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: 22

1995 Yugoslav Team Selection Test, Problem 1
Tags: number theory , rational number , rational , rational numbers
Determine all triples $(x,y,z)$ of positive rational numbers with $x\le y\le z$ such that $x+y+z,\frac1x+\frac1y+\frac1z$, and xyz are natural numbers.

2019 India Regional Mathematical Olympiad, 1
Tags: RMO , algebra , rational numbers
Suppose $x$ is a non zero real number such that both $x^5$ and $20x+\frac{19}{x}$ are rational numbers. Prove that $x$ is a rational number.

PEN F Problems, 2
Tags: trigonometry , algebra , polynomial , rational numbers
Find all $x$ and $y$ which are rational multiples of $\pi$ with $0<x<y<\frac{\pi}{2}$ and $\tan x+\tan y =2$.

PEN F Problems, 5
Tags: floor function , number theory , relatively prime , rational numbers
Prove that there is no positive rational number $x$ such that \[x^{\lfloor x\rfloor }=\frac{9}{2}.\]

PEN F Problems, 10
Tags: irrational number , rational numbers
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.

PEN F Problems, 1
Tags: geometry , rectangle , induction , rational numbers
Suppose that a rectangle with sides $ a$ and $ b$ is arbitrarily cut into $ n$ squares with sides $ x_{1},\ldots,x_{n}$. Show that $ \frac{x_{i}}{a}\in\mathbb{Q}$ and $ \frac{x_{i}}{b}\in\mathbb{Q}$ for all $ i\in\{1,\cdots, n\}$.

PEN F Problems, 3
Tags: trigonometry , rational numbers , IMO Shortlist
Let $ \alpha$ be a rational number with $ 0 < \alpha < 1$ and $ \cos (3 \pi \alpha) \plus{} 2\cos(2 \pi \alpha) \equal{} 0$. Prove that $ \alpha \equal{} \frac {2}{3}$.

PEN F Problems, 12
Tags: trigonometry , rational numbers
Does there exist a circle and an infinite set of points on it such that the distance between any two points of the set is rational?

PEN F Problems, 16
Tags: rational numbers
Prove that for any distinct rational numbers $a, b, c$, the number \[\frac{1}{(b-c)^{2}}+\frac{1}{(c-a)^{2}}+\frac{1}{(a-b)^{2}}\] is the square of some rational number.

PEN F Problems, 7
Tags: search , rational numbers
If $x$ is a positive rational number, show that $x$ can be uniquely expressed in the form \[x=a_{1}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots,\] where $a_{1}a_{2},\cdots$ are integers, $0 \le a_{n}\le n-1$ for $n>1$, and the series terminates. Show also that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^{6}$.

KoMaL A Problems 2021/2022, A. 822
Tags: number theory , rational numbers , Elliptic Curves
Is it possible to find $p,q,r\in\mathbb Q$ such that $p+q+r=0$ and $pqr=1$? [i]Proposed by Máté Weisz, Cambridge[/i]

2019 Poland - Second Round, 3
Tags: functions , rational numbers , algebra , number theory , function
Let $f(t)=t^3+t$. Decide if there exist rational numbers $x, y$ and positive integers $m, n$ such that $xy=3$ and: \begin{align*} \underbrace{f(f(\ldots f(f}_{m \ times}(x))\ldots)) = \underbrace{f(f(\ldots f(f}_{n \ times}(y))\ldots)). \end{align*}

PEN F Problems, 15
Tags: trigonometry , algebra , polynomial , rational numbers
Find all rational numbers $k$ such that $0 \le k \le \frac{1}{2}$ and $\cos k \pi$ is rational.

PEN F Problems, 8
Tags: algebra , polynomial , analytic geometry , function , induction , blogs , rational numbers
Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.

1991 IMO Shortlist, 19
Tags: trigonometry , rational numbers , IMO Shortlist
Let $ \alpha$ be a rational number with $ 0 < \alpha < 1$ and $ \cos (3 \pi \alpha) \plus{} 2\cos(2 \pi \alpha) \equal{} 0$. Prove that $ \alpha \equal{} \frac {2}{3}$.

PEN F Problems, 11
Tags: rational numbers
Let $S=\{x_0, x_1, \cdots, x_n\} \subset [0,1]$ be a finite set of real numbers with $x_{0}=0$ and $x_{1}=1$, such that every distance between pairs of elements occurs at least twice, except for the distance $1$. Prove that all of the $x_i$ are rational.

PEN F Problems, 13
Tags: factorial , rational numbers
Prove that numbers of the form \[\frac{a_{1}}{1!}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots,\] where $0 \le a_{i}\le i-1 \;(i=2, 3, 4, \cdots)$ are rational if and only if starting from some $i$ on all the $a_{i}$'s are either equal to $0$ ( in which case the sum is finite) or all are equal to $i-1$.

PEN F Problems, 9
Tags: geometry , 3D geometry , rational numbers
Prove that every positive rational number can be represented in the form \[\frac{a^{3}+b^{3}}{c^{3}+d^{3}}\] for some positive integers $a, b, c$, and $d$.

2006 MOP Homework, 1
Tags: set , rational numbers , rational , number theory , strong induction
Let $S$ be a set of rational numbers with the following properties: (a) $\frac12$ is an element in $S$, (b) if $x$ is in $S$, then both $\frac{1}{x+1}$ and $\frac{x}{x+1}$ are in $S$. Prove that $S$ contains all rational numbers in the interval $(0, 1)$.

PEN F Problems, 14
Tags: rational numbers
Let $k$ and $m$ be positive integers. Show that \[S(m, k)=\sum_{n=1}^{\infty}\frac{1}{n(mn+k)}\] is rational if and only if $m$ divides $k$.

PEN F Problems, 6
Tags: rational numbers
Let $x, y, z$ non-zero real numbers such that $xy$, $yz$, $zx$ are rational. [list=a] [*] Show that the number $x^{2}+y^{2}+z^{2}$ is rational. [*] If the number $x^{3}+y^{3}+z^{3}$ is also rational, show that $x$, $y$, $z$ are rational. [/list]

PEN F Problems, 4
Tags: trigonometry , quadratics , rational numbers
Suppose that $\tan \alpha =\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove the number $\tan \beta$ for which $\tan 2\beta =\tan 3\alpha$ is rational only when $p^2 +q^2$ is the square of an integer.