Which of the following groups are cyclic?
(A) $\mathbb { Z } / 2 \mathbb { Z } \oplus \mathbb { Z } / 9 \mathbb { Z }$
(B) $\mathbb { Z } / 3 \mathbb { Z } \oplus \mathbb { Z } / 9 \mathbb { Z }$
(C) Every group of order 18.
(D) $\left( \mathbb { Q } ^ { \times } , \cdot \right)$

Let $p , q$ be distinct prime numbers and let $\zeta _ { p } , \zeta _ { q }$ denote (any) primitive $p$-th and $q$-th roots of unity, respectively. Choose all the correct statements.
(A) $\zeta _ { 13 } \notin \mathbb { Q } \left( \zeta _ { 31 } \right)$.
(B) If $p$ divides $q - 1$, then $\zeta _ { p } \in \mathbb { Q } \left( \zeta _ { q } \right)$.
(C) If $\zeta _ { p } \in \mathbb { Q } \left( \zeta _ { q } \right)$, then $p - 1$ divides $q - 1$.
(D) If there exists a field homomorphism $\mathbb { Q } \left( \zeta _ { p } \right) \longrightarrow \mathbb { Q } \left( \zeta _ { q } \right)$, then $p - 1$ divides $q - 1$.

Let $f , g : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R }$ be functions. Let $F = ( f , g ) : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R } ^ { 2 }$. Assume that $F$ is infinitely differentiable and that $F ( 0,0 ) = ( 0,0 )$. Suppose further that the function $f g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R }$ is everywhere non-negative. Then
(A) $f _ { x } ( 0,0 ) = 0 , f _ { y } ( 0,0 ) = 0$.
(B) $g _ { x } ( 0,0 ) = 0 , g _ { y } ( 0,0 ) = 0$.
(C) The image of $F$ is not dense in $\mathbb { R } ^ { 2 }$.
(D) $\operatorname { det } J ( 0,0 ) = 0$ where $J$ is the matrix of first partial derivatives (i.e., the jacobian matrix).

Let $f : \mathbb { R } _ { \geq 0 } \longrightarrow \mathbb { R }$ be the function
$$f ( x ) = \begin{cases} 1 , & x = 0 \\ x ^ { - x } , & x > 0 \end{cases}$$
Determine whether the following statement is true:
$$\int _ { 0 } ^ { 1 } f ( x ) \mathrm { d } x = \sum _ { i = 0 } ^ { \infty } n ^ { - n }$$

(A) (3 marks) Let $G$ be a group such that $| G | = p ^ { a } d$ with $a \geq 1$ and $( p , d ) = 1$. Let $P$ be a Sylow $p$-subgroup and let $Q$ be any $p$-subgroup of $G$ such that $Q$ is not a subgroup of $P$. Show that $P Q$ is not a subgroup of $G$.
(B) (7 marks) Let $\Gamma$ be a group that is the direct product of its Sylow subgroups. Show that every subgroup of $\Gamma$ also satisfies the same property.

(A) (5 marks) Let $n \geq 2$ be an integer. Let $V$ be the $\mathbb { R }$-vector-space of homogeneous real polynomials in three variables $X , Y , Z$ of degree $n$. Let $p = ( 1,0,0 )$. Let
$$W = \left\{ f \in V \left\lvert \, f ( p ) = \frac { \partial f } { \partial X } ( p ) \right. \right\}$$
Determine the dimension of $V / W$.
(B) (5 marks) A linear transformation $T : \mathbb { R } ^ { 9 } \longrightarrow \mathbb { R } ^ { 9 }$ is defined on the standard basis $e _ { 1 } , \ldots , e _ { 9 }$ by
$$\begin{aligned} & T e _ { i } = e _ { i - 1 } , \quad i = 3 , \ldots , 9 \\ & T e _ { 2 } = e _ { 3 } \\ & T e _ { 1 } = e _ { 1 } + e _ { 3 } + e _ { 8 } . \end{aligned}$$
Determine the nullity of $T$.

