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.

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Found problems: 102

PEN G Problems, 9
Tags: irrational number , trigonometry , algebra , polynomial
Show that $\cos \frac{\pi}{7}$ is irrational.

PEN G Problems, 27
Tags: irrational number , limit , calculus , function , derivative , integration
Let $1<a_{1}<a_{2}<\cdots$ be a sequence of positive integers. Show that \[\frac{2^{a_{1}}}{{a_{1}}!}+\frac{2^{a_{2}}}{{a_{2}}!}+\frac{2^{a_{3}}}{{a_{3}}!}+\cdots\] is irrational.

2001 Poland - Second Round, 1
Tags: modular arithmetic , arithmetic sequence , irrational number , number theory
Find all integers $n\ge 3$ for which the following statement is true: Any arithmetic progression $a_1,\ldots ,a_n$ with $n$ terms for which $a_1+2a_2+\ldots+na_n$ is rational contains at least one rational term.

2007 Croatia Team Selection Test, 2
Tags: floor function , irrational number , number theory
Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.

KoMaL A Problems 2024/2025, A. 903
Tags: irrational number , continued fraction , number theory , floor function
Let the irrational number \[\alpha =1-\cfrac{1}{2a_1-\cfrac{1}{2a_2-\cfrac{1}{2a_3-\cdots}}}\] where coefficients $a_1, a_2, \ldots$ are positive integers, infinitely many of which are greater than $1$. Prove that for every positive integer $N$ at least half of the numbers $\lfloor \alpha\rfloor, \lfloor 2\alpha\rfloor, \ldots, \lfloor N\alpha\rfloor$ are even. [i]Proposed by Géza Kós, Budapest[/i]

2008 Mathcenter Contest, 5
Tags: irrational number , algebra
There are $6$ irrational numbers. Prove that there are always three of them, suppose $a,b,c$ such that $a+b$,$b+c$,$c+a$ are irrational numbers. [i](Erken)[/i]

1959 AMC 12/AHSME, 34
Tags: vieta , irrational number , complex number , imaginary numbers , quadratic
Let the roots of $x^2-3x+1=0$ be $r$ and $s$. Then the expression $r^2+s^2$ is: $ \textbf{(A)}\ \text{a positive integer} \qquad\textbf{(B)}\ \text{a positive fraction greater than 1}\qquad\textbf{(C)}\ \text{a positive fraction less than 1}$ $\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}$

PEN G Problems, 1
Tags: irrational number , floor function
Find the smallest positive integer $n$ such that \[0< \sqrt[4]{n}-\lfloor \sqrt[4]{n}\rfloor < 0.00001.\]

1953 Putnam, B7
Tags: irrational number , expansion
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 $p_0 , p_1 , p_2.$

2002 District Olympiad, 2
Tags: algebra , rational , irrational number
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.

2017 Romania Team Selection Test, P2
Tags: irrational number , polynomial , number theory
Determine all intergers $n\geq 2$ such that $a+\sqrt{2}$ and $a^n+\sqrt{2}$ are both rational for some real number $a$ depending on $n$

1967 AMC 12/AHSME, 9
Tags: trapezoid , irrational number , geometry
Let $K$, in square units, be the area of a trapezoid such that the shorter base, the altitude, and the longer base, in that order, are in arithmetic progression. Then: $\textbf{(A)}\ K \; \text{must be an integer} \qquad \textbf{(B)}\ K \; \text{must be a rational fraction} \\ \textbf{(C)}\ K \; \text{must be an irrational number} \qquad \textbf{(D)}\ K\; \text{must be an integer or a rational fraction} \qquad$ $\textbf{(E)}\ \text{taken alone neither} \; \textbf{(A)} \; \text{nor} \; \textbf{(B)} \; \text{nor} \; \textbf{(C)} \; \text{nor} \; \textbf{(D)} \; \text{is true}$

PEN G Problems, 6
Tags: irrational number
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, 3
Tags: irrational number
Prove that there exist positive integers $ m$ and $ n$ such that \[ \left\vert\frac{m^{2}}{n^{3}}\minus{}\sqrt{2001}\right\vert <\frac{1}{10^{8}}.\]

2007 Croatia Team Selection Test, 2
Tags: floor function , irrational number , number theory
Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.

2011 IFYM, Sozopol, 1
Tags: number theory , irrational number
Prove that for $\forall n>1$, $n\in \mathbb{N}$ , there exist infinitely many pairs of positive irrational numbers $a$ and $b$, such that $a^n=b$.

2024 Australian Mathematical Olympiad, P8
Tags: irrational number , number theory
Let $r=0.d_0d_1d_2\ldots$ be a real number. Let $e_n$ denote the number formed by the digits $d_n, d_{n-1}, \ldots, d_0$ written from left to right (leading zeroes are permitted). Given that $d_0=6$ and for each $n \geq 0$, $e_n$ is equal to the number formed by the $n+1$ rightmost digits of $e_n^2$. Show that $r$ is irrational.

1987 National High School Mathematics League, 3
Tags: irrational number
In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are rational numbers, we call it rational point. If $a$ is an irrational number, then in all lines that passes $(a,0)$, $\text{(A)}$There are infinitely many lines, on which there are at least two rational points. $\text{(B)}$There are exactly $n(n\geq2)$ lines, on which there are at least two rational points. $\text{(C)}$There are exactly 1 line, on which there are at least two rational points. $\text{(D)}$Every line passes at least one rational point.

2017 Thailand Mathematical Olympiad, 1
Tags: algebra , irrational number , irrational
Let $p$ be a prime. Show that $\sqrt[3]{p} +\sqrt[3]{p^5} $ is irrational.

1987 IMO Longlists, 78
Tags: modular arithmetic , algebra , polynomial , irrational number , number theory
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]

1995 Tournament Of Towns, (478) 2
Tags: irrational number , number theory , algebra
Let $p$ be the product of $n$ real numbers $x_1$, $x_2$,$...$, $x_n$. Prove that if $p - x_k$ is an odd integer for $k = 1, 2,..., n$, then each of the numbers $x_1$, $x_2$,$...$, $x_n$is irrational. (G Galperin)

2018 District Olympiad, 1
Tags: algebra , irrational number
Show that $$\sqrt{n + \left[ \sqrt{n} +\frac12\right]}$$ is an irrational number, for every positive integer $n$.

2016 Nigerian Senior MO Round 2, Problem 7
Tags: irrational number , induction , algebra
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 $n \geq 1$.

2014 South East Mathematical Olympiad, 6
Tags: irrational number , number theory
Let $a,b$ and $c$ be integers and $r$ a real number such that $ar^2+br+c=0$ with $ac\not =0$.Prove that $\sqrt{r^2+c^2}$ is an irrational number

2021 Brazil Undergrad MO, Problem 3
Tags: power , real analysis , irrational number , approximation , perfect square
Find all positive integers $k$ for which there is an irrational $\alpha>1$ and a positive integer $N$ such that $\left\lfloor\alpha^{n}\right\rfloor$ is of the form $m^2-k$ com $m \in \mathbb{Z}$ for every integer $n>N$.