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.

An alien script has $n$ letters $b_{1}, \ldots, b_{n}$. For some $k < n/2$ assume that all words formed by any of the $k$ letters (written left to right) are meaningful. These words are called $k$-words. Such a $k$-word is considered sacred if:
i) no letter appears twice and,
ii) if a letter $b_{i}$ appears in the word then the letters $b_{i-1}$ and $b_{i+1}$ do not appear. (Here $b_{n+1} = b_{1}$ and $b_{0} = b_{n}$.)
For example, if $n = 7$ and $k = 3$ then $b_{1}b_{3}b_{6}, b_{3}b_{1}b_{6}, b_{2}b_{4}b_{6}$ are sacred 3-words. On the other hand $b_{1}b_{7}b_{4}, b_{2}b_{2}b_{6}$ are not sacred. What is the total number of sacred $k$-words? Use your formula to find the answer for $n = 10$ and $k = 4$.

Imagine the unit square in the plane to be a carrom board. Assume the striker is just a point, moving with no friction (so it goes forever), and that when it hits an edge, the angle of reflection is equal to the angle of incidence, as in real life. If the striker ever hits a corner it falls into the pocket and disappears. The trajectory of the striker is completely determined by its starting point $(x, y)$ and its initial velocity $\overrightarrow{(p, q)}$.
If the striker eventually returns to its initial state (i.e. initial position and initial velocity), we define its bounce number to be the number of edges it hits before returning to its initial state for the first time.
For example, the trajectory with initial state $[(.5, .5); \overrightarrow{(1,0)}]$ has bounce number 2 and it returns to its initial state for the first time in 2 time units. And the trajectory with initial state $[(.25, .75); \overrightarrow{(1,1)}]$ has bounce number 4.
(a) Suppose the striker has initial state $[(.5, .5); \overrightarrow{(p, q)}]$. If $p > q \geq 0$ then what is its velocity after it hits an edge for the first time? What if $q > p \geq 0$?
(b) Draw a trajectory with bounce number 5 or justify why it is impossible.
(c) Consider the trajectory with initial state $[(x, y); \overrightarrow{(p, 0)}]$ where $p$ is a positive integer. In how much time will the striker first return to its initial state?
(d) What is the bounce number for the initial state $[(x, y); \overrightarrow{(p, q)}]$ where $p, q$ are relatively prime positive integers, assuming the striker never hits a corner?

Let $G$ be a group of order 6. Let $C_1, C_2, \ldots, C_k$ be the distinct conjugacy classes of $G$. Which of the following sequences of integers are possible values of $\left(\left|C_1\right|, \left|C_2\right|, \ldots, \left|C_k\right|\right)$?
(A) $(1,1,1,1,1,1)$;
(B) $(1,5)$;
(C) $(3,3)$;
(D) $(1,2,3)$.

Let $R = \mathbb{F}_2[X]$. Choose the correct statement(s) from below:
(A) $R$ has uncountably many maximal ideals;
(B) Every maximal ideal of $R$ has infinitely many elements;
(C) For all maximal ideals $\mathfrak{m}$ of $R$, $R/\mathfrak{m}$ is a finite field;
(D) For every integer $n$, every ideal of $R$ has only finitely many elements of degree $\leq n$.

Which of the following spaces are connected?
(A) $\left\{(x,y) \in \mathbb{R}^2 \mid xy = 1\right\}$ as a subspace of $\mathbb{R}^2$;
(B) The set of upper triangular matrices as a subspace of $M_n(\mathbb{R})$;
(C) The set of invertible diagonal matrices as a subspace of $M_n(\mathbb{R})$;
(D) $\left\{(x,y,z) \in \mathbb{R}^3 \mid z \geq 0, z^2 \geq x^2 + y^2\right\}$ as a subspace of $\mathbb{R}^3$.

Let $A$ be an $n \times n$ nilpotent real matrix $A$. Define $$e^A = I_n + A + \frac{1}{2!}A^2 + \frac{1}{3!}A^3 + \cdots$$ Choose the correct statement(s) from below:
(A) For every real number $t$, $e^{tA}$ is invertible;
(B) There exists a basis of $\mathbb{R}^n$ such that $e^A$ is upper-triangular;
(C) There exist $B, P \in \mathrm{GL}_n(\mathbb{R})$ such that $B = Pe^AP^{-1}$ and $\operatorname{trace}(B) = 0$;
(D) There exists a basis of $\mathbb{R}^n$ such that $A$ is lower-triangular.

Let $f(w,x,y,z) = wz - xy$. Choose the correct statement(s) from below:
(A) The directional derivative at $(1,0,0,1)$ in the direction $(a,b,c,d)$ is 0 if $a + d = 0$;
(B) The directional derivative at $(1,0,0,1)$ in the direction $(a,b,c,d)$ is 0 only if $a + d = 0$;
(C) The vector $(0,-1,-1,0)$ is normal to $f^{-1}(1)$ at the point $(1,0,0,1)$;
(D) The set of points $(a,b,c,d)$ where the total derivative of $f$ is zero is finite.

Choose the correct statement(s) from below:
(A) There exists a subfield $F$ of $\mathbb{C}$ such that $F \nsubseteq \mathbb{R}$ and $F \simeq \mathbb{Q}[X]/(2X^3 - 3X^2 + 6)$;
(B) For every irreducible cubic polynomial $f(X) \in \mathbb{Q}[X]$, there exists a subfield $F$ of $\mathbb{C}$ such that $F \nsubseteq \mathbb{R}$ and $F \simeq \mathbb{Q}[X]/f(X)$;
(C) There exists a subfield $F$ of $\mathbb{R}$ such that $F \simeq \mathbb{Q}[X]/(2X^3 - 3X^2 + 6)$;
(D) For every irreducible cubic polynomial $f(X) \in \mathbb{Q}[X]$, there exists a subfield $F$ of $\mathbb{R}$ such that $F \simeq \mathbb{Q}[X]/f(X)$.

For a continuous function $f : [0,1] \longrightarrow \mathbb{R}$, define $a_n(f) = \int_0^1 x^n f(x)\,\mathrm{d}x$. Choose the correct statement(s) from below:
(A) The sequence $\{a_n(f)\}$ is bounded for every continuous function $f : [0,1] \longrightarrow \mathbb{R}$;
(B) The sequence $\{a_n(f)\}$ is Cauchy for every continuous function $f : [0,1] \longrightarrow \mathbb{R}$;
(C) The sequence $\{a_n(f)\}$ converges to 0 for every continuous function $f : [0,1] \longrightarrow \mathbb{R}$;
(D) There exists a continuous function $f : [0,1] \longrightarrow \mathbb{R}$ such that the sequence $\{a_n(f)\}$ is divergent.