Let $F$ be a field and $R$ a subring of $F$ that is not a field. Let $x$ be a variable. Let $S = \left\{ a _ { 0 } + a _ { 1 } x + \cdots + a _ { n } x ^ { n } \mid n \geq 0 \text{ and } a _ { 0 } \in R , a _ { 1 } , \cdots a _ { n } \in F \right\}$.
(A) (2 marks) Show that, with the natural operations of addition and multiplication of polynomials, $S$ is an integral domain.
(B) (4 marks) Let $I = \{ f ( x ) \in S \mid f ( 0 ) = 0 \}$. Determine whether $I$ is a prime ideal.
(C) (4 marks) Determine whether $S$ is a PID.

(A) (6 marks) Let $f , g : [ 0,1 ] \mapsto \mathbb { R }$ be monotonically increasing continuous functions. Show that
$$\left( \int _ { 0 } ^ { 1 } f ( x ) d x \right) \left( \int _ { 0 } ^ { 1 } g ( x ) d x \right) \leq \int _ { 0 } ^ { 1 } f ( x ) g ( x ) d x$$
(Hint: try double integrals.)
(B) (4 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R }$ be an infinitely differentiable function such that $f ( 1 ) = f ( 0 ) = 0$. Also, suppose that for some $n > 0$, the first $n$ derivatives of $f$ vanish at zero. Then prove that for the $( n + 1 )$ th derivative of $f$, $f ^ { ( n + 1 ) } ( x ) = 0$ for some $x \in ( 0,1 )$.

(A) (5 marks) Consider the euclidean space $\mathbb { R } ^ { n }$ with the usual norm and dot product. Let $\mathrm { x } , \mathrm { y } \in \mathbb { R } ^ { n }$ be such that
$$\| \mathrm { x } + t \mathrm { y } \| \geq \| \mathrm { x } \| , \text { for all } t \in \mathbb { R }$$
Show that $\mathbf { x } \cdot \mathbf { y } = 0$.
(B) (5 marks) Consider the vector field $\vec { v } = \left( v _ { x } , v _ { y } \right)$ (with components $\left( v _ { x } , v _ { y } \right)$ ) on $\mathbb { R } ^ { 2 }$:
$$v _ { x } ( x , y ) = x - y , v _ { y } ( x , y ) = y + x$$
Compute the line integral of $\vec { v }$ along the unit circle (counterclockwise). Is there a function $f$ such that $\vec { v } = \operatorname { grad } f$?

Denote by $V$ the $\mathbb { Q }$-vector-space $\mathbb { Q } [ X ]$ (polynomial ring in one variable $X$ ). Show that $V ^ { * }$ is not isomorphic to $V$.

Let $f$ be a non-constant entire function with $f ( 0 ) = 0$. Let $u$ and $v$ be the real and imaginary parts of $f$ respectively. Let $R > 0$ and
$$B = \sup \{ u ( z ) : | z | = R \}$$
(A) (2 marks) Show that $B > 0$.
(B) (2 marks) Consider the function
$$F ( z ) : = \frac { f ( z ) } { z ( 2 B - f ( z ) ) }$$
Show that $F$ is analytic on the open ball with radius $R$ and continuous on the boundary $\{ z : | z | = R \}$.
(C) (3 marks) Show that $\sup \{ | F ( z ) | : | z | = R \} \leq \frac { 1 } { R }$.
(D) (3 marks) Show that
$$\sup \left\{ | f ( z ) | : | z | = \frac { R } { 2 } \right\} \leq 2 B$$

A list of $k$ elements, possibly with repeats, is given. The goal is to find if there is a majority element. This is defined to be an element $x$ such that the number of times $x$ occurs in the list is strictly greater than $\frac{k}{2}$. (Note that there need not be such an element, but if it is there, it must be unique.) A celebrated efficient way to do this task uses two functions $f$ and $m$ with domain $\{1, 2, \ldots, k\}$. The functions are defined inductively as follows.
Define $f(1) =$ first element of the list, $m(1) = 1$. Assuming $f$ and $m$ are defined for all inputs from 1 to $i$, define $$f(i+1) = \begin{cases} f(i) & \text{if } m(i) > 0 \\ (i+1)^{\text{th}} \text{ element of the list} & \text{if } m(i) = 0 \end{cases}$$ $$m(i+1) = \begin{cases} m(i) - 1 & \text{if } m(i) > 0 \text{ and } (i+1)^{\text{th}} \text{ element of the list is other than } f(i) \\ m(i) + 1 & \text{otherwise} \end{cases}$$
(a) For the example of length 15 given below, write a sequence of 15 letters showing the values of $f(i)$ and a sequence of 15 numbers directly underneath showing the values of $m(i)$ for $i = 1, 2, \ldots, 15$.
$$\text{a a b a b c c b b b a b b c b}$$
(b) Prove that in general the list can be divided into two disjoint parts A and B such that

