Let $R$ be a commutative ring with 1 and $I$ and $J$ ideals of $R$. Choose the correct statement(s) from below:
(A) If $I$ or $J$ is maximal then $IJ = I \cap J$;
(B) If $IJ = I \cap J$, then $I$ or $J$ is maximal;
(C) If $IJ = I \cap J$, then $1 \in I + J$;
(D) If $1 \in I + J$ then $IJ = I \cap J$.

Let $(X, d)$ and $(Y, \rho)$ be metric spaces and $f : X \longrightarrow Y$ a homeomorphism. Choose the correct statement(s) from below:
(A) If $B \subseteq Y$ is compact, then $f^{-1}(B)$ is compact;
(B) If $B \subseteq Y$ is bounded, then $f^{-1}(B)$ is bounded;
(C) If $B \subseteq Y$ is connected, then $f^{-1}(B)$ is connected;
(D) If $\{y_n\}$ is Cauchy in $Y$, then $\{f^{-1}(y_n)\}$ is Cauchy in $X$.

Fix a non-negative integer $d$. Let $$\mathcal{A}_d := \{A \subseteq \mathbb{C} : A \text{ is the zero-set of a polynomial of degree } \leq d \text{ in } \mathbb{C}[X]\}.$$ Let $\mathcal{T}$ be the coarsest topology on $\mathbb{C}$ in which $A$ is closed for every $A \in \mathcal{A}_d$.
(A) Determine whether $\mathcal{T}$ is Hausdorff.
(B) Show that for every polynomial $f(X) \in \mathbb{C}[X]$, the function $\mathbb{C} \longrightarrow \mathbb{C}$ defined by $z \mapsto f(z)$ is continuous, where $\mathbb{C}$ (on both the sides) is given the topology $\mathcal{T}$.

A compactification of a topological space $X$ is a compact topological space $Y$ which contains a dense subspace homeomorphic to $X$. Let $X = (0,1]$, in the subspace topology of $\mathbb{R}$ and $f : X \longrightarrow \mathbb{R},\ x \mapsto \sin\frac{1}{x}$. Show the following:
(A) $Y := [0,1]$ is a compactification of $X$, but $f$ does not extend to a continuous function $Y \longrightarrow \mathbb{R}$, i.e., there does not exist a continuous function $g : Y \longrightarrow \mathbb{R}$ such that $\left.g\right|_X = f$.
(B) $X$ is homeomorphic to the set $X_1 := \left\{\left.\left(t, \sin\frac{1}{t}\right)\right\rvert\, t \in X\right\} \subseteq \mathbb{R}^2$.
(C) The closure $Y_1$ of $X_1$ in $\mathbb{R}^2$ is a compactification of $X$.
(D) $f$ extends to a continuous function $Y_1 \longrightarrow \mathbb{R}$.

Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Define $g(0) = f(0)$ and $g(x) = \max\{f(y) \mid 0 \leq y \leq x\}$ for $0 < x \leq 1$. Show that $g$ is well-defined and that $g$ is a monotone continuous function.

[12 points] Throughout this problem we are interested in real valued functions $f$ satisfying two conditions: at each $x$ in its domain, $f$ is continuous and $f(x^{2}) = f(x)^{2}$. Prove the following independent statements about such functions. The hints below may be useful.
(i) There is a unique such function $f$ with domain $[0,1]$ and $f(0) \neq 0$.
(ii) If the domain of such $f$ is $(0, \infty)$, then ($f(x) = 0$ for every $x$) OR ($f(x) \neq 0$ for every $x$).
(iii) There are infinitely many such $f$ with domain $(0, \infty)$ such that $\int_{0}^{\infty} f(x)\, dx < 1$.
Hints: (1) Suppose a number $a$ and a sequence $x_{n}$ are in the domain of a continuous function $f$ and $x_{n}$ converges to $a$. Then $f(x_{n})$ must converge to $f(a)$. For example $f(0.5^{n}) \rightarrow f(0)$ and $f(2^{\frac{1}{n}}) \rightarrow f(1)$ if all the mentioned points are in the domain of $f$. In parts (i) and (ii) suitable sequences may be useful. (2) Notice that $f(x) = x^{r}$ satisfies $f(x^{2}) = f(x)^{2}$.

