1A. Consider a sequence of functions $f _ { n } \colon [ a , b ] \to \mathbb { R }$ with the property that for each $x \in [ a , b ]$ there is an open interval $I _ { x }$ containing x such that $( f _ { n } ) _ { n \geq 1 }$ converges uniformly in $I _ { x } \cap [ a , b ]$ Show that $( f _ { n } ) _ { n \geq 1 }$ converges uniformly in $[ a , b ]$

2A. Find a countable abelian group whose endomorphism ring has the same cardinality as the set of real numbers. Justify your answer.

3A. Let $a _ { 1 } , \ldots , a _ { n } , b _ { 1 } , \ldots , b _ { m }$ be distinct complex numbers, let $r _ { 1 } , \ldots , r _ { n }$ be nonnegative integers, and let $c _ { 1 } , \ldots , c _ { m }$ be complex numbers. Prove that if $m \leq r _ { 1 } + \cdot \cdot \cdot + r _ { n } + 1$ , then there exists a rational function $F ( z ) \in \mathbb { C } ( z )$ satisfying all of the following:

1. $F ( z )$ is holomorphic at ∞ and everywhere in C except possibly at $a _ { 1 } , \ldots , a _ { n }$

2. ${ \mathrm { o r d } } _ { z = a _ { i } } F ( z ) \geq - r _ { i }$

3. $F ( b _ { j } ) = c _ { j }$ for $j = 1 , \ldots , m$

4A. For which positive integers n is it true that every invertible $2 \times 2$ matrix A with real entries can be expressed as the n-th power of another $2 \times 2$ matrix with real entries?

5A. Suppose $f \colon  { \mathbb { R } } \to  { \mathbb { C } }$ satisfies $f ^ { \prime } ( t ) + 2 i t f ( t ) = e ^ { 2 i t }$ and $f ( 0 ) = 0$ . Compute

$$
\operatorname* { l i m } _ { t \to + \infty } e ^ { i t ^ { 2 } } ( f ( t ) - f ( - t ) ) .
$$

You may assume $\textstyle \int _ { 0 } ^ { \infty } e ^ { - t ^ { 2 } } d t = { \sqrt { \pi } } / 2$

6A. For which pairs of integers $( a , b )$ is the quotient ring $\mathbb { Z } [ x ] / ( x ^ { 2 } + a x + b )$ isomorphic (as a ring) to the direct product of rings $\mathbb { Z } \times \mathbb { Z } ?$

7A. Evaluate $\int _ { 0 } ^ { \infty } { \frac { \sin { x } } { x } } d x$

8A. Let V and W be finite-dimensional vector spaces over a field k. Let $f \colon V ^ { n } \to W$ be a function such that

(a) For each fixed $i \in \{ 1 , \ldots , n \}$ and fixed $v _ { 1 } , \dots , v _ { i - 1 } , v _ { i + 1 } , \dots , v _ { n } \in V$ , the map

$$
x \mapsto f ( v _ { 1 } , \ldots , v _ { i - 1 } , x , v _ { i + 1 } , \ldots , v _ { n } )
$$

is a k-linear transformation; and

(b) $f ( v _ { 1 } , \ldots , v _ { n } ) = 0$ whenever $v _ { i } = v _ { i + 1 }$ for some $i \in \{ 1 , \ldots , n - 1 \}$

Prove that either dim $V \geq n$ or f is identically zero.

