Let $ABC$ be an equilateral triangle with side length 2. Point $A'$ is chosen on side $BC$ such that the length of $A'B$ is $k < 1$. Likewise points $B'$ and $C'$ are chosen on sides $CA$ and $AB$ with $CB' = AC' = k$. Line segments are drawn from points $A', B', C'$ to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled $PQR$. Show that $PQR$ is an equilateral triangle with side length $\dfrac{4(1-k)}{\sqrt{k^{2}-2k+4}}$.

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.

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 $U = \mathbb{R}$.

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 $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Determine (with appropriate justification) the following limit: $$\lim_{n \longrightarrow \infty} \int_0^1 nx^n f(x)\,\mathrm{d}x$$

Let $f$ be a real valued continuous function defined on $\mathbb{R}$ satisfying $$f'\left(\tan^{2}\theta\right) = \cos 2\theta + \tan\theta \sin 2\theta, \text{ for all real numbers } \theta.$$ If $f'(0) = -\cos\frac{\pi}{12}$ then find $f(1)$.

$X$ is said to have the universal extension property if for every normal space $Y$ and every closed subset $A \subset Y$ and every continuous function $f : A \longrightarrow X$, $f$ extends to a continuous function from $Y$ to $X$. You may assume, without proof, that $\mathbb{R}^{2}$ has the universal extension property.
(A) Prove or find a counterexample: If $X$ has the universal extension property, then $X$ is connected.
(B) Give an example (with justification) of a compact subset $X$ of $\mathbb{R}^{2}$ that does not have the universal extension property.
(C) Let $X = \{(x, \sin x) \mid x \in \mathbb{R}\}$. Then show that $X$ has the universal extension property.

Let $p$ be a prime number and $q$ a power of $p$. Let $K$ be an algebraic closure of $\mathbb{F}_{q}$. Say that a polynomial $f(X) \in K[X]$ is a $q$-polynomial if it is of the form
$$f(X) = \sum_{i=0}^{n} a_{i} X^{q^{i}}$$
Let $f(X)$ be a $q$-polynomial of degree $q^{n}$, with $a_{0} \neq 0$. Show that the set of zeros of $f(X)$ is an $n$-dimensional vector-space over $\mathbb{F}_{q}$.

Let $f : [ 0,1 ] \longrightarrow [ 0,1 ]$ be a continuous function. Which of the following is/are true?
(A) For every continuous $g : [ 0,1 ] \longrightarrow \mathbb { R }$ with $g ( 0 ) = 0$ and $g ( 1 ) = 1$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$.
(B) For every continuous $g : [ 0,1 ] \longrightarrow \mathbb { R }$ with $g ( 0 ) < 0$ and $g ( 1 ) > 1$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$.
(C) For every continuous $g : [ 0,1 ] \longrightarrow \mathbb { R }$ with $0 < g ( 0 ) < 1$ and $0 < g ( 1 ) < 1$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$.
(D) For every continuous $g : [ 0,1 ] \longrightarrow [ 0,1 ]$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$.

Let $I , J$ be nonempty open intervals in $\mathbb { R }$. Let $f : I \longrightarrow J$ and $g : J \longrightarrow \mathbb { R }$ be functions. Let $h : I \longrightarrow \mathbb { R }$ be the composite function $g \circ f$. Pick the correct statement(s) from below.
(A) If $f , g$ are continuous, then $h$ is continuous.
(B) If $f , g$ are uniformly continuous, then $h$ is uniformly continuous.
(C) If $h$ is continuous, then $f$ is continuous.
(D) If $h$ is continuous, then $g$ is continuous.

Let $A , B$ be non-empty subsets of $\mathbb { R } ^ { 2 }$. Pick the correct statement(s) from below:
(A) If $A$ is compact, $B$ is open and $A \cup B$ is compact, then $A \cap B \neq \varnothing$.
(B) If $A$ and $B$ are path-connected and $A \cap B \neq \varnothing$ then $A \cup B$ is path-connected.
(C) If $A$ and $B$ are connected and open and $A \cap B \neq \varnothing$, then $A \cap B$ is connected.
(D) If $A$ is countable with $| A | \geq 2$, then $A$ is not connected.

Pick the correct statement(s) from below.
(A) $X = \prod _ { n = 1 } ^ { \infty } X _ { n }$ where $X _ { n } = \left\{ 1,2 , \ldots , 2 ^ { n } \right\}$ for $n \geq 1$ is not compact in the product topology.
(B) $Y = \prod _ { n = 1 } ^ { \infty } Y _ { n }$ where $Y _ { n } = \left[ 0,2 ^ { n } \right] \subseteq \mathbb { R }$ for $n \geq 1$ is path-connected in the product topology.
(C) $Z = \prod _ { n = 1 } ^ { \infty } Z _ { n }$ where $Z _ { n } = \left( 0 , \frac { 1 } { n } \right) \subseteq \mathbb { R }$ for $n \geq 1$ is compact in the product topology.
(D) $P = \prod _ { n = 1 } ^ { \infty } P _ { n }$ where $P _ { n } = \{ 0,1 \}$ for $n \geq 1$ (with product topology) is homeomorphic to $( 0,1 )$.