Let $U = \left\{(x, y) \in \mathbb{R}^{2} \mid x < y^{2} < 4\right\}$ and $V = \left\{(x, y) \in \mathbb{R}^{2} \mid 0 < xy < 4\right\}$, both taken with the subspace topology from $\mathbb{R}^{2}$. Which of the following statement(s) is/are true?
(A) There exists a non-constant continuous map $V \longrightarrow \mathbb{R}$ whose image is not an interval.
(B) Image of $U$ under any continuous map $U \longrightarrow \mathbb{R}$ is bounded.
(C) There exists an $\epsilon > 0$ such that given any $p \in V$ the open ball $B_{\epsilon}(p)$ with centre $p$ and radius $\epsilon$ is contained in $V$.
(D) If $C$ is a closed subset of $\mathbb{R}^{2}$ which is contained in $U$, then $C$ is compact.

Consider the function $f : \mathbb{R}^{2} \longrightarrow \mathbb{R}$ given by
$$f(x, y) = \left(1 - \cos \frac{x^{2}}{y}\right) \sqrt{x^{2} + y^{2}}$$
for $y \neq 0$ and $f(x, 0) = 0$. (The square root is chosen to be non-negative). Pick the correct statement(s) from below:
(A) $f$ is continuous at $(0,0)$.
(B) $f$ is an open map.
(C) $f$ is differentiable at $(0,0)$.
(D) $f$ is a bounded function.

Which of the following functions are uniformly continuous on $\mathbb{R}$?
(A) $f(x) = x$;
(B) $f(x) = x^{2}$;
(C) $f(x) = (\sin x)^{2}$;
(D) $f(x) = e^{-|x|}$.

Let $U$ and $V$ be non-empty open connected subsets of $\mathbb{C}$ and $f : U \longrightarrow V$ an analytic function. Which of the following statement(s) is/are true?
(A) $f^{\prime}(z) \neq 0$ for every $z \in U$.
(B) If $f$ is bijective, then $f^{\prime}(z) \neq 0$ for every $z \in U$.
(C) If $f^{\prime}(z) \neq 0$ for every $z \in U$, then $f$ is bijective.
(D) If $f^{\prime}(z) \neq 0$ for every $z \in U$, then $f$ is injective.

Let $U$ denote the unit open disc centred at 0. Let $f : U \backslash \{0\} \longrightarrow \mathbb{C}$ be an analytic function. Assume that $\lim_{z \longrightarrow 0} z f(z) = 0$.
(A) $\lim_{z \longrightarrow 0} |f(z)|$ exists and is in $\mathbb{R}$.
(B) $f$ has a pole of order 1 at 0.
(C) $zf(z)$ has a zero of order 1 at 0.
(D) There exists an analytic function $g : U \longrightarrow \mathbb{C}$ such that $g(z) = f(z)$ for every $z \in U \backslash \{0\}$.

Let $(X, d)$ be a compact metric space. For $x \in X$ and $\epsilon > 0$, define $B_{\epsilon}(x) := \{y \in X \mid d(x, y) < \epsilon\}$. For $C \subseteq X$ and $\epsilon > 0$, define $B_{\epsilon}(C) := \cup_{x \in C} B_{\epsilon}(x)$. Let $\mathcal{K}$ be the set of non-empty compact subsets of $X$. For $C, C^{\prime} \in \mathcal{K}$, define $\delta\left(C, C^{\prime}\right) = \inf\{\epsilon \mid C \subseteq B_{\epsilon}\left(C^{\prime}\right)$ and $C^{\prime} \subseteq B_{\epsilon}(C)\}$. Show that $(\mathcal{K}, \delta)$ is a compact metric space.

Let $f$ be a non-constant entire function with $f(z) \neq 0$ for all $z \in \mathbb{C}$. Consider the set $U = \{z : |f(z)| < 1\}$. Show that all connected components of $U$ are unbounded.

Let $F \subseteq \mathbb{R}^{3}$ be a non-empty finite set, and $X = \mathbb{R}^{3} \backslash F$, taken with the subspace topology of $\mathbb{R}^{3}$. Show that $X$ is homeomorphic to a complete metric space. (Hint: Look for a suitable continuous function from $X$ to $\mathbb{R}$.)

Show that there is no differentiable function $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that $f(0) = 1$ and $f^{\prime}(x) \geq (f(x))^{2}$ for every $x \in \mathbb{R}$.

Let $a_{1}, \ldots, a_{n}$ be distinct complex numbers. Show that the functions $e^{a_{1} z}, \ldots, e^{a_{n} z}$ are linearly independent over $\mathbb{C}$.

