Let $a _ { 0 }$ and $a _ { 1 }$ be complex numbers and define $a _ { n } = 2 a _ { n - 1 } + a _ { n - 2 }$ for $n \geq 2$.
(A) Show that there are polynomials $p ( z ) , q ( z ) \in \mathbb { C } [ z ]$ such that $q ( 0 ) \neq 0$ and $\sum _ { n \geq 0 } a _ { n } z ^ { n }$ is the Taylor series expansion (around 0) of $\frac { p ( z ) } { q ( z ) }$.
(B) Let $a _ { 0 } = 1$ and $a _ { 1 } = 2$. Show that there exist complex numbers $\beta _ { 1 } , \beta _ { 2 } , \gamma _ { 1 } , \gamma _ { 2 }$ such that $$a _ { n } = \beta _ { 1 } \gamma _ { 1 } ^ { n + 1 } + \beta _ { 2 } \gamma _ { 2 } ^ { n + 1 }$$ for all $n$.

By a simple group, we mean a group $G$ in which the only normal subgroups are $\left\{ 1 _ { G } \right\}$ and $G$. Pick the correct statement(s) from below.
(A) No group of order 625 is simple.
(B) $\mathrm { GL } ( 2 , \mathbb { R } )$ is simple.
(C) Let $G$ be a simple group of order 60. Then $G$ has exactly six subgroups of order 5 .
(D) Let $G$ be a group of order 60. Then $G$ has exactly seven subgroups of order 3 .

Let $f : \mathbb { R } \longrightarrow ( 0 , \infty )$ be an infinitely differentiable function with $\int _ { - \infty } ^ { \infty } f ( t ) d t = 1$. Pick the correct statement(s) from below.
(A) $f ( t )$ is bounded.
(B) $\lim _ { | t | \rightarrow \infty } f ^ { \prime } ( t ) = 0$.
(C) There exists $t _ { 0 } \in \mathbb { R }$ such that $f \left( t _ { 0 } \right) \geq f ( t )$ for all $t \in \mathbb { R }$.
(D) $f ^ { \prime \prime } ( a ) = 0$ for some $a \in \mathbb { R }$.

Let $\mathcal { P } _ { n } = \{ f ( x ) \in \mathbb { R } [ x ] \mid \operatorname { deg } f ( x ) \leq n \}$, considered as an ($n + 1$)-dimensional real vector space. Let $T$ be the linear operator $f \mapsto f + \frac { \mathrm { d } f } { \mathrm {~d} x }$ on $\mathcal { P } _ { n }$. Pick the correct statement(s) from below.
(A) $T$ is invertible.
(B) $T$ is diagonalizable.
(C) $T$ is nilpotent.
(D) $( T - I ) ^ { 2 } = ( T - I )$ where $I$ is the identity map.

Pick the correct statement(s) from below.
(A) There exists a finite commutative ring $R$ of cardinality 100 such that $r ^ { 2 } = r$ for all $r \in R$.
(B) There is a field $K$ such that the additive group ( $K , +$ ) is isomorphic to the multiplicative group ( $K ^ { \times } , \cdot$ ).
(C) An irreducible polynomial in $\mathbb { Q } [ x ]$ is irreducible in $\mathbb { Z } [ x ]$.
(D) A monic polynomial of degree $n$ over a commutative ring $R$ has at most $n$ roots in $R$.

Pick the correct statement(s) from below.
(A) If $f$ is continuous and bounded on $( 0,1 )$, then $f$ is uniformly continuous on $( 0,1 )$.
(B) If $f$ is uniformly continuous on $( 0,1 )$, then $f$ is bounded on $( 0,1 )$.
(C) If $f$ is continuous on $( 0,1 )$ and $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$ and $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ exists, then $f$ is uniformly continuous on $( 0,1 )$.
(D) Product of a continuous and a uniformly continuous function on $[ 0,1 ]$ is uniformly continuous.

Let $X$ be the metric space of real-valued continuous functions on the interval $[ 0,1 ]$ with the ``supremum distance'': $$d ( f , g ) = \sup \{ | f ( x ) - g ( x ) | : x \in [ 0,1 ] \} \text { for all } f , g \in X$$ Let $Y = \{ f \in X : f ( [ 0,1 ] ) \subset [ 0,1 ] \}$ and $Z = \left\{ f \in X : f ( [ 0,1 ] ) \subset \left[ 0 , \frac { 1 } { 2 } \right) \cup \left( \frac { 1 } { 2 } , 1 \right] \right\}$. Pick the correct statement(s) from below.
(A) $Y$ is compact.
(B) $X$ and $Y$ are connected.
(C) $Z$ is not compact.
(D) $Z$ is path-connected.