• Part A contains $m(k)$ elements of the list each of which is $f(k)$.
• Part B contains the remaining $k - m(k)$ elements of the list and B can be written as disjoint union of pairs such that the two elements in each pair are distinct.

(c) If there is a majority element, show that it must be $f(k)$. You may assume part (b) even if you did not do it.
(d) Assuming $f(k)$ is the majority element, answer the following two questions. Show by examples that the number of occurrences of $f(k)$ in the list does not determine the value of $m(k)$. Can the value of $m(k)$ be anything in $\{0, \ldots, k\}$? Find constraints if any on the possible values of $m(k)$.
(e) Now assume instead that an element occurs exactly $\frac{k}{2}$ times in the list. Is it necessary that $f(k)$ is such an element?

A region in $\mathbb { C }$ is a non-empty open connected set. Select all the statement(s) that are true.
(A) Let $f$ be a function on a region $\Omega$ such that the integral of $f$ along the boundary of any closed triangle in $\Omega$ is zero. Then $f$ is analytic on $\Omega$.
(B) There exist a region $\Omega$ containing the real interval $( 0,1 )$ and a non-zero analytic function $f : \Omega \rightarrow \mathbb { C }$ such that $f \left( \frac { 1 } { n } \right) = 0$ for all positive integers $n$.
(C) Let $f$ be an analytic function on $\mathbb { C } \backslash \{ 0 \}$ with an essential singularity at $z = 0$. Then $\lim _ { z \rightarrow 0 } | f ( z ) | = \infty$.
(D) Every bounded analytic function on $\mathbb { C } \backslash \{ 0 \}$ is constant.

Let $u$ and $v$ be real-valued functions on $\mathbb { R } ^ { 2 }$ defined as follows:
$$\begin{aligned} & u ( x , y ) = \begin{cases} \frac { x ^ { 3 } - 3 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } & \text { if } ( x , y ) \neq ( 0,0 ) \\ 0 & \text { otherwise } \end{cases} \\ & v ( x , y ) = \begin{cases} \frac { y ^ { 3 } - 3 y x ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } & \text { if } ( x , y ) \neq ( 0,0 ) \\ 0 & \text { otherwise } \end{cases} \end{aligned}$$
Let $f : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R } ^ { 2 }$ be the function $f ( x , y ) = ( u ( x , y ) , v ( x , y ) )$. Pick the correct statement(s) from below.
(A) $\frac { \partial u } { \partial x } , \frac { \partial u } { \partial y } , \frac { \partial v } { \partial x }$ and $\frac { \partial v } { \partial y }$ exist at $( 0,0 )$.
(B) $\frac { \partial u } { \partial x }$ is continuous at $( 0,0 )$.
(C) For every fixed $( a , b ) \neq ( 0,0 ) \in \mathbb { R } ^ { 2 }$, the function $t \mapsto f ( t a , t b )$ is a differentiable function (of $t$ ).
(D) $f$ is differentiable at $( 0,0 )$.

Let $G$ (respectively, $H$ ) be a Sylow 2-subgroup (respectively, Sylow 7-subgroup) of the symmetric group $S _ { 17 }$. Pick the correct statement(s) from below.
(A) The order of $G$ is $2 ^ { 15 }$.
(B) $H$ is abelian.
(C) $G$ has a subgroup isomorphic to $\mathbb { Z } / 8 \mathbb { Z } \times \mathbb { Z } / 8 \mathbb { Z }$.
(D) If $\sigma \in S _ { 17 }$ has order 4 , then $\sigma$ is a 4-cycle.

