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

2023 Romanian Master of Mathematics Shortlist, A1
Tags: algebra , rational number , polynomial
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 $P(x + y)$.

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

2019 Thailand TST, 2
Tags: algebra , rational number , set
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 $S$.

1986 IMO Longlists, 67
Tags: algebra , decimal representation , rational number , number theory
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 $a = 0.1d_1(2)d_2(2) \cdots d_{r(2)}(2)d_1(3) . . . d_{r(3)}(3)d_1(4) \cdots .$

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

2018 IMO Shortlist, A3
Tags: algebra , rational number , set
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 $S$.

2018 Greece Junior Math Olympiad, 1
Tags: rational number , algebra
a) Does there exist a real number $x$ such that $x+\sqrt{3}$ and $x^2+\sqrt{3}$ are both rationals? b) Does there exist a real number $y$ such that $y+\sqrt{3}$ and $y^3+\sqrt{3}$ are both rationals?

PEN F Problems, 1
Tags: geometry , rectangle , induction , rational number
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\}$.

2000 Poland - Second Round, 1
Tags: fraction , algebra , 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.

2017 Peru MO (ONEM), 3
Tags: algebra , rational number
The infinity sequence $r_{1},r_{2},...$ of rational numbers it satisfies that: $\prod_{i=1}^ {k}r_{i}=\sum_{i=1}^{k} r_{i}$. For all natural k. Show that $\frac{1}{r_{n}}-\frac{3}{4}$ is a square of rationale number for all natural $n\geq3$

2001 Moldova National Olympiad, Problem 5
Tags: algebra , polynomial , rational number
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.

1986 IMO Shortlist, 2
Tags: algebra , decimal representation , rational number , number theory
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 Shortlist, 2
Tags: induction , algebra , series summation , rational number
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 $10^6.$

1983 IMO Shortlist, 10
Tags: polynomial , approximation , number theory , rational number , algebra
Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that \[ \left|P(x)-\frac pq \right| < \frac{1}{q^2}\]for all $x \in I.$

1995 Yugoslav Team Selection Test, Problem 1
Tags: rational number , number theory , rational
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.

1967 IMO Shortlist, 2
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).

2016 Poland - Second Round, 1
Tags: geometry , rational number , algebra
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.

PEN F Problems, 4
Tags: rational number , trigonometry , quadratic
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.

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 $10^{6}$.

1962 Putnam, A6
Tags: rational number , set
Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $ab$ and $a+b$, and having the property that for every rational number $r$ exactly one of the following three statements is true: $$r\in S,\;\; -r\in S,\;\;r =0.$$ Prove that $S$ is the set of all positive rational numbers.

1990 IMO Longlists, 1
Tags: analytic geometry , vector , number theory , rational number
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.

2004 Finnish National High School Mathematics Competition, 2
Tags: algebra , rational number , integrality
$a, b$ and $c$ are positive integers and \[\frac{a\sqrt{3} + b}{b\sqrt{3} + c}\] is a rational number. Show that \[\frac{a^2 + b^2 + c^2}{a + b + c}\] is an integer.

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

2018 Brazil Team Selection Test, 1
Tags: irrational number , rational number , blackboard , algebra , combinatorics
The numbers $1- \sqrt{2}$, $\sqrt{2}$ and $1+\sqrt{2}$ are written on a blackboard. Every minute, if $x, y, z$ are the numbers written, then they are erased and the numbers, $x^2 + xy + y^2$, $y^2 + yz + z^2$ and $z^2 + zx + x^2$ are written. Determine whether it is possible for all written numbers to be rational numbers after a finite number of minutes.