Let $X : = \left\{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid z \leq 0 \right.$, or $\left. x , y \in \mathbb { Q } \right\}$ with subspace topology. Pick the correct statement(s) from below.
(A) $X$ is not locally connected but path connected.
(B) There exists a surjective continuous function $X \longrightarrow \mathbb { Q } _ { \geq 0 }$ (the set of non-negative rational numbers, with the subspace topology of $\mathbb { R }$ ).
(C) Let $S$ be the set of all points $p \in X$ having a compact neighbourhood (i.e. there exists a compact $K \subset X$ containing $p$ in its interior). Then $S$ is open.
(D) The closed and bounded subsets of $X$ are compact.

Consider the complex polynomial $P ( x ) = x ^ { 6 } + i x ^ { 4 } + 1$. (Here $i$ denotes a square-root of $-1$.) Pick the correct statement(s) from below.
(A) $P$ has at least one real zero.
(B) $P$ has no real zeros.
(C) $P$ has at least three zeros of the form $\alpha + i \beta$ with $\beta < 0$.
(D) $P$ has exactly three zeros $\alpha + i \beta$ with $\beta > 0$.

Let $f ( z ) = \sum _ { n \geq 0 } a _ { n } z ^ { n }$ be an analytic function on the open unit disc $D$ around 0 with $a _ { 1 } \neq 0$. Suppose that $\sum _ { n \geq 2 } \left| n a _ { n } \right| < \left| a _ { 1 } \right|$. Then which of the following are true?
(A) There are only finitely many such $f$.
(B) $\left| f ^ { \prime } ( z ) \right| > 0$ for all $z \in D$.
(C) If $z , w \in D$ are such that $z \neq w$ and $f ( z ) = f ( w )$, then $a _ { 1 } = - \sum _ { n \geq 2 } a _ { n } \left( z ^ { n - 1 } + z ^ { n - 2 } w + \cdots + w ^ { n - 1 } \right)$.
(D) $f$ is one-one on $D$.

Consider the function $S ( a )$ defined by the limit below: $$S ( a ) : = \lim _ { n \rightarrow \infty } \frac { 1 ^ { a } + 2 ^ { a } + 3 ^ { a } + \cdots + n ^ { a } } { ( n + 1 ) ^ { a - 1 } [ ( n a + 1 ) + ( n a + 2 ) + \cdots + ( n a + n ) ] }$$ Find the sum of all values $a$ such that $S ( a ) = \frac { 1 } { 60 }$.

Let $U$ and $V$ be non-empty open connected subsets of $\mathbb { C }$ and $f : U \longrightarrow V$ an analytic function. Suppose that for all compact subsets $K$ of $V , f ^ { - 1 } ( K )$ is compact. Show that $f ( U ) = V$.

Let $G$ be a finite group that has a non-trivial subgroup $N$ (i.e. $\left\{ 1 _ { G } \right\} \neq N \neq G$ ) that is contained in every non-trivial subgroup of $G$. Show that
(A) $G$ is a $p$-group for some prime number $p$;
(B) $N$ is a normal subgroup of $G$.

Let $f$ be an entire function such that $f$ maps the open unit ball $D$ around 0 to itself. Suppose further that $f ( 0 ) = 0$ and $f ( 1 ) = 1$. Show that $f ^ { \prime } ( 1 ) \in \mathbb { R }$ and that $\left| f ^ { \prime } ( 1 ) \right| \geq 1$.

Let $F$ be a field such that it has a finite non-Galois extension field. Let $V$ be a finite-dimensional vector-space over $F$. Let $V _ { 1 } , \ldots , V _ { r }$ be proper subspaces of $V$. Prove or disprove the following assertion: $V \neq \bigcup _ { i = 1 } ^ { r } V _ { i }$.

For a ring homomorphism $R \longrightarrow S$ (of commutative rings) and an ideal $I$ of $R$, the fibre over $I$ is the ring $S / I S$, i.e., the quotient of $S$ by the $S$-ideal generated by the image of $I$ in $S$. Let $S = \mathbb { C } [ X , Y ] / ( X Y - 1 )$ and $R = \mathbb { C } [ x + \alpha y ]$ where $\alpha \in \mathbb { C }$ and $x , y$ are the images of $X , Y$ in $S$. Consider the ring homomorphism $R \subseteq S$. Let $I = ( x + \alpha y - \beta ) R$, where $\beta \in \mathbb { C }$. For each nonnegative integer $n$, determine the set of ( $\alpha , \beta$ ) such that the fibre over $I$ has exactly $n$ maximal ideals.

