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

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Found problems: 102

2016 India IMO Training Camp, 1
Tags: irrational number , algebra
Suppose $\alpha, \beta$ are two positive rational numbers. Assume for some positive integers $m,n$, it is known that $\alpha^{\frac 1n}+\beta^{\frac 1m}$ is a rational number. Prove that each of $\alpha^{\frac 1n}$ and $\beta^{\frac 1m}$ is a rational number.

PEN G Problems, 19
Tags: induction , irrational number , geometry
Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with a rational area.

1957 AMC 12/AHSME, 36
Tags: calculus , derivative , irrational number
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$

2018 District Olympiad, 2
Tags: irrational number
Show that the number \[\sqrt[n]{\sqrt{2019} + \sqrt{2018}} + \sqrt[n]{\sqrt{2019} - \sqrt{2018}}\] is irrational for any $n\ge 2$.

2016 International Zhautykov Olympiad, 3
Tags: number theory , irrational number
We call a positive integer $q$ a $convenient \quad denominator$ for a real number $\alpha$ if $\displaystyle |\alpha - \dfrac{p}{q}|<\dfrac{1}{10q}$ for some integer $p$. Prove that if two irrational numbers $\alpha$ and $\beta$ have the same set of convenient denominators then either $\alpha+\beta$ or $\alpha- \beta$ is an integer.

1989 IMO Shortlist, 9
Tags: number theory , algebra , irrational number , equation
$ \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.$

2012 USAJMO, 4
Tags: floor function , analytic geometry , irrational number , induction
Let $\alpha$ be an irrational number with $0<\alpha < 1$, and draw a circle in the plane whose circumference has length $1$. Given any integer $n\ge 3$, define a sequence of points $P_1, P_2, \ldots , P_n$ as follows. First select any point $P_1$ on the circle, and for $2\le k\le n$ define $P_k$ as the point on the circle for which the length of arc $P_{k-1}P_k$ is $\alpha$, when travelling counterclockwise around the circle from $P_{k-1}$ to $P_k$. Suppose that $P_a$ and $P_b$ are the nearest adjacent points on either side of $P_n$. Prove that $a+b\le n$.

PEN G Problems, 16
Tags: inequalities , algebra , integration , irrational number , polynomial
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 \[\left\vert \pi-a_{n}\right\vert < \frac{1}{4^{5n-1}}, \; n=1,2, \cdots.\]

2005 Miklós Schweitzer, 2
Tags: number theory with sequences , integer sequence , periodic function , irrational number , ceiling function , number theory
Let $(a_{n})_{n \ge 1}$ be a sequence of integers satisfying the inequality \[ 0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1 \] for all $n \ge 2$. Prove that the sequence $(a_{n})$ is periodic. Any Hints or Sols for this hard problem?? :help:

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

2003 Portugal MO, 6
Tags: algebra , natural number , irrational number , al , number theory
Given six irrational numbers, will it be possible to choose three such that the sum of any two of these three is irrational?

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

1967 IMO Shortlist, 4
Tags: number theory , approximation , irrational number
Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.

2012 District Olympiad, 1
Tags: algebra , irrational number , number theory
Let $a_1, a_2, ... , a_{2012}$ be odd positive integers. Prove that the number $$A=\sqrt{a^2_1+ a^2_2+ ...+ a^2_{2012}-1}$$ is irrational.

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

PEN G Problems, 13
Tags: irrational number
It is possible to show that $ \csc\frac{3\pi}{29}\minus{}\csc\frac{10\pi}{29}\equal{} 1.999989433...$. Prove that there are no integers $ j$, $ k$, $ n$ with odd $ n$ satisfying $ \csc\frac{j\pi}{n}\minus{}\csc\frac{k\pi}{n}\equal{} 2$.

2000 Saint Petersburg Mathematical Olympiad, 11.7
Tags: algebra , set , irrational number
It is known that for irrational numbers $\alpha$, $\beta$, $\gamma$, $\delta$ and for any positive integer $n$ the following is true: $$[n\alpha]+[n\beta]=[n\gamma]+[n\delta]$$ Does this mean that sets $\{\alpha,\beta\}$ and $\{\gamma,\delta\}$ are equal? (As usual $[x]$ means the greatest integer not greater than $x$).

1998 South africa National Olympiad, 1
Tags: prime factorization , rational number , irrational number , logarithm , number theory
Is $\log_{10}{8}$ rational?

2016 USA Team Selection Test, 1
Tags: binary representation , irrational number , number theory
Let $\sqrt 3 = 1.b_1b_2b_3 \dots _{(2)}$ be the binary representation of $\sqrt 3$. Prove that for any positive integer $n$, at least one of the digits $b_n$, $b_{n+1}$, $\dots$, $b_{2n}$ equals $1$.

1987 IMO Shortlist, 23
Tags: number theory , polynomial , irrational number , algebra , modular arithmetic
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]

2012 Ukraine Team Selection Test, 6
Tags: algebra , irrational number , irrational , digit , decimal representation
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.

KoMaL A Problems 2024/2025, A. 903
Tags: floor function , irrational number , continued fraction , number theory
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]

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

2000 Romania National Olympiad, 1
Tags: irrational number , number theory
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. $

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