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: 61

PEN F Problems, 12
Tags: trigonometry , rational number
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?

Russian TST 2016, P3
Tags: number theory , rational number
Prove that any rational number can be represented as a product of four rational numbers whose sum is zero.

PEN F Problems, 7
Tags: rational number , search
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 $10^{6}$.

2016 Poland - Second Round, 1
Tags: algebra , rational number , geometry
Point $P$ lies inside triangle of sides of length $3, 4, 5$. Show that if distances between $P$ and vertices of triangle are rational numbers then distances from $P$ to sides of triangle are rational numbers too.

2020 Bundeswettbewerb Mathematik, 2
Tags: rational number , algebra , number theory , system of equations
Prove that there are no rational numbers $x,y,z$ with $x+y+z=0$ and $x^2+y^2+z^2=100$.

KoMaL A Problems 2021/2022, A. 822
Tags: number theory , elliptic curves , rational number
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]

1967 IMO Shortlist, 2
Tags: algebra , series summation , rational number , induction
If $x$ is a positive rational number show that $x$ can be uniquely expressed in the form $x = \sum^n_{k=1} \frac{a_k}{k!}$ where $a_1, a_2, \ldots$ are integers, $0 \leq a_n \leq n - 1$, for $n > 1,$ and the series terminates. Show that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^6.$

PEN F Problems, 9
Tags: geometry , 3d geometry , rational number
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 $d$.

2012 Polish MO Finals, 1
Tags: number theory , rational number , power of number
Decide, whether exists positive rational number $w$, which isn't integer, such that $w^w$ is a rational number.

2019 Switzerland Team Selection Test, 3
Tags: algebra , set , rational number
Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

1990 IMO Longlists, 1
Tags: analytic geometry , number theory , rational number , vector
Prove that on the coordinate plane it is impossible to draw a closed broken line such that [i](i)[/i] the coordinates of each vertex are rational; [i](ii)[/i] the length each of its edges is 1; [i](iii)[/i] the line has an odd number of vertices.

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

1991 IMO Shortlist, 19
Tags: trigonometry , rational number
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}$.

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

2019 India IMO Training Camp, P1
Tags: rational number , set , algebra
Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

2023 Romanian Master of Mathematics Shortlist, A1
Tags: algebra , polynomial , rational number
Determine all polynomials $P$ with real coefficients satisfying the following condition: whenever $x$ and $y$ are real numbers such that $P(x)$ and $P(y)$ are both rational, so is $P(x + y)$.

2000 Poland - Second Round, 1
Tags: algebra , fraction , rational number
Decide, whether every positive rational number can present in the form $\frac{a^2 + b^3}{c^5 + d^7}$, where $a, b, c, d$ are positive integers.

1973 Putnam, B2
Tags: complex number , rational number
Let $z=x+yi$ be a complex number with $x$ and $y$ rational and with $|z|=1.$ Prove that the number $|z^{2n} -1|$ is rational for every integer $n$.

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

2013 Poland - Second Round, 5
Tags: polynomial , rational number , algebra
Let $W(x)$ be a polynomial of integer coefficients such that for any pair of different rational number $r_1$, $r_2$ dependence $W(r_1) \neq W(r_2)$ is true. Decide, whether the assuptions imply that for any pair of different real numbers $t_1$, $t_2$ dependence $W(t_1) \neq W(t_2)$ is true.

PEN F Problems, 11
Tags: rational number
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.

1986 IMO Shortlist, 2
Tags: algebra , number theory , decimal representation , rational number
Let $f(x) = x^n$ where $n$ is a fixed positive integer and $x =1, 2, \cdots .$ Is the decimal expansion $a = 0.f (1)f(2)f(3) . . .$ rational for any value of $n$ ? The decimal expansion of a is defined as follows: If $f(x) = d_1(x)d_2(x) \cdots d_{r(x)}(x)$ is the decimal expansion of $f(x)$, then $a = 0.1d_1(2)d_2(2) \cdots d_{r(2)}(2)d_1(3) . . . d_{r(3)}(3)d_1(4) \cdots .$

1967 IMO Longlists, 14
Tags: number theory , approximation , decimal representation , rational number , irrational number
Which fractions $ \dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\sqrt{2}$ (without using any table).

2001 Moldova National Olympiad, Problem 5
Tags: polynomial , rational number , algebra
Let $a,b,c,d$ be real numbers. Prove that the set $M=\left\{ax^3+bx^2+cx+d|x\in\mathbb R\right\}$ contains no irrational numbers if and only if $a=b=c=0$ and $d$ is rational.

PEN F Problems, 13
Tags: factorial , rational number
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 $i-1$.