Curated collections

Let $n$ be a positive integer, and let $p(x)$ be a polynomial of degree $n$ with integer coefficients. Prove that $$\max_{0 \le x \le 1} \bigl| p(x) \bigr| > \frac{1}{e^n}.$$

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuously differentiable function that satisfies $f'(t) > f(f(t))$ for all $t \in \mathbb{R}$. Prove that $f(f(f(t))) \le 0$ for all $t \ge 0$.

Does there exist a finite group $G$ with a normal subgroup $H$ such that $|\operatorname{Aut} H| > |\operatorname{Aut} G|$?

For $R > 1$ let $D_R = \{ (a, b) \in \mathbb{Z}^2 : 0 < a^2 + b^2 < R \}$. Compute $$\lim_{R \to \infty} \sum_{(a,b) \in D_R} \frac{(-1)^{a+b}}{a^2 + b^2}.$$

Let $\mathbb{R}$ be the set of real numbers. Prove that there is no function $f : \mathbb{R} \to \mathbb{R}$ with $f(0) > 0$, and such that $$f(x + y) \ge f(x) + y f(f(x)) \quad \text{for all } x, y \in \mathbb{R}.$$

For each positive integer $n$, let $f_n(\vartheta) = \sin\vartheta \cdot \sin(2\vartheta) \cdot \sin(4\vartheta) \cdots \sin(2^n \vartheta)$. For all real $\vartheta$ and all $n$, prove that $$|f_n(\vartheta)| \le \frac{2}{\sqrt{3}} |f_n(\pi/3)|.$$

Compare $\tan(\sin x)$ and $\sin(\tan x)$ for all $x \in \left( 0, \frac{\pi}{2} \right)$.

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 $n = 1$.

Prove that $$\int_0^1 \int_0^1 \frac{dx\,dy}{x^{-1} + |\ln y| - 1} \le 1.$$

Let $\mathbb{R}^+$ be the set of positive real numbers. Find all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that for all $x, y \in \mathbb{R}^+$ $$f(x) f(y f(x)) = f(x + y).$$

Determine whether or not there exist 15 integers $m_1, \dots, m_{15}$ such that $$\tag{1} \sum_{k=1}^{15} m_k \cdot \arctan(k) = \arctan(16).$$

Let $a, b, c, d, e > 0$ be real numbers such that $a^2 + b^2 + c^2 = d^2 + e^2$ and $a^4 + b^4 + c^4 = d^4 + e^4$. Compare the numbers $a^3 + b^3 + c^3$ and $d^3 + e^3$.

Let $(a_n)_{n \in \mathbb{N}}$ be the sequence defined by $$a_0 = 1, \qquad a_{n+1} = \frac{1}{n+1} \sum_{k=0}^{n} \frac{a_k}{n - k + 2}.$$ Find the limit $$\lim_{n \to \infty} \sum_{k=0}^{n} \frac{a_k}{2^k},$$ if it exists.