Let $0 < a < b$. Prove that $$\int_a^b (x^2 + 1) e^{-x^2} dx \ge e^{-a^2} - e^{-b^2}.$$

Compute the sum of the series $$\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1 \cdot 2 \cdot 3 \cdot 4} + \frac{1}{5 \cdot 6 \cdot 7 \cdot 8} + \cdots.$$

Define the sequence $x_1, x_2, \dots$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \ge 1$. Compute $$\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdots x_n}{x_{n+1}}.$$

Let $a, b$ be two integers and suppose that $n$ is a positive integer for which the set $$\mathbb{Z} \setminus \{ a x^n + b y^n \mid x, y \in \mathbb{Z} \}$$ is finite. Prove that $n = 1$.

Suppose that $a, b, c$ are real numbers in the interval $[-1, 1]$ such that $$1 + 2abc \ge a^2 + b^2 + c^2.$$ Prove that $$1 + 2 (abc)^n \ge a^{2n} + b^{2n} + c^{2n}$$ for all positive integers $n$.

(a) A sequence $x_1, x_2, \dots$ of real numbers satisfies $$x_{n+1} = x_n \cos x_n \quad \text{for all } n \ge 1.$$ Does it follow that this sequence converges for all initial values $x_1$? (5 points)

(b) A sequence $y_1, y_2, \dots$ of real numbers satisfies $$y_{n+1} = y_n \sin y_n \quad \text{for all } n \ge 1.$$ Does it follow that this sequence converges for all initial values $y_1$? (5 points)

Let $a_0, a_1, \dots, a_n$ be positive real numbers such that $a_{k+1} - a_k \ge 1$ for all $k = 0, 1, \dots, n-1$. Prove that $$1 + \frac{1}{a_0} \left( 1 + \frac{1}{a_1 - a_0} \right) \cdots \left( 1 + \frac{1}{a_n - a_0} \right) \le \left( 1 + \frac{1}{a_0} \right) \left( 1 + \frac{1}{a_1} \right) \cdots \left( 1 + \frac{1}{a_n} \right).$$

Denote by $S_n$ the group of permutations of the sequence $(1, 2, \dots, n)$. Suppose that $G$ is a subgroup of $S_n$, such that for every $\pi \in G \setminus \{e\}$ there exists a unique $k \in \{1, 2, \dots, n\}$ for which $\pi(k) = k$. (Here $e$ is the unit element in the group $S_n$.) Show that this $k$ is the same for all $\pi \in G \setminus \{e\}$.

Let $A$ be a symmetric $m \times m$ matrix over the two-element field all of whose diagonal entries are zero. Prove that for every positive integer $n$ each column of the matrix $A^n$ has a zero entry.

Suppose that for a function $f : \mathbb{R} \to \mathbb{R}$ and real numbers $a < b$ one has $f(x) = 0$ for all $x \in (a, b)$. Prove that $f(x) = 0$ for all $x \in \mathbb{R}$ if $$\sum_{k=0}^{p-1} f \left( y + \frac{k}{p} \right) = 0$$ for every prime number $p$ and every real number $y$.