Consider the following subsets of $\mathbb { R } ^ { 2 } : X _ { 1 } = \left\{ \left. \left( x , \sin \frac { 1 } { x } \right) \right\rvert \, 0 < x < 1 \right\} , X _ { 2 } = [ 0,1 ] \times \{ 0 \}$, and $X _ { 3 } = \{ ( 0,1 ) \}$. Then,
(a) $X _ { 1 } \cup X _ { 2 } \cup X _ { 3 }$ is a connected set;
(b) $X _ { 1 } \cup X _ { 2 } \cup X _ { 3 }$ is a path-connected set;
(c) $X _ { 1 } \cup X _ { 2 } \cup X _ { 3 }$ is not path-connected, but $X _ { 1 } \cup X _ { 2 }$ is path-connected;
(d) $X _ { 1 } \cup X _ { 2 }$ is not path-connected, but every open neighbourhood of a point in this set contains a smaller open neighbourhood which is path-connected.

For a set $A \subset \mathbb { R }$, denote by $\operatorname { Cl } ( A )$ the closure of $A$, and by $\operatorname { Int } ( A )$ the interior of $A$. There is a set $A \subset \mathbb { R }$ such that
(a) $A , Cl ( A )$, and $\operatorname { Int } ( A )$ are pairwise distinct;
(b) $A , Cl ( A ) , \operatorname { Int } ( A )$, and $\operatorname { Cl } ( \operatorname { Int } ( A ) )$ are pairwise distinct;
(c) $A , \operatorname { Cl } ( A ) , \operatorname { Int } ( A )$, and $\operatorname { Int } ( \operatorname { Cl } ( A ) )$ are pairwise distinct;
(d) $A , Cl ( A ) , \operatorname { Int } ( A ) , \operatorname { Int } ( Cl ( A ) )$, and $\operatorname { Cl } ( \operatorname { Int } ( A ) )$ are pairwise distinct.

Let $f , g : [ 0,1 ] \rightarrow \mathbb { R }$ be given by $$\begin{gathered} f ( x ) : = \begin{cases} x ^ { 2 } & \text { if } x \text { is rational, } \\ 0 & \text { if } x \text { is irrational; } \end{cases} \\ g ( x ) : = \begin{cases} 1 / q & \text { if } x = \frac { p } { q } \text { is rational, with } \operatorname { gcd } ( p , q ) = 1 , \\ 0 & \text { if } x \text { is irrational. } \end{cases} \end{gathered}$$ Then,
(a) $g$ is Riemann integrable, but not $f$;
(b) both $f$ and $g$ are Riemann integrable;
(c) the Riemann integral $\int _ { 0 } ^ { 1 } f ( x ) d x = 0$;
(d) the Riemann integral $\int _ { 0 } ^ { 1 } g ( x ) d x = 0$.

Let $C$ be the ellipse $24 x ^ { 2 } + x y + 5 y ^ { 2 } + 3 x + 2 y + 1 = 0$. Then, the line integral $\oint \left( x ^ { 2 } y \, d y + x y ^ { 2 } \, d x \right)$
(a) lies in $( 0,1 )$;
(b) is 1;
(c) is either 1 or $-1$ depending on whether $C$ is traversed clockwise or counterclockwise;
(d) is 0.

Let $f : X \rightarrow Y$ be a nonconstant continuous map of topological spaces. Which of the following statements are true?
(a) If $Y = \mathbb { R }$ and $X$ is connected then $X$ is uncountable.
(b) If $X$ is Hausdorff then $f ( X )$ is Hausdorff.
(c) If $X$ is compact then $f ( X )$ is compact.
(d) If $X$ is connected then $f ( X )$ is connected.

Let $X$ be a set with the property that for any two metrics $d _ { 1 }$, and $d _ { 2 }$ on $X$, the identity map $$id : \left( X , d _ { 1 } \right) \rightarrow \left( X , d _ { 2 } \right)$$ is continuous. Which of the following are true?
(a) $X$ must be a singleton.
(b) $X$ can be any finite set.
(c) $X$ cannot be infinite.
(d) $X$ may be infinite but not uncountable.

Let $G$ be a finite group, $p$ the smallest prime divisor of $| G |$, and $x \in G$ an element of order $p$. Suppose $h \in G$ is such that $h x h ^ { - 1 } = x ^ { 10 }$. Show that $p = 3$.

Compute the integral $$\int _ { - \infty } ^ { \infty } \frac { x } { \left( x ^ { 2 } + 2 x + 2 \right) \left( x ^ { 2 } + 4 \right) } d x$$

Show that there does not exist an analytic function $f$ defined in the open unit disk for which $f \left( \frac { 1 } { n } \right) = 2 ^ { - n }$.