Let $Q$ be the space of infinite sequences $$\mathbf { x } : = \left( x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots \right)$$ of real numbers $x _ { n } \in [ 0,1 ]$, with the product topology coming from the identification $Q = [ 0,1 ] ^ { \mathbb { N } }$. ($[ 0,1 ]$ has the euclidean topology.) Let $S : Q \longrightarrow \mathbb { R }$ be the map $$S ( \mathbf { x } ) : = \sum _ { n } \frac { x _ { n } } { n ^ { 2 } } .$$ (A) Let $Q _ { 2 } : = \left\{ \left( y _ { 1 } , y _ { 2 } , \ldots , y _ { n } , \ldots \right) \left\lvert \, 0 \leq y _ { n } \leq \frac { 1 } { n } \right. \right\}$. Show that $Q _ { 2 }$ is compact.
(B) Let $D : Q _ { 2 } \longrightarrow Q$ be the map $$\left( y _ { 1 } , y _ { 2 } , \ldots , y _ { n } , \ldots \right) \mapsto \left( y _ { 1 } , 2 y _ { 2 } , \ldots , n y _ { n } , \ldots \right)$$ Show that $D$ is a homeomorphism. (Hint: first show that $Q$ is Hausdorff.)
(C) Show that $S \circ D : Q _ { 2 } \longrightarrow \mathbb { R }$ is continuous. (Hint: Show that there is a suitable inner-product space $( L , \langle - , - \rangle )$ and a vector $\mathbf { a } \in L$ such that $( S \circ D ) ( \mathbf { x } ) = \langle \mathbf { x } , \mathbf { a } \rangle$ for each $\mathbf { x } \in Q _ { 2 }$.)
(D) Show that $S$ is continuous.

Let $E$ be a finite extension of the field $\mathbb { Q }$. We say that a homomorphism of fields $\phi : E \longrightarrow \mathbb { C }$ is real if $\phi ( E ) \subset \mathbb { R }$. Prove or disprove each of the following assertions:
(A) For each prime number $p$, the field $\mathbb { Q } \left( p ^ { 1 / 12 } \right)$ has exactly one real embedding in $\mathbb { C }$. ($p ^ { 1 / 12 }$ is the unique real number $r > 0$ such that $r ^ { 12 } = p$.)
(B) If $[ E : \mathbb { Q } ] = 11$, there exists a real embedding of $E$.
(C) If $E$ is a Galois extension of $\mathbb { Q }$ and $[ E : \mathbb { Q } ] = 11$, then every embedding $E \longrightarrow \mathbb { C }$ is a real embedding.

A continuous map $f : A \longrightarrow B$ between two metric spaces $( A , d _ { A } )$, $( B , d _ { B } )$ is said to be a bilipschitz map if there exists a real number $\lambda > 0$ such that $( 1 / \lambda ) d _ { A } \left( a _ { 0 } , a _ { 1 } \right) \leq d _ { B } \left( f \left( a _ { 0 } \right) , f \left( a _ { 1 } \right) \right) \leq \lambda d _ { A } \left( a _ { 0 } , a _ { 1 } \right)$ for each $a _ { 0 } , a _ { 1 } \in A$.
Let $X = \mathbb { R } ^ { 2 } \backslash \{ 0 \}$ and $Y = \mathbb { S } ^ { 1 } \times \mathbb { R } = \left\{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid x ^ { 2 } + y ^ { 2 } = 1 \right\}$. Let $d _ { X }$ (respectively, $d _ { Y }$ ) be the euclidean metric on $X$ induced from $\mathbb { R } ^ { 2 }$ (respectively, on $Y$ induced from $\mathbb { R } ^ { 3 }$ ). Let $f : X \longrightarrow Y$ be a bilipschitz map.
(A) Let $R > 0$ and $C _ { R } \subseteq X$ the circle of radius $R$ with centre at 0. Let $\bar { f } : X \longrightarrow \mathbb { R }$ be the composite of $f$ and the projection from $Y = \mathbb { S } ^ { 1 } \times \mathbb { R }$ to its second factor $\mathbb { R }$. Let $L _ { R }$ be the length of the interval $\bar { f } \left( C _ { R } \right) \subseteq \mathbb { R }$. Let $a , b \in X$ be such that $\bar { f } ( b ) = \bar { f } ( a ) + L _ { R }$. Show that $d _ { X } ( a , b ) \geq ( 2 R - 2 \lambda ) / \lambda ^ { 2 }$.
(B) Let $C _ { 1 }$ and $C _ { 2 }$ be the two arcs of $C _ { R }$, joining $a$ and $b$. Show that there exists $x _ { 1 } \in C _ { 1 }$ and $x _ { 2 } \in C _ { 2 }$ such that $\bar { f } \left( x _ { 1 } \right) = \bar { f } \left( x _ { 2 } \right) = \frac { f ( a ) + f ( b ) } { 2 }$. Show that $d _ { Y } \left( f \left( x _ { 1 } \right) , f \left( x _ { 2 } \right) \right) \leq 2$.
(C) Show that for all sufficiently large $R , d _ { Y } \left( f \left( x _ { 1 } \right) , f \left( x _ { 2 } \right) \right) > 2$. (Hint: Let $a _ { i } \in C _ { i }$ be such that $d _ { X } \left( a , a _ { i } \right) = R / \lambda ^ { 2 }$; show that $d _ { X } \left( x _ { 1 } , x _ { 2 } \right) \geq d _ { X } \left( a _ { 1 } , a _ { 2 } \right)$.)
(D) What is your conclusion about $f$?

