This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 30

PEN G Problems, 23
Tags: calculus , derivative , Irrational numbers
Let $f(x)=\prod_{n=1}^{\infty} \left( 1 + \frac{x}{2^n} \right)$. Show that at the point $x=1$, $f(x)$ and all its derivatives are irrational.

PEN G Problems, 12
Tags: trigonometry , number theory , relatively prime , Irrational numbers
An integer-sided triangle has angles $ p\theta$ and $ q\theta$, where $ p$ and $ q$ are relatively prime integers. Prove that $ \cos\theta$ is irrational.

PEN G Problems, 6
Tags: Putnam , irrational number , Irrational numbers
Prove that for any irrational number $\xi$, there are infinitely many rational numbers $\frac{m}{n}$ $\left( (m,n) \in \mathbb{Z}\times \mathbb{N}\right)$ such that \[\left\vert \xi-\frac{n}{m}\right\vert < \frac{1}{\sqrt{5}m^{2}}.\]

PEN G Problems, 10
Tags: trigonometry , Irrational numbers
Show that $\frac{1}{\pi} \arccos \left( \frac{1}{\sqrt{2003}} \right)$ is irrational.

PEN G Problems, 5
Tags: Irrational numbers
Let $ a, b, c$ be integers, not all equal to $ 0$. Show that \[ \frac{1}{4a^{2}\plus{}3b^{2}\plus{}2c^{2}}\le\vert\sqrt[3]{4}a\plus{}\sqrt[3]{2}b\plus{}c\vert.\]