Olympiad problems

2007 Germany Team Selection Test, 3

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$. [i]Proposed by J.P. Grossman, Canada[/i]

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.

1983 Kurschak Competition, 1

Tags: algebra , rational

Let $x, y$ and $z$ be rational numbers satisfying $$x^3 + 3y^3 + 9z^3 - 9xyz = 0.$$ Prove that $x = y = z = 0$.

2014 IMAC Arhimede, 4

Tags: polynomial , algebra , sum of powers , sum , rational

Let $n$ be a natural number and let $P (t) = 1 + t + t^2 + ... + t^{2n}$. If $x \in R$ such that $P (x)$ and $P (x^2)$ are rational numbers, prove that $x$ is rational number.

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)

2004 Estonia Team Selection Test, 4

Tags: trigonometry , sum , rational , algebra

Denote $f(m) =\sum_{k=1}^m (-1)^k cos \frac{k\pi}{2 m + 1}$ For which positive integers $m$ is $f(m)$ rational?

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.

2017 Czech-Polish-Slovak Junior Match, 6

Tags: integer , rational , number theory

On the board are written $100$ mutually different positive real numbers, such that for any three different numbers $a, b, c$ is $a^2 + bc$ is an integer. Prove that for any two numbers $x, y$ from the board , number $\frac{x}{y}$ is rational.

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

2015 Estonia Team Selection Test, 3

Tags: rational , number theory

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?

2020 Canadian Mathematical Olympiad Qualification, 8

Tags: radical , rational , equation , algebra

Find all pairs $(a, b)$ of positive rational numbers such that $\sqrt[b]{a}= ab$

2009 Greece JBMO TST, 3

Tags: algebra , rational , integration , radical

Given are the non zero natural numbers $a,b,c$ such that the number $\frac{a\sqrt2+b\sqrt3}{b\sqrt2+c\sqrt3}$ is rational. Prove that the number $\frac{a^2+b^2+c^2}{a+b+c}$ is an integer .

VMEO II 2005, 10

Tags: rational , algebra

a) Prove that for any positive integer $m > 2$, the equation $$y^3 = x^3_1 + x^3_2 + ... + x^3_m$$ always has a positive integer solution. b) Given a positive integer $n > 1$ and suppose $n \ne 3$. Prove that every rational number $x > 0$ can be expressed as $$x =\frac{a^3_1 + a^3_2 + ... + a^3_n}{b^3_1 + b^3_2 + ... + b^3_n}$$ where $a_i, b_i$ $(i = 1, . . . , n)$ are positive integers.

2010 Ukraine Team Selection Test, 12

Tags: algebra , rational , sum , reciprocal sum , number theory

Is there a positive integer $n$ for which the following holds: for an arbitrary rational $r$ there exists an integer $b$ and non-zero integers $a _1, a_2, ..., a_n$ such that $r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}$ ?

VII Soros Olympiad 2000 - 01, 10.2

Tags: trigonometry , algebra , rational

Let $a$ and $ b$ be acute corners. Prove that if $\sin a$, $\sin b$, and $\sin (a + b)$ are rational numbers, then $\cos a$, $\cos b$, and $\cos (a + b)$ are also rational numbers.

2022 Junior Balkan Team Selection Tests - Moldova, 4

Tags: number theory , rational , fraction

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 $p$.

2019 Argentina National Olympiad, 4

Tags: rational , algebra

If we have a set $M$ of $2019$ real numbers such that for every even $a$, $b$ of numbers of $M$ it is verified that $a^2+b \sqrt2$ is a rational number. Show that for all $a$ of $M$, $a\sqrt2$ is a rational number.

2012 Junior Balkan Team Selection Tests - Romania, 2

Tags: algebra , rational , equation

Let $x$ and $y$ be two rational numbers and $n$ be an odd positive integer. Prove that, if $x^n - 2x = y^n - 2y$, then $x = y$.

1997 Bundeswettbewerb Mathematik, 2

Tags: rational , number theory , analytic geometry

Show that for any rational number $a$ the equation $y =\sqrt{x^2 +a}$ has infinitely many solutions in rational numbers $x$ and $y$.

1989 ITAMO, 1

Tags: algebra , equation , rational

Determine whether the equation $x^2 +xy+y^2 = 2$ has a solution $(x,y)$ in rational numbers.

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.

2001 Abels Math Contest (Norwegian MO), 1b

Tags: rational , algebra

Suppose that $x$ and $y$ are positive real numbers such that $x^3, y^3$ and $x + y$ are all rational numbers. Show that the numbers $xy, x^2+y^2, x$ and $y$ are also rational

1979 Swedish Mathematical Competition, 2

Tags: radical , rational , algebra

Find rational $x$ in $(3,4)$ such that $\sqrt{x-3}$ and $\sqrt{x+1}$