Let $f$ be a real valued continuous function on $[ 0,2 ]$ which is differentiable at every point except possibly at $x = 1$. Suppose that $\lim _ { x \rightarrow 1 } f ^ { \prime } ( x ) = 2013$. Show that $f$ is differentiable at $x = 1$.

(a) Show that there exists no bijective map $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 3 }$ such that $f$ and $f ^ { - 1 }$ are differentiable.
(b) Let $f : \mathbb { R } ^ { m } \rightarrow \mathbb { R } ^ { n }$ be a differentiable map such that the derivative $D f ( x )$ is surjective for all $x$. Is $f$ surjective?

Let $K _ { 1 } \supset K _ { 2 } \supset \ldots$ be a sequence of connected compact subsets of $\mathbb { R } ^ { 2 }$. Is it true that their intersection $K = \cap _ { i = 1 } ^ { \infty } K _ { i }$ is connected also? Provide either a proof or a counterexample.

Let $A$ be a subset of $\mathbb { R } ^ { 2 }$ with the property that every continuous function $f : A \rightarrow \mathbb { R }$ has a maximum in $A$. Prove that $A$ is compact.

For sets $A$ and $B$, let $f : A \rightarrow B$ and $g : B \rightarrow A$ be functions such that $f ( g ( x ) ) = x$ for each $x$. For each statement below, write whether it is TRUE or FALSE. a) The function $f$ must be one-to-one.
Answer: $\_\_\_\_$ b) The function $f$ must be onto.
Answer: $\_\_\_\_$ c) The function $g$ must be one-to-one.
Answer: $\_\_\_\_$ d) The function $g$ must be onto.
Answer: $\_\_\_\_$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function, where $\mathbb { R }$ is the set of real numbers. For each statement below, write whether it is TRUE or FALSE. a) If $| f ( x ) - f ( y ) | \leq 39 | x - y |$ for all $x , y$ then $f$ must be continuous everywhere.
Answer: $\_\_\_\_$ b) If $| f ( x ) - f ( y ) | \leq 39 | x - y |$ for all $x , y$ then $f$ must be differentiable everywhere.
Answer: $\_\_\_\_$ c) If $| f ( x ) - f ( y ) | \leq 39 | x - y | ^ { 2 }$ for all $x , y$ then $f$ must be differentiable everywhere.
Answer: $\_\_\_\_$ d) If $| f ( x ) - f ( y ) | \leq 39 | x - y | ^ { 2 }$ for all $x , y$ then $f$ must be constant.
Answer: $\_\_\_\_$

Let $S$ be a circle with center $O$. Suppose $A , B$ are points on the circumference of $S$ with $\angle A O B = 120 ^ { \circ }$. For triangle $A O B$, let $C$ be its circumcenter and $D$ its orthocenter (i.e., the point of intersection of the three lines containing the altitudes). For each statement below, write whether it is TRUE or FALSE. a) The triangle $A O C$ is equilateral.
Answer: $\_\_\_\_$ b) The triangle $A B D$ is equilateral.
Answer: $\_\_\_\_$ c) The point $C$ lies on the circle $S$.
Answer: $\_\_\_\_$ d) The point $D$ lies on the circle $S$.
Answer: $\_\_\_\_$

A polynomial $f ( x )$ with real coefficients is said to be a sum of squares if we can write $f ( x ) = p _ { 1 } ( x ) ^ { 2 } + \cdots + p _ { k } ( x ) ^ { 2 }$, where $p _ { 1 } ( x ) , \ldots , p _ { k } ( x )$ are polynomials with real coefficients. For each statement below, write whether it is TRUE or FALSE. a) If a polynomial $f ( x )$ is a sum of squares, then the coefficient of every odd power of $x$ in $f ( x )$ must be 0.
Answer: $\_\_\_\_$ b) If $f ( x ) = x ^ { 2 } + p x + q$ has a non-real root, then $f ( x )$ is a sum of squares.
Answer: $\_\_\_\_$ c) If $f ( x ) = x ^ { 3 } + p x ^ { 2 } + q x + r$ has a non-real root, then $f ( x )$ is a sum of squares.
Answer: $\_\_\_\_$ d) If a polynomial $f ( x ) > 0$ for all real values of $x$, then $f ( x )$ is a sum of squares.
Answer: $\_\_\_\_$

