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: 106

2022 Junior Balkan Team Selection Tests - Moldova, 4
Tags: fraction , number theory , rational
Rational number $\frac{m}{n}$ admits representation $$\frac{m}{n} = 1+ \frac12+\frac13 + ...+ \frac{1}{p-1}$$ where p $(p > 2)$ is a prime number. Show that the number $m$ is divisible by $p$.

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.

1989 ITAMO, 1
Tags: equation , rational , algebra
Determine whether the equation $x^2 +xy+y^2 = 2$ has a solution $(x,y)$ in rational numbers.

2015 Estonia Team Selection Test, 3
Tags: number theory , rational
Let $q$ be a fixed positive rational number. Call number $x$ [i]charismatic [/i] if there exist a positive integer $n$ and integers $a_1, a_2, . . . , a_n$ such that $x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} ...(q + n)^{a_n}$. a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic. b) Is it true for every $q$ that, for every charismatic number $x$, the number $x + 1$ is charismatic, too?

2009 Dutch IMO TST, 5
Tags: geometry , rational , polygon , equal segments , analytic geometry
Suppose that we are given an $n$-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that $n$ is even.

2000 Regional Competition For Advanced Students, 4
Tags: rational , number theory , algebra , sequence , recurrence relation
We consider the sequence $\{u_n\}$ defined by recursion $u_{n+1} =\frac{u_n(u_n + 1)}{n}$ for $n \ge 1$. (a) Determine the terms of the sequence for $u_1 = 1$. (b) Show that if a member of the sequence is rational, then all subsequent members are also rational numbers. (c) Show that for every natural number $K$ there is a $u_1 > 1$ such that the first $K$ terms of the sequence are natural numbers.

2006 Abels Math Contest (Norwegian MO), 3
Tags: integer , number theory , diophantine , rational
(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection. (b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.

2008 Mathcenter Contest, 4
Tags: number theory , algebra , rational , integer
Let $a,b$ and $c$ be positive integers that $$\frac{a\sqrt{3}+b}{b\sqrt3+c}$$ is a rational number, show that $$\frac{a^2+b^2+c^2}{a+b+ c}$$ is an integer. [i](Anonymous314)[/i]

1994 Tournament Of Towns, (414) 2
Tags: algebra , rational , sequence , periodic
Consider a sequence of numbers between $0$ and $1$ in which the next number after $x$ is $1 - |1 - 2x|$. ($|x| = x$ if$ x \ge 0$, $|x| = -x$ if $x < 0$.) Prove that (a) if the first number of the sequence is rational, then the sequence will be periodic (i.e. the terms repeat with a certain cycle length after a certain term in the sequence); (b) if the sequence is periodic, then the first number is rational. (G Shabat)

1978 Chisinau City MO, 166
Tags: analytic geometry , irrational , rational , circles
It is known that at least one coordinate of the center $(x_0, y_0)$ of the circle $(x -x_0)^2+ (y -y_0)^2 = R^2$ is irrational. Prove that on the circle itself there are at most two points with rational coordinates.

2013 Balkan MO Shortlist, N9
Tags: number theory , rational
Let $n\ge 2$ be a given integer. Determine all sequences $x_1,...,x_n$ of positive rational numbers such that $x_1^{x_2}=x_2^{x_3}=...=x_{n-1}^{x_n}=x_n^{x_1}$

2017 Junior Balkan Team Selection Tests - Romania, 3
Tags: number theory , rational
Determine the integers $x$ and $y$ for which $\sqrt{4^x + 5^y}$ is rational.

2011 QEDMO 8th, 3
Tags: number theory , rational
Show that every rational number $r$ can be written as the sum of numbers in the form $\frac{a}{p^k}$ where $p$ is prime, $a$ is an integer and $k$ is natural.

VMEO IV 2015, 10.1
Tags: algebra , rational
Given a real number $\alpha$ satisfying $\alpha^3 = \alpha + 1$. Determine all $4$-tuples of rational numbers $(a, b, c, d)$ satisfying: $a\alpha^2 + b\alpha+ c = \sqrt{d}.$

2020 India National Olympiad, 5
Tags: number theory , tiling , descent , trigonometry , niven theorem , rational , galois theory
Infinitely many equidistant parallel lines are drawn in the plane. A positive integer $n \geqslant 3$ is called frameable if it is possible to draw a regular polygon with $n$ sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon. (a) Show that $3, 4, 6$ are frameable. (b) Show that any integer $n \geqslant 7$ is not frameable. (c) Determine whether $5$ is frameable. [i]Proposed by Muralidharan[/i]

1975 IMO, 5
Tags: trigonometry , algebra , point set , rational , roots of unity
Can there be drawn on a circle of radius $1$ a number of $1975$ distinct points, so that the distance (measured on the chord) between any two points (from the considered points) is a rational number?

2002 All-Russian Olympiad Regional Round, 11.1
Tags: number theory , algebra , rational , irrational
The real numbers $x$ and $y$ are such that for any distinct odd primes $p$ and $q$ the number $x^p + y^q$ is rational. Prove that $x$ and $y$ are rational numbers.

2017 Federal Competition For Advanced Students, P2, 4
Tags: maximum , inequalities , three variable inequality , rational , algebra
(a) Determine the maximum $M$ of $x+y +z$ where $x, y$ and $z$ are positive real numbers with $16xyz = (x + y)^2(x + z)^2$. (b) Prove the existence of infinitely many triples $(x, y, z)$ of positive rational numbers that satisfy $16xyz = (x + y)^2(x + z)^2$ and $x + y + z = M$. Proposed by Karl Czakler

1992 ITAMO, 6
Tags: perfect cube , rational , number theory
Let $a$ and $b$ be integers. Prove that if $\sqrt[3]{a}+\sqrt[3]{b}$ is a rational number, then both $a$ and $b$ are perfect cubes.

2008 Postal Coaching, 2
Tags: altitude , sidelenghts , rational , geometry
Does there exist a triangle $ABC$ whose sides are rational numbers and $BC$ equals to the altitude from $A$?

1995 Singapore MO Open, 1
Tags: polynomial , rational , algebra , root
Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer

2017 Junior Regional Olympiad - FBH, 5
Tags: rational , sqaure roots
Find all positive integers $a$ and $b$ such that number $p=\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ is rational number

VMEO III 2006 Shortlist, A10
Tags: algebra , rational , sequence , floor function
Let ${a_n}$ be a sequence defined by $a_1=2$, $a_{n+1}=\left[ \frac {3a_n}{2}\right]$ $\forall n \in \mathbb N$ $0.a_1a_2...$ rational or irrational?

1955 Moscow Mathematical Olympiad, 316
Tags: algebra , polynomial , divisor , integer polynomial , rational
Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.

VMEO IV 2015, 12.1
Tags: rational , algebra
Given a set $S \subset R^+$, $S \ne \emptyset$ such that for all $a, b, c \in S$ (not necessarily distinct) then $a^3 + b^3 + c^3 - 3abc$ is rational number. Prove that for all $a, b \in S$ then $\frac{a - b}{a + b}$ is also rational.