4. What do the following function calls return? Briefly explain your answer.
(a) SecondBest $( [ 2,2,3 ] , 3 )$
(b) SecondBest $( [ 3,2,2 ]$, 3)
5. Give one example of an input array A with exactly 3 elements for which the call SecondBest (A, 3) returns a wrong answer. What is this wrong answer? What is the correct answer? 6. Let $F$ be the set of points on the plane with coordinates ( $x , y$ ) such that $| | x | - | y | | + | x | + | y | = 4$. What is the number of points in $F$ with integer coordinates? 7. We select three points at random on the circumference of a circle. What is the probability that $\triangle A B C$ contains the centre $O$ in the interior? 8. Let $\alpha$ be a fixed real number and let $f$ be a function from $\mathbb { R } ^ { 2 }$ to $\mathbb { R } ^ { 2 }$ defined as
$$f ( x , y ) = ( x \cos \alpha - y \sin \alpha , x \sin \alpha + y \cos \alpha ) .$$
Write down an expression for $f ^ { 10 } ( x , y )$, where $f ^ { 10 }$ denotes the function obtained by composing $f$ with itself 10 times. 9. "I've defeated more than 10 FIDE masters," a chess player boasted. "Surely, it must be fewer than 10," said the referee. "Well, I suppose it was at least one," said the Grand Master. If exactly one of them spoke the truth, how many FIDE masters did the chess player actually defeated? 10. A supplier of art material has four reams of handmade paper, three boxes of acrylic colors and two printing blocks. The two artists in the shop want to buy one item each, but insist on having the same kind of art material. How many items does the supplier have to take out to be sure that the artists' demand is met? 11. In a farmer's stable three animals, a donkey, a cow and a horse had to share two types of feed bags, one containing hay and the other containing grain, as follows:

• If the donkey ate grain then the horse ate what the cow ate.
• If the horse ate grain then then donkey ate what the cow did not eat.
• If the cow ate hay, then the donkey ate what the horse ate.

Among all the assignments of feedbags that satisfy the above condition which animal always gets to eat from the same feedbag? 12. There are 7 elevators in a large shopping mall, each stopping at ground floor and at most six other floors. If at least 3 elevators stop at each floor and if it is possible to go from any floor to any other floor without changing elevators, what is the maximum number of floors in the mall? 13. Consider the following expression
$$\sqrt { x + 2 \sqrt { x - 1 } } + \sqrt { x - 2 \sqrt { x - 1 } }$$
Find two real numbers $a , b$ such that the above expression is constant for $a \leq x \leq b$. 14. You are given a strange, analogue wall clock whose hour and minute hands are identical. Both the hands move continuously and there is no second hand. How many times are there from noon to midnight when it is not possible to tell what time is it by looking at the clock at that instant? For example, at 06 : 00 pm one can be sure that the upper hand is the minute hand, for otherwise at $12 : 30 \mathrm { pm }$ the top hand is between 12 and 1 . However, a bit after $1 : 15 \mathrm { pm }$ and $3 : 06 \mathrm { pm }$ the clock looks identical and you wouldn't be able to tell the exact time. 15. Does there exist a polynomial $q ( x )$ with integer coefficients such that $q ( 1 ) = 2$ and $q ( 3 ) = 5$ ? Given an example if there is one. Justify, if there is not. 16. Find the number of all 4-digit natural numbers formed with exactly two distinct digits. 17. An electronic card shuffling machine always rearranges the cards in the same way relative to the order in which they are placed in it. One iteration of shuffling means that the cards are placed in the machine and they are taken out after the rearrangement. Two iterations means that you place the cards in the machine immediately after the first iteration. The ace through the king of hearts are arranged in order with the ace on top and the king at the bottom. After 2 iterations the order of the cards, from top to bottom, is
$$10,9 , Q , 8 , K , 3,4 , A , 5 , J , 6,2,7 .$$
What will be the order after 13 iterations?

Questions 18-20 are based on the following description.