The Frattini subgroup of a finite group $G$ is the intersection of all its proper maximal subgroups. Let $p$ be a prime number. Show that the Frattini subgroup of $\mathbb{Z} / p^{n}$, $n \geq 2$, is generated by $p$.

Show that there is no polynomial $p ( x )$ for which $\cos ( \theta ) = p ( \sin \theta )$ for all angles $\theta$ in some nonempty interval.
Hint: Note that $x$ and $| x |$ are different functions but their values are equal on an interval (as $x = | x |$ for all $x \geq 0$). You may want to show as a first step that this cannot happen for two polynomials, i.e., if polynomials $f$ and $g$ satisfy $f ( x ) = g ( x )$ for all $x$ in some interval, then $f$ and $g$ must be equal as polynomials, i.e., in each degree they must have the same coefficient.

[15 points] Two distinct real numbers $r$ and $s$ are said to form a good pair $(r, s)$ if
$$r^3 + s^2 = s^3 + r^2$$
(i) Find a good pair $(a, \ell)$ with the largest possible value of $\ell$. Find a good pair $(s, b)$ with the smallest possible value $s$. For every good pair $(c, d)$ other than the two you found, show that there is a third real number $e$ such that $(d, e)$ and $(c, e)$ are also good pairs.
(ii) Show that there are infinitely many good pairs of rational numbers.
Hints (use these or your own method): The function $f(x) = x^3 - x^2$ may be useful. If $(r, s)$ is a good pair, can you express $s$ in terms of $r$? You may use that there are infinitely many right triangles with integer sides such that no two of these triangles are similar to each other.

Throughout this question every mentioned function is required to be a differentiable function from $\mathbb { R }$ to $\mathbb { R }$. The symbol $\circ$ denotes composition of functions.
(a) Suppose $f \circ f = f$. Then for each $x$, one must have $f ^ { \prime } ( x ) = $ \_\_\_\_ or $f ^ { \prime } ( f ( x ) ) = $ \_\_\_\_. Complete the sentence and justify.
(b) For a non-constant $f$ satisfying $f \circ f = f$, it is known and you may assume that the range of $f$ must have one of the following forms: $\mathbb { R } , ( - \infty , b ] , [ a , \infty )$ or $[ a , b ]$. Show that in fact the range must be all of $\mathbb { R }$ and deduce that there is a unique such function $f$. (Possible hints: For each $y$ in the range of $f$, what can you say about $f ( y )$? If the range has a maximum element $b$ what can you say about the derivative of $f$?)
(c) Suppose that $g \circ g \circ g = g$ and that $g \circ g$ is a non-constant function. Show that $g$ must be onto, $g$ must be strictly increasing or strictly decreasing and that there is a unique such increasing $g$.

(a) For non-negative numbers $a, b, c$ and any positive real number $r$ prove the following inequality and state precisely when equality is achieved. $$a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r(c-a)(c-b) \geq 0$$ Hint: Assuming $a \geq b \geq c$ do algebra with just the first two terms. What about the third term? What if the assumption is not true?
(b) As a special case obtain an inequality with $a^4 + b^4 + c^4 + abc(a+b+c)$ on one side.
(c) Show that if $abc = 1$ for positive numbers $a, b, c$, then $$a^4 + b^4 + c^4 + a^3 + b^3 + c^3 + a + b + c \geq \frac{a^2+b^2}{c} + \frac{b^2+c^2}{a} + \frac{c^2+a^2}{b} + 3.$$

Translate the statement "For every real number $x$, there exists a real number $y$ such that $x ^ { 3 } + 3 y - 2 = 0$.

1. Given sets $A = \{ 1,2,3 \} , B = \{ 2,3 \}$, then
A. $\mathrm { A } = \mathrm { B }$
B. $\mathrm { A } \cap \mathrm { B } = \varnothing$
C. $A \subset B$
D. $B \subset A$

``$\mathrm { x } = 1$'' is ``$\mathrm { x } ^ { 2 } - 2 x + 1 = 0$'' a
(A) necessary and sufficient condition
(B) sufficient but not necessary condition
(C) necessary but not sufficient condition
(D) neither sufficient nor necessary condition

Let $\alpha , \beta$ be two planes. Then a necessary and sufficient condition for $\alpha \parallel \beta$ is
A. There are infinitely many lines in $\alpha$ that are parallel to $\beta$
B. There are two intersecting lines in $\alpha$ that are parallel to $\beta$
C. $\alpha$ and $\beta$ are both parallel to the same line
D. $\alpha$ and $\beta$ are both perpendicular to the same plane