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 $S_n$.

Let $A = \{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1\}$ and $B = \{(x,y) \in \mathbb{R}^2 \mid 1 < x^2 + y^2 < 2\}$, both taken with the subspace topology of $\mathbb{R}^2$. Choose the correct statement(s) from below:
(A) Every continuous function from $A$ to $\mathbb{R}$ has bounded image;
(B) There exists a non-constant continuous function from $B$ to $\mathbb{N}$ (in the subspace topology of $\mathbb{R}$);
(C) For every surjective continuous function from $A \cup B$ to a topological space $X$, $X$ has at most two connected components;
(D) $B$ is homeomorphic to the unit circle.

Let $(X, d)$ be a metric space. Choose the correct statement(s) from below:
(A) There exists a metric $\tilde{d}$ on $X$ such that $d$ and $\tilde{d}$ define the same topology and such that $\tilde{d}$ is bounded (i.e., there exists a real number $M$ such that $\tilde{d}(x,y) < M$ for all $x,y \in X$).
(B) Every closed subset of $X$ that is bounded with respect to $d$ is compact;
(C) $X$ is connected;
(D) For every $x \in X$, there exists $y \in X$ such that $d(x,y)$ is a non-zero rational number.

Which of the following are equivalence relations on $\mathbb{R}$?
(A) $a \sim b$ if and only if $|a - b| \leq 25$;
(B) $a \sim b$ if and only if $a - b$ is rational;
(C) $a \sim b$ if and only if $a - b$ is irrational;
(D) $a \sim b$ if and only if $f(a) = f(b)$ for every continuous $f : \mathbb{R} \longrightarrow \mathbb{R}$.

Let $f, g : \mathbb{R}^2 \longrightarrow \mathbb{R}$ be two differentiable functions such that $f(x+1, y) = f(x, y+1) = f(x,y)$ and $g(x+1, y) = g(x, y+1) = g(x,y)$ for all $(x,y) \in \mathbb{R}^2$. Choose the correct statement(s) from below:
(A) $f$ is uniformly continuous;
(B) $f$ is bounded;
(C) The function $(f,g) : \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ is differentiable;
(D) If $\partial f/\partial x = \partial g/\partial y$ and $\partial f/\partial y = -\partial g/\partial x$, then the function $\mathbb{C} \longrightarrow \mathbb{C}$ sending $(x + \imath y) \longrightarrow f(x,y) + \imath g(x,y)$ (with $x, y \in \mathbb{R}$) is constant.

Consider the equation $$\frac{1}{a} + \frac{1}{b} = \frac{1}{a+b}$$ Choose the correct statement(s) from below:
(A) There exists $(a,b) \in \mathbb{R}^2$ satisfying the above equation;
(B) There exists $(a,b) \in \mathbb{C}^2$ satisfying the above equation;
(C) There exists $(a,b) \in \mathbb{C}^2$ with $a = b$ satisfying the above equation;
(D) There exists $(a,b) \in (\mathbb{F}_3)^2$ with $a = b$ satisfying the above equation.

Let $p = (0,0)$, $q = (0,1)$, $r = (\imath, 0)$ be points of $\mathbb{C}^2$. What is the dimension of the $\mathbb{C}$-vector space $$\{ f(X,Y) \in \mathbb{C}[X,Y] \mid \deg f \leq 2 \text{ and } f(p) = f(q) = f(r) = 0 \}$$ where by $\deg f$, we mean the total degree of the polynomial $f$?

Let $(X, \tau)$ be a topological space and $d : X \times X \longrightarrow \mathbb{R}_{\geq 0}$ a continuous function where $X \times X$ has the product topology and $\mathbb{R}_{\geq 0}$ is the set of non-negative real numbers, with the subspace topology of the usual topology of $\mathbb{R}$. Assume that $d^{-1}(0) = \{(x,x) \mid x \in X\}$, and that $d(x,y) \leq d(x,z) + d(y,z)$ for all $x,y,z \in X$. Show the following:
(A) $(X, \tau)$ is Hausdorff.
(B) The sets $B_{x,\epsilon} := \{y \in X \mid d(x,y) < \epsilon\}$, $0 < \epsilon \in \mathbb{R}$ is the basis for a topology $\tau'$ on $X$.
(C) $\tau'$ is coarser than $\tau$ (i.e., every set open in $\tau'$ is open in $\tau$).

(A) Let $f$ be an entire function such that $|f(z)| \leq |z|$. Show that $f$ is a polynomial of degree $\leq 1$.
(B) Let $\Gamma$ be a closed differentiable contour oriented counterclockwise and let $$\int_{\Gamma} \bar{z} \, \mathrm{d}z = A$$ What is the integral $$\int_{\Gamma} (x + y) \, \mathrm{d}z$$ (where $x$ and $y$, respectively, are the real and imaginary parts of $z$) in terms of $A$?

Let $f_n, f$ be real-valued functions on $[0,1]$ with $f$ continuous. Suppose that for all convergent sequences $\{x_n : n \geq 1\} \subseteq [0,1]$ with $x = \lim_{n \to \infty} x_n$ one has $$\lim_{n \to \infty} f_n(x_n) = f(x).$$ Show that $f_n$ converges to $f$ uniformly.

(A) Show that for any positive rational number $r$, the sequence $\left\{\frac{\log n}{n^r} : n \geq 1\right\}$ is bounded.
(B) Show that the series $$\sum_{n \geq 10} \frac{(\log n)^2 (\log \log n)}{n^2}$$ is convergent.