Let $\mathbb { R } =$ the set of real numbers. A continuous function $f : \mathbb { R } \rightarrow \mathbb { R }$ satisfies $f ( 1 ) = 1$, $f ( 2 ) = 4 , f ( 3 ) = 9$ and $f ( 4 ) = 16$. Answer the independent questions below by choosing the correct option from the given ones. a) Which of the following values must be in the range of $f$?
Options: 5 25 both neither
Answer: $\_\_\_\_$ b) Suppose $f$ is differentiable. Then which of the following intervals must contain an $x$ such that $f ^ { \prime } ( x ) = 2 x$?
Options: $( 1,2 )$ $( 2,4 )$ both neither
Answer: $\_\_\_\_$ c) Suppose $f$ is twice differentiable. Which of the following intervals must contain an $x$ such that $f ^ { \prime \prime } ( x ) = 2$?
Options: $(1,2)$ $(2,4)$ both neither
Answer: $\_\_\_\_$ d) Suppose $f$ is a polynomial, then which of the following are possible values of its degree?
Options: 3 4 both neither
Answer: $\_\_\_\_$

Suppose $f ( x )$ is a function from $\mathbb { R }$ to $\mathbb { R }$ such that $f ( f ( x ) ) = f ( x ) ^ { 2013 }$. Show that there are infinitely many such functions, of which exactly four are polynomials. (Here $\mathbb { R } =$ the set of real numbers.)

Let $\alpha , \beta$ and $c$ be positive numbers less than 1 , with $c$ rational and $\alpha , \beta$ irrational.
(A) The number $\alpha + \beta$ must be irrational.
(B) The infinite sum $\sum _ { i = 0 } ^ { \infty } \alpha c ^ { i } = \alpha + \alpha c + \alpha c ^ { 2 } + \cdots$ must be irrational.
(C) The value of the integral $\int _ { 0 } ^ { \pi } ( \beta \cos x + c ) d x$ must be irrational.

Consider the integral $I = \int _ { 1 } ^ { \infty } e ^ { a x ^ { 2 } + b x + c } d x$, where $a , b , c$ are constants. Some combinations of values for these constants are given below and you have to decide in each case whether the integral $I$ converges.
(A) $I$ converges for $a = - 1 \quad b = 10 \quad c = 100$.
(B) $I$ converges for $a = 1 \quad b = - 10 \quad c = - 100$.
(C) $I$ converges for $a = 0 \quad b = - 1 \quad c = 100$.
(D) $I$ converges for $a = 0 \quad b = 0 \quad c = - 100$.

The total length of all 12 sides of a rectangular box is 60. (i) Write the possible values of the volume of the box. Your answer should be an interval. Now suppose in addition that the surface area of the box is given to be 56. Find, if you can, (ii) the length of the longest diagonal of the box (iii) the volume of the box.

Find the area of the region in the XY plane consisting of all points in the set
$$\left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 144 \text { and } \sin ( 2 x + 3 y ) \leq 0 \right\} .$$

Let $x$ be a real number such that $x ^ { 2014 } - x ^ { 2004 }$ and $x ^ { 2009 } - x ^ { 2004 }$ are both integers. Show that $x$ is an integer. (Hint: it may be useful to first prove that $x$ is rational.)

(i) How many functions are there from the set $\{ 1 , \ldots , k \}$ to the set $\{ 1 , \ldots , n \}$ ?
(ii) Let $P _ { k }$ denote the set of all subsets of $\{ 1 , \ldots , k \}$. Find a formula for the number of functions $f$ from $P _ { k }$ to $\{ 1 , \ldots , n \}$ such that $f ( A \cup B ) =$ the larger of the two integers $f ( A )$ and $f ( B )$. Your answer need not be a closed formula but it should be simple enough to use for given values of $n$ and $k$, e.g., to see that for $k = 3$ and $n = 4$ there are 100 such functions. Example: When $k = 2$, the set $P _ { 2 }$ contains 4 elements: the empty set $\phi , \{ 1 \} , \{ 2 \}$ and $\{ 1,2 \}$. The function $f$ given by $\phi \rightarrow 2 , \{ 1 \} \rightarrow 3 , \{ 2 \} \rightarrow 4 , \{ 1,2 \} \rightarrow 4$ satisfies the given condition. But the function $g$ given by $\phi \rightarrow 2 , \{ 1 \} \rightarrow 3 , \{ 2 \} \rightarrow 4 , \{ 1,2 \} \rightarrow 5$ does not, because $g ( \{ 1 \} \cup \{ 2 \} ) = g ( \{ 1,2 \} ) = 5 \neq$ the larger of $g ( \{ 1 \} )$ and $g ( \{ 2 \} ) = \max ( 3,4 ) = 4$.