Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. Then $f'(x)$ is continuous.

There is a continuous onto function from the unit sphere in $\mathbb{R}^3$ to the complex plane $\mathbb{C}$.

$f : \mathbb{C} \rightarrow \mathbb{C}$ is an entire function such that the function $g(z)$ given by $g(z) = f\left(\frac{1}{z}\right)$ has a pole at $0$. Then $f$ is a surjective map.

Every finite group of order 17 is abelian.

Let $n \geq 2$ be an integer. Given an integer $k$ there exists an $n \times n$ matrix $A$ with integer entries such that $\operatorname{det} A = k$ and the first row of $A$ is $(1, 2, \ldots, n)$.

There is a finite Galois extension of $\mathbb{R}$ whose Galois group is nonabelian.

There is a non-constant continuous function from the open unit disc $$D = \{ z \in \mathbb{C} \mid |z| < 1 \}$$ to $\mathbb{R}$ which takes only irrational values.

There is a field of order 121.

Let $\alpha, \beta$ be two complex numbers with $\beta \neq 0$, and $f(z)$ a polynomial function on $\mathbb{C}$ such that $f(z) = \alpha$ whenever $z^5 = \beta$. What can you say about the degree of the polynomial $f(z)$?

Let $f, g : \mathbb{Z}/5\mathbb{Z} \rightarrow S_5$ be two non-trivial group homomorphisms. Show that there is a $\sigma \in S_5$ such that $f(x) = \sigma g(x) \sigma^{-1}$, for every $x \in \mathbb{Z}/5\mathbb{Z}$.

Suppose $f$ is continuous on $[0, \infty)$, differentiable on $(0, \infty)$ and $f(0) \geq 0$. Suppose $f'(x) \geq f(x)$ for all $x \in (0, \infty)$. Show that $f(x) \geq 0$ for all $x \in (0, \infty)$.

If $f$ and $g$ are continuous functions on $[0,1]$ satisfying $f(x) \geq g(x)$ for every $0 \leq x \leq 1$, and if $\int_0^1 f(x)\, dx = \int_0^1 g(x)\, dx$, then show that $f = g$.

Let $\{a_n\}$ and $\{b_n\}$ be sequences of complex numbers such that each $a_n$ is non-zero, $\lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} b_n = 0$, and such that for every natural number $k$, $$\lim_{n \rightarrow \infty} \frac{b_n}{a_n^k} = 0$$ Suppose $f$ is an analytic function on a connected open subset $U$ of $\mathbb{C}$ which contains $0$ and all the $a_n$. Show that if $f(a_n) = b_n$ for every natural number $n$, then $b_n = 0$ for every natural number $n$.

Let $T : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be an orthogonal transformation such that $\operatorname{det} T = 1$ and $T$ is not the identity linear transformation. Let $S \subset \mathbb{R}^3$ be the unit sphere, i.e., $$S = \left\{ (x, y, z) \mid x^2 + y^2 + z^2 = 1 \right\}$$ Show that $T$ fixes exactly two points on $S$.

Compute $$\int_0^{\infty} \frac{x^{1/3}}{1 + x^2}\, dx$$

Suppose $\varphi = (\varphi_2, \ldots, \varphi_n) : \mathbb{R}^n \rightarrow \mathbb{R}^{n-1}$ is a $C^2$ function, i.e. all second order partial derivatives of the $\varphi_i$ exist and are continuous. Show that the symbolic determinant $$\left| \begin{array}{cccc} \frac{\partial}{\partial x_1} & \frac{\partial \varphi_2}{\partial x_1} & \ldots & \frac{\partial \varphi_n}{\partial x_1} \\ \vdots & \vdots & & \vdots \\ \frac{\partial}{\partial x_n} & \frac{\partial \varphi_2}{\partial x_n} & \ldots & \frac{\partial \varphi_n}{\partial x_n} \end{array} \right|$$ vanishes identically.

Evaluate:
(a) $\lim _ { x \rightarrow 1 } \frac { n - \sum _ { k = 1 } ^ { n } x ^ { k } } { 1 - x }$
(b) $\lim _ { x \rightarrow 0 } \frac { e ^ { - 1 / x } } { x }$

There is a sequence of open intervals $I _ { n } \subset \mathbb { R }$ such that $\bigcap _ { n = 1 } ^ { \infty } I _ { n } = [ 0,1 ]$.

The set $S$ of real numbers of the form $\frac { m } { 10 ^ { n } }$ with $m , n \in \mathbb { Z }$ and $n \geq 0$ is a dense subset of $\mathbb { R }$.

There is a continuous bijection from $\mathbb { R } ^ { 2 } \rightarrow \mathbb { R }$.

There is a bijection between $\mathbb { Q }$ and $\mathbb { Q } \times \mathbb { Q }$.

If $\left\{ a _ { n } \right\} _ { n = 1 } ^ { \infty } , \left\{ b _ { n } \right\} _ { n = 1 } ^ { \infty }$ are two sequences of positive real numbers with the first converging to zero, and the second diverging to $\infty$, then the sequence of complex numbers $c _ { n } = a _ { n } e ^ { i b _ { n } }$ also converges to zero.

For any polynomial $f ( x )$ with real coefficients and of degree 2011 , there is a real number $b$ such that $f ( b ) = f ^ { \prime } ( b )$.

If $f : [ 0,1 ] \rightarrow [ - \pi , \pi ]$ is a continuous bijection then it is a homeomorphism.

For any $n \geq 2$ there is an $n \times n$ matrix $A$ with real entries such that $A ^ { 2 } = A$ and trace $( A ) = n + 1$.