In June 2017, a cyberattack named NotPetya spread all over the world. Big companies like Marex, Merck, and FedEx's TNT Express lost a lot of money because of this attack.
\begin{table}[h]

Company	\begin{tabular}{ c } Pre-attack Average
Monthly Revenue
(USD million)

&

Post-Attack Average

Monthly Revenue

(USD million)

\hline Marex & 1000 & 700 Merc & 800 & 480 TNT Expanse & 500 & 250 \hline \end{tabular}
Table 1: Financial impact of NotPetya attack on three major companies
\end{table}

1. Which company experienced the smallest percentage decrease in total revenue over a one month period relative to its estimated total revenue loss over the same period?
2. Calculate the overall percentage decrease in revenue across all three companies in one month period following the NotPetya attack.
3. Assuming the monthly revenue of Marex follows a Gaussian distribution with a standard deviation of 75 , (i) what is the probability that Marex's revenue will be less than $\$ 850$ million in the pre-attack scenario, and (ii) what is the probability that Marex's revenue will be more than $\$ 850$ million in the post-attack scenario?

\begin{table}[h]

$z$	$P ( Z \leq z )$
- 2	0.0227
- 1	0.1586
0	0.5000
1	0.8413
2	0.9772

Table 2: Note that $Z \sim N ( 0,1 )$
\end{table}

Let $p \geq 3$ be a prime number and $V$ be an $n$-dimensional vector space over $\mathbb { F } _ { p }$. Let $T : V \rightarrow V$ be a linear transformation. Select all the true statement(s) from below.
(A) $T$ has an eigenvalue in $\mathbb { F } _ { p }$.
(B) If $T ^ { p - 1 } = I$, then the minimal polynomial of $T$ has distinct roots in $\mathbb { F } _ { p }$.
(C) If $T \neq I$ and $T ^ { p - 1 } = I$, then the characteristic polynomial of $T$ has distinct roots in $\mathbb { F } _ { p }$.
(D) If $T ^ { p - 1 } = I$, then $T$ is diagonalizable over $\mathbb { F } _ { p }$.

Let $X$ be a subset of $\mathbb { R } ^ { 3 }$. We say that $X$ has property $S$ if it contains at least two elements and every Cauchy sequence in $X$ has a limit point in $X$. Pick the correct statement(s) from below.
(A) If $X$ has property $S$ then it must be compact.
(B) If $X$ has property $S$ then it must be closed.
(C) Suppose that $X$ has property $S$ and it further satisfies the following condition: if $\left( a _ { 1 } , b _ { 1 } , c _ { 1 } \right) , \left( a _ { 2 } , b _ { 2 } , c _ { 2 } \right) \in X$, then $\left( a _ { 1 } + a _ { 2 } , b _ { 1 } + b _ { 2 } , c _ { 1 } + c _ { 2 } \right) \in X$. Then $X$ is dense in $\mathbb { R } ^ { 3 }$.
(D) Suppose that $X$ has property $S$ and it further satisfies the following condition: if $( a , b , c ) \in X$, then $\left( \frac { a } { 2 } , \frac { b } { 2 } , \frac { c } { 2 } \right) \in X$. Then $X$ is dense in $\mathbb { R } ^ { 3 }$.

Let $f$ be a continuous real-valued function on $[ 0,1 ]$ such that
$$\int _ { 0 } ^ { 1 } f ( x ) d x = \int _ { 0 } ^ { 1 } x f ( x ) d x = 0$$
Pick the correct statement(s) from below.
(A) $f$ must have a zero in $[ 0,1 ]$.
(B) $f$ has at least two zeros, counted with multiplicity, in $[ 0,1 ]$.
(C) If $f \not\equiv 0$, then $f$ has exactly two zeros in $[ 0,1 ]$.
(D) $f \equiv 0$.

Which of the following statement(s) are true?
(A) Every prime ideal of a finite commutative ring with unity is maximal.
(B) A commutative ring with unity whose set of all ideals is countably infinite is necessarily a countable ring.
(C) Let $R$ be a unique factorisation domain and $K$ be its field of fractions. There exists an irreducible element $\alpha \in R$ and an element $\beta \in K$ such that $\beta ^ { 2 } = \alpha$.
(D) Every subring $R$ (with unity) of $\mathbb { Q }$ with $\mathbb { Z } \varsubsetneqq R \varsubsetneqq \mathbb { Q }$ has infinitely many prime ideals.