For a group $G$, let $\operatorname{Aut}(G)$ denote the group of group automorphisms of $G$. (The group operation of $\operatorname{Aut}(G)$ is composition.) Let $p$ be a prime number. Show that the multiplicative group $\mathbb{F}_p \setminus \{0\}$ is isomorphic to $\operatorname{Aut}((\mathbb{F}_p, +))$ under the map $a \mapsto [b \mapsto ab]$ ($a \in \mathbb{F}_p \setminus \{0\}$, $b \in \mathbb{F}_p$).

Let $\mathbb{k}$ be a field, $X$ an indeterminate and $R = \mathbb{k}[X]/(X^7 - 1)$. Determine the set $$\left\{\dim_{\mathbb{k}} R/\mathfrak{m} \mid \mathfrak{m} \text{ is a maximal ideal in } R\right\}$$ in the following three cases: $\mathbb{k} = \mathbb{Q}$; $\mathbb{k} = \mathbb{C}$; $\mathbb{k}$ is a field of characteristic 7.

For a $3 \times 3$ matrix $A$, say that a point $p$ on the unit sphere centred at the origin in $\mathbb{R}^3$ is a pole of $A$ if $Ap = p$. Denote by $\mathrm{SO}_3$ the subgroup of $\mathrm{GL}_3(\mathbb{R})$ consisting of all the orthogonal matrices with determinant 1.
(A) Show that if $A \in \mathrm{SO}_3$, then $A$ has a pole.
(B) Let $G$ be a subgroup of $\mathrm{SO}_3$. Show that $G$ acts on the set $$\left\{p \in \mathbb{S}^2 \mid p \text{ is a pole for some matrix } A \in G\right\}.$$

Let $f : X \longrightarrow Y$ be a continuous surjective map such that for every closed $A \subseteq X$, $f(A)$ is closed in $Y$. Show that if $Y$ and all the fibres $f^{-1}(y)$, $y \in Y$ are compact, then $X$ is compact. Show that if $Y$ is Hausdorff and $X$ is compact, then $Y$ and all the fibres $f^{-1}(y)$, $y \in Y$ are compact.

Let $\mathbb{k}$ be an algebraically closed uncountable field and $\mathfrak{m}$ a maximal ideal in the polynomial ring $R := \mathbb{k}[x_1, \ldots, x_n]$ in the indeterminates $x_1, \ldots, x_n$. Show that the composite map $\mathbb{k} \longrightarrow R \longrightarrow R/\mathfrak{m}$ is a field isomorphism. You may use without proof the following fact from linear algebra: If a vector space has a countable spanning set, it cannot have a linearly independent uncountable set in it. (Hint: If $t$ is transcendental over $\mathbb{k}$, then consider the set $\left\{\left.\frac{1}{t - \alpha}\right\rvert\, \alpha \in \mathbb{k}\right\}$.)

Prove that for every $z \in \mathbb{C}$, the series $\sum_{n=1}^{\infty} \frac{\sin(z/n)}{n}$ converges. For $z \in \mathbb{C}$, let $f(z) = \sum_{n=1}^{\infty} \frac{\sin(z/n)}{n}$. Prove that $f$ is entire.

Consider the following function defined for all real numbers $x$ $$f(x) = \frac{2018}{100 + e^{x}}$$ How many integers are there in the range of $f$?

List every solution of the following equation. You need not simplify your answer(s). $$\sqrt[3]{x+4} - \sqrt[3]{x} = 1$$

How many non-congruent triangles are there with integer lengths $a \leq b \leq c$ such that $a + b + c = 20$?

Consider a sequence of polynomials with real coefficients defined by $$p_{0}(x) = \left(x^{2}+1\right)\left(x^{2}+2\right) \cdots\left(x^{2}+1009\right)$$ with subsequent polynomials defined by $p_{k+1}(x) := p_{k}(x+1) - p_{k}(x)$ for $k \geq 0$. Find the least $n$ such that $$p_{n}(1) = p_{n}(2) = \cdots = p_{n}(5000)$$

Recall that $\arcsin(t)$ (also known as $\sin^{-1}(t)$) is a function with domain $[-1,1]$ and range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. Consider the function $f(x) := \arcsin(\sin(x))$ and answer the following questions as a series of four letters (T for True and F for False) in order.
(a) The function $f(x)$ is well defined for all real numbers $x$.
(b) The function $f(x)$ is continuous wherever it is defined.
(c) The function $f(x)$ is differentiable wherever it is continuous.

Answer the following questions
(a) A natural number $k$ is called stable if there exist $k$ distinct natural numbers $a_{1}, \ldots, a_{k}$, each $a_{i} > 1$, such that $$\frac{1}{a_{1}} + \cdots + \frac{1}{a_{k}} = 1.$$ Show that if $k$ is stable then $k+1$ is also stable. Using this or otherwise, find all stable numbers.
(b) Let $f$ be a differentiable function defined on a subset $A$ of the real numbers. Define $$f^{*}(y) := \max_{x \in A}\{yx - f(x)\}$$ whenever the above maximum is finite.
For the function $f(x) = -\ln(x)$, determine the set of points for which $f^{*}$ is defined and find an expression for $f^{*}(y)$ involving only $y$ and constants.

Answer the following questions
(a) Find all real solutions of the equation $$\left(x^{2}-2x\right)^{x^{2}+x-6} = 1$$ Explain why your solutions are the only solutions.
(b) The following expression is a rational number. Find its value. $$\sqrt[3]{6\sqrt{3}+10} - \sqrt[3]{6\sqrt{3}-10}$$

Let $f$ be a function on nonnegative integers defined as follows $$f(2n) = f(f(n)) \quad \text{and} \quad f(2n+1) = f(2n)+1$$
(a) If $f(0) = 0$, find $f(n)$ for every $n$.
(b) Show that $f(0)$ cannot equal 1.
(c) For what nonnegative integers $k$ (if any) can $f(0)$ equal $2^{k}$?