Found problems: 102
Let $a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that \[\left\vert a+b\sqrt{2}+c\sqrt{3}\right\vert > \frac{1}{10^{21}}.\]
Prove that if $g \ge 2$ is an integer, then two series \[\sum_{n=0}^{\infty}\frac{1}{g^{n^{2}}}\;\; \text{and}\;\; \sum_{n=0}^{\infty}\frac{1}{g^{n!}}\] both converge to irrational numbers.
Prove that $(2+\sqrt{3})^{2n}+(2-\sqrt{3})^{2n}$ is an even integer and that $(2+\sqrt{3})^{2n}-(2-\sqrt{3})^{2n}=w\sqrt{3}$ for some positive integer $w$, for all integers $n \geq 1$.
$\alpha,\beta$ are positive irrational numbers and $[\alpha[\beta x]]=[\beta[\alpha x]]$ for every positive $x$. Prove that $\alpha=\beta$
Let be two natural primes $ 1\le q \le p. $ Prove that $ \left( \sqrt{p^2+q} +p\right)^2 $ is irrational and its fractional part surpasses $ 3/4. $
Show that $e=\sum^{\infty}_{n=0} \frac{1}{n!}$ is irrational.
Let $m$ be equal to $1$ or $2$ and $n<10799$ be a positive integer. Determine all such $n$ for which $\sum_{k=1}^{n}\frac{1}{\sin{k}\sin{(k+1)}}=m\frac{\sin{n}}{\sin^{2}{1}}$.
Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$,
\[[r^m] \equiv -1 \pmod k .\]
[i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients.
[i]Proposed by Yugoslavia.[/i]
Let $\alpha=0.d_{1}d_{2}d_{3} \cdots$ be a decimal representation of a real number between $0$ and $1$. Let $r$ be a real number with $\vert r \vert<1$. [list=a][*] If $\alpha$ and $r$ are rational, must $\sum_{i=1}^{\infty} d_{i}r^{i}$ be rational? [*] If $\sum_{i=1}^{\infty} d_{i}r^{i}$ and $r$ are rational, $\alpha$ must be rational? [/list]
The set $ S$ is a finite subset of $ [0,1]$ with the following property: for all $ s\in S$, there exist $ a,b\in S\cup\{0,1\}$ with $ a, b\neq s$ such that $ s \equal{}\frac{a\plus{}b}{2}$. Prove that all the numbers in $ S$ are rational.
Show that the cube roots of three distinct primes cannot be terms in an arithmetic progression.
Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.
Let $ a, b, c$ be integers, not all equal to $ 0$. Show that
\[ \frac{1}{4a^{2}\plus{}3b^{2}\plus{}2c^{2}}\le\vert\sqrt[3]{4}a\plus{}\sqrt[3]{2}b\plus{}c\vert.\]
a) Let $x$ be a real number such that $x^2+x$ and $x^3+2x$ are rational numbers. Show that $x$ is a rational number.
b) Show that there exist irrational numbers $x$ such that $x^2+x$and $x^3-2x$ are rational.
Show that $\tan \left( \frac{\pi}{m} \right)$ is irrational for all positive integers $m \ge 5$.
$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such
\[ \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}.\]
Prove that $ c_n \equal{} 0$ implies $ n \equal{} 0.$
Prove that among any $6$ irrational numbers you can pick three numbers $a$, $b$ and $c$ such that numbers $a+b$, $b+c$ and $c+a$ are irrational
Prove that if $ \alpha$ and $ \beta$ are positive irrational numbers satisfying $ \frac{1}{\alpha}\plus{}\frac{1}{\beta}\equal{} 1$, then the sequences
\[ \lfloor\alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 3\alpha\rfloor,\cdots\]
and
\[ \lfloor\beta\rfloor,\lfloor 2\beta\rfloor,\lfloor 3\beta\rfloor,\cdots\]
together include every positive integer exactly once.
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.
Prove that for any positive integers $ a$ and $ b$
\[ \left\vert a\sqrt{2}\minus{}b\right\vert >\frac{1}{2(a\plus{}b)}.\]
For each integer $n \ge 1$, prove that there is a polynomial $P_{n}(x)$ with rational coefficients such that $x^{4n}(1-x)^{4n}=(1+x)^{2}P_{n}(x)+(-1)^{n}4^{n}$. Define the rational number $a_{n}$ by \[a_{n}= \frac{(-1)^{n-1}}{4^{n-1}}\int_{0}^{1}P_{n}(x) \; dx,\; n=1,2, \cdots.\] Prove that $a_{n}$ satisfies the inequality \[\left\vert \pi-a_{n}\right\vert < \frac{1}{4^{5n-1}}, \; n=1,2, \cdots.\]
For the positive integer $k$ we denote by the $a_n$ , the $k$ from the left digit in the decimal notation of the number $2^n$ ($a_n = 0$ if in the notation of the number $2^n$ less than the digits). Consider the infinite decimal fraction $a = \overline{0, a_1a_2a_3...}$. Prove that the number $a$ is irrational.
If the parabola $ y \equal{} \minus{}x^2 \plus{} bx \minus{} 8$ has its vertex on the $ x$-axis, then $ b$ must be:
$ \textbf{(A)}\ \text{a positive integer}\qquad \\
\textbf{(B)}\ \text{a positive or a negative rational number}\qquad \\
\textbf{(C)}\ \text{a positive rational number}\qquad \\
\textbf{(D)}\ \text{a positive or a negative irrational number}\qquad \\
\textbf{(E)}\ \text{a negative irrational number}$
Let $w\in (0,1)$ be an irrational number. Prove that $w$ has a unique convergent expansion of the form
$$w= \frac{1}{p_0} - \frac{1}{p_0 p_1 } + \frac{1}{ p_0 p_1 p_2 } - \frac{1}{p_0 p_1 p_2 p_3 } +\ldots,$$
where $1\leq p_0 < p_1 < p_2 <\ldots $ are integers. If $w= \frac{1}{\sqrt{2}},$ find $p_0 , p_1 , p_2.$
If $ x \plus{} y \equal{} 1$, then the largest value of $ xy$ is:
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 0.5\qquad
\textbf{(C)}\ \text{an irrational number about }{0.4}\qquad
\textbf{(D)}\ 0.25\qquad
\textbf{(E)}\ 0$