Which of the following statement(s) are true?
(A) If $F _ { 1 } , F _ { 2 }$ are finite field extensions of $\mathbb { Q }$ such that $\left[ F _ { 1 } : \mathbb { Q } \right] = \left[ F _ { 2 } : \mathbb { Q } \right]$, then $F _ { 1 } , F _ { 2 }$ are isomorphic as fields.
(B) If $F _ { 1 } , F _ { 2 }$ are finite field extensions of $\mathbb { R }$ such that $\left[ F _ { 1 } : \mathbb { R } \right] = \left[ F _ { 2 } : \mathbb { R } \right]$, then $F _ { 1 } , F _ { 2 }$ are isomorphic as fields.
(C) Let $\mathbb { F }$ be a finite field. If $F _ { 1 } , F _ { 2 }$ be finite field extensions of $\mathbb { F }$ such that $\left[ F _ { 1 } : \mathbb { F } \right] = \left[ F _ { 2 } : \mathbb { F } \right]$, then $F _ { 1 } , F _ { 2 }$ are isomorphic as fields.
(D) Let $\omega \in \mathbb { C }$ be a primitive cube root of unity and let $\sqrt [ 3 ] { 2 } \in \mathbb { R }$ be a cube root of 2. Let $K = \mathbb { Q } ( \omega , \sqrt [ 3 ] { 2 } )$. If $F _ { 1 } , F _ { 2 }$ are subfields of $K$ such that $\left[ K : F _ { 1 } \right] = \left[ K : F _ { 2 } \right] = 2$, then $F _ { 1 } = F _ { 2 }$.

Let $X = \{ \alpha \in \mathbb { C } \mid \alpha$ satisfies a monic polynomial over $\mathbb { Q } \}$. (I.e., $X$ is the algebraic closure of $\mathbb { Q }$ in $\mathbb { C }$.) Endow $X$ with the subspace topology of the euclidean metric topology from $\mathbb { C }$. Pick the correct statement(s) from below.
(A) $X$ is closed.
(B) $X$ is complete.
(C) $X$ is unbounded.
(D) $X$ is connected.

Let $n \geq 3$ be an integer. Write $D _ { 2 n }$ for the dihedral group with $2 n$ elements. Show that the automorphism group of $D _ { 2 n }$ has at most $n \varphi ( n )$ elements. (Here $\varphi ( n )$ is the number of positive integers that are relatively prime to $n$.)

Let $\operatorname { SL } ( 2 , \mathbb { R } )$ be the group of $2 \times 2$ matrices with real entries and determinant 1, endowed with the subspace topology of $\mathbb { R } ^ { 4 }$. Consider the continuous map $f : \operatorname { SL } ( 2 , \mathbb { R } ) \longrightarrow \mathbb { C }$ given by
$$f \left( \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right] \right) = \frac { a i + b } { c i + d }$$
(A) (4 marks) Show that $f$ maps SL$( 2 , \mathbb { R } )$ onto the upper half plane $H = \{ z : \operatorname { Im } ( z ) > 0 \}$.
(B) (6 marks) Assume the following two facts:
(i) For all $M , N \in \mathrm { SL } ( 2 , \mathbb { R } ) , f ( M ) = f ( N )$ if and only if $M ^ { - 1 } N$ is an orthogonal matrix.
(ii) The map $f$ is an open map.
Now show that for every compact $K \subseteq H , f ^ { - 1 } ( K )$ is compact.

12. In the following code, A is an array indexed from 0 whose elements are all positive integers, and n is the number of elements in A . It is given that n is at least 2 . The operator $*$ denotes multiplication.
\begin{verbatim} function foo(A,n) { if A[0] > A[1] { first = A[0]; second = A[1]; } else { first = A[1]; second = A[0]; } for i from 2 to (n-1) { if A[i] > second { if A[i] > first { second = first; first = A[i]; } else { second = A[i]; } } } return(first * second); } \end{verbatim}
If $\mathrm { A } = [ 15,7,16,12,17,14,16,4,13,12 ]$, what will $\mathrm { f } _ { \mathrm { o } } \circ ( \mathrm { A } , 10 )$ return?
(a) 28
(b) 144
(c) 180
(d) 272