What is the nullity of $T$?

Let $f \in \mathbb{R}[x,y]$ be such that there exists a non-empty open set $U \subseteq \mathbb{R}^{2}$ such that $f(x,y) = 0$ for every $(x,y) \in U$. Show that $f = 0$.

Let $A \in M_{n \times n}(\mathbb{C})$.
(a) Suppose that $A^{2} = 0$. Show that $\lambda$ is an eigenvalue of $(I_{n}+A)$ if and only if $\lambda = 1$. ($I_{n}$ is the $n \times n$ identity matrix.)
(b) Suppose that $A^{2} = -1$. Determine (with proof) whether $A$ is diagonalizable.

Let $f$ be a non-constant entire function satisfying the following conditions:
(a) $f(0) = 0$;
(b) For every positive real number $M$, the set $\{z : |f(z)| < M\}$ is connected.
Prove that $f(z) = cz^{n}$ for some constant $c$ and positive integer $n$.

Let $\left(a_{mn}\right)_{m \geq 1, n \geq 1}$ be a double sequence of real numbers such that
(a) For every $n$, $b_{n} := \lim_{m \rightarrow \infty} a_{mn}$ exists;
(b) For all strictly increasing sequences $\left(m_{k}\right)_{k \geq 1}$ and $\left(n_{k}\right)_{k \geq 1}$ of positive integers, $\lim_{k \rightarrow \infty} a_{m_{k}n_{k}} = 1$.
Show that the sequence $\left(b_{n}\right)_{n \geq 1}$ converges to $1$.

Let $f \in \mathbb{C}[x,y]$ be such that $f(x,y) = f(y,x)$. Show that there is a $g \in \mathbb{C}[x,y]$ such that $f(x,y) = g(x+y, xy)$.

Let $X$ be a topological space and $f : X \longrightarrow [0,1]$ be a closed continuous surjective map such that $f^{-1}(a)$ is compact for every $0 \leq a \leq 1$. Prove or disprove: $X$ is compact. (A map is said to be closed if it takes closed sets to closed sets.)

Determine the cardinality of set of subrings of $\mathbb{Q}$. (Hint: For a set $P$ of positive prime numbers, consider the smallest subring of $\mathbb{Q}$ that contains $\left\{\left.\frac{1}{p}\right\rvert\, p \in P\right\}$.)

Let $$f(x) = \sum_{n \geq 1} \frac{\sin\left(\frac{x}{n}\right)}{n}$$ Show that $f$ is continuous. Determine (with justification) whether $f$ is differentiable.

Let $m$ and $n$ be positive integers and $0 \leq k \leq \min\{m,n\}$ an integer. Prove or disprove: The subspace of $M_{m \times n}(\mathbb{C})$ consisting of all matrices of rank equal to $k$ is connected. (You may use the following fact: For $t \geq 2$, $\mathrm{GL}_{t}(\mathbb{C})$ is connected.)

We say that two subsets $X$ and $Y$ of $\mathbb{R}$ are order-isomorphic if there is a bijective map $\phi : X \longrightarrow Y$ such that for every $x_1 \leq x_2 \in X$, $\phi(x_1) \leq \phi(x_2)$, where '$\leq$' denotes the usual order on $\mathbb{R}$. Choose the correct statement(s) from below:
(A) $\mathbb{N}$ and $\mathbb{Z}$ are not order-isomorphic;
(B) $\mathbb{N}$ and $\mathbb{Q}$ are order-isomorphic;
(C) $\mathbb{Z}$ and $\mathbb{Q}$ are order-isomorphic;
(D) The sets $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ are pairwise non-order-isomorphic.

Four children $\mathrm{K}$, $\mathrm{L}$, $\mathrm{M}$ and R are about to run a race. They make some predictions as follows.
K says: M will win. Myself will come second. R says: M will come second. L will be third. M says: L will be last. R will be second. After the race, it turns out that each person has made exactly one correct and one incorrect prediction. Write the result of the race in the order from first to the last.

Let $x_n = \left(1 - \frac{1}{n}\right) \sin \frac{n\pi}{3}$, $n \geq 1$. Write $l = \liminf x_n$ and $s = \limsup x_n$. Choose the correct statement(s) from below:
(A) $-\frac{\sqrt{3}}{2} \leq l < s \leq \frac{\sqrt{3}}{2}$;
(B) $-\frac{1}{2} \leq l < s \leq \frac{1}{2}$;
(C) $l = -1$ and $s = 1$;
(D) $l = s = 0$.

A country's GDP grew by $7.8\%$ within a period. During the same period the country's per-capita-GDP ($=$ ratio of GDP to the total population) increased by $10\%$. During this period, the total population of the country increased/decreased by $\_\_\_\_$ \%. (Choose the correct option and fill in the blank if possible.)