9A. Let $f \colon  { \mathbb { R } } ^ { n } \to  { \mathbb { R } } ^ { n }$ be a differentiable function, and let L be a nonnegative real number. Prove that the following are equivalent:

(i) For every $x , y \in \mathbb { R } ^ { n }$

$$
( f ( x ) - f ( y ) ) . ( x - y ) \leq L | x - y | ^ { 2 }
$$

(ii) For every $x , v \in \mathbb { R } ^ { n }$

$$
D f ( x ) v . v \leq L | v | ^ { 2 } ,
$$

where $D f ( x )$ is the derivative of f at x, and . denotes the standard inner product of vectors in $\mathbb { R } ^ { n }$

1B. Let F be a field (of arbitrary characteristic). Suppose g is a nonnegative integer, and polynomials $a ( x ) , b ( x ) \in F [ x ]$ satisfy deg $a ( x ) \leq g$ and deg $b ( x ) = 2 g + 1$ . Prove that the polynomial $y ^ { 2 } + a ( x ) y + b ( x )$ is irreducible over $F ( x )$

2B. Find the maximum possible value of $\vert f ^ { \prime } ( 1 ) \vert$ given that f is holomorphic on an open neighborhood of $\{ z \in \mathbb { C } : | z | \leq 2 \}$ and satisfies $| f ( z ) | \leq 1$ when $| z | = 2$ •

3B. Let A be a $d \times d$ matrix with complex entries. Assume that every eigenvalue of A has absolute value 1. Prove that there exists a constant $c \in \mathbb { R }$ independent of n such that

$$
\| A ^ { n } x \| \leq c n ^ { d - 1 } \| x \|
$$

for all $n \geq 1$ and $\boldsymbol { x } \in \mathbb { C } ^ { d }$ . Here $\| x \| : = ( | x _ { 1 } | ^ { 2 } + \cdot \cdot \cdot + | x _ { d } | ^ { 2 } ) ^ { 1 / 2 }$ for all $( x _ { 1 } , \ldots , x _ { d } ) \in \mathbb { C } ^ { d }$

4B. Let $a _ { 1 } , \ldots , a _ { n }$ be positive real numbers. Let $\Delta$ be the set of points $\mathbf { x } \in \mathbb { R } ^ { n }$ satisfying the conditions

$$
\sum _ { i = 1 } ^ { n } a _ { i } x _ { i } = 1 , \quad x _ { i } > 0 { \mathrm { ~ f o r ~ a l l ~ } } i .
$$

Prove that the function $\scriptstyle \log ( \prod _ { i = 1 } ^ { n } x _ { i } )$ has a unique maximum on $\Delta$ and find the point where it occurs.

5B. Let $n _ { 1 } , \ldots , n _ { r }$ be integers ≥ 2. Prove that there is a finite group G containing elements $g _ { 1 } , \ldots , g _ { r }$ such that $g _ { i }$ has exact order $n _ { i }$ for each $i ,$ and $g _ { i } g _ { j } \neq g _ { j } g _ { i }$ for $i \neq j$

6B. Let $( u _ { n } ( x , y ) ) _ { n \geq 1 }$ be a sequence of functions that are defined and harmonic for $( x , y )$ in an open neighborhood of the upper half plane $\mathbb { R } \times \mathbb { R } _ { \geq 0 }$ . Suppose that $\begin{array} { r } { \frac { \partial u _ { n } } { \partial y } ( x , 0 ) = \dot { 0 } } \end{array}$ for all $x \in \mathbb { R }$ , and $u _ { n } ( x , 0 )$ converges to 0 as $n \to \infty$ uniformly for $x \in \mathbb { R }$ . Must $u _ { n } ( x , y )  0$ as $n \to \infty$ for every $( x , y ) \in \mathbb { R } \times \mathbb { R } _ { > 0 } ?$

7B. Let A and B be $n \times n$ matrices with complex entries, such that $A B - B A$ is a linear combination of A and B. Prove that there exists a nonzero vector v that is an eigenvector of both A and B.

8B. For each real number x, compute

$$
\operatorname* { l i m } _ { n \to \infty } n \left( \left( 1 + { \frac { x } { n } } \right) ^ { n } - e ^ { x } \right) .
$$

9B. Let $S _ { 4 }$ be the group of permutations of $\{ 1 , 2 , 3 , 4 \}$ . Determine the order of the automorphism group $\mathrm { A u t } ( S _ { 4 } )$ . Justify your answer.