Let $R$ be an integral domain containing $\mathbb { C }$ such that it is a finite-dimensional $\mathbb { C }$-vector-space. Pick the correct statement(s) from below.
(A) For every $a \in R$, the set $\left\{ 1 , a , a ^ { 2 } , \ldots \right\}$ is linearly dependent over $\mathbb { C }$.
(B) $R$ is a field.
(C) $R = \mathbb { C }$.
(D) The transcendence degree of $R$ over $\mathbb { C }$ is 1 .

Let $R$ be a euclidean domain that is not a field. Let $d : R \backslash \{ 0 \} \longrightarrow \mathbb { N }$ be the euclidean size (degree) function. Write $R ^ { \times }$for the invertible elements of $R$. Pick the correct statements from below.
(A) $R = R ^ { \times } \cup \{ 0 \}$.
(B) There exists $a \in R \backslash \left( R ^ { \times } \cup \{ 0 \} \right)$ such that $d ( a ) = \inf \left\{ d ( r ) \mid r \in R \backslash \left( R ^ { \times } \cup \{ 0 \} \right) \right\}$.
(C) With $a$ defined as above, for all $r \in R$, there exists $u \in R ^ { \times } \cup \{ 0 \}$ such that $a$ divides ( $r - u$ ).
(D) With $a$ defined as above, the ideal generated by $a$ is a maximal ideal.

Let $X$ be a compact topological space. Let $f : X \longrightarrow \mathbb { R }$ be a function satisfying $f ^ { - 1 } ( [ n , \infty ) )$ is closed for all $n \in \mathbb { N }$. Pick the correct statements from below.
(A) $f$ is continuous.
(B) $f ( U )$ is open for each open subset $U$ of $X$.
(C) $f ( U )$ is closed for each closed subset $U$ of $X$.
(D) $f$ is bounded above.

Let $f : [ 0,1 ] \longrightarrow \mathbb { R }$ be a continuous function and $E \subseteq [ 0,1 ]$. Which of the following are true?
(A) If $E$ is closed, then $f ( E )$ is closed.
(B) If $E$ is open, then $f ( E )$ is open.
(C) If $E$ is a countable union of closed sets, then $f ( E )$ is a countable union of closed sets.
(D) If $f$ injective and $E$ is a countable intersection of open sets, then $f ( E )$ is a countable intersection of open sets.

Let $a _ { n } , n \geq 1$, be a sequence of positive real numbers such that $a _ { n } \longrightarrow \infty$ as $n \longrightarrow \infty$. Then which of the following are true?
(A) There exists a natural number $M$ such that
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( a _ { n } \right) ^ { M } } \in \mathbb { R }$$
(B)
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( n ^ { 2 } a _ { n } \right) } \in \mathbb { R } .$$
(C)
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( n a _ { n } \right) } \in \mathbb { R }$$
(D) For all positive real numbers $R$,
$$\sum _ { n = 1 } ^ { \infty } \frac { R ^ { n } } { \left( a _ { n } \right) ^ { n } } \in \mathbb { R } .$$

Let $A$ be the ring of all entire functions under point-wise addition and multiplication. Then which of the following are true?
(A) $A$ does not have non-zero nilpotent elements.
(B) In the group of the units of $A$ (under multiplication), every element other than 1 has infinite order.
(C) For every $f \in A$, there is a sequence of polynomials which converges to $f$ uniformly on compact sets.
(D) The ideal generated by $z$ and $\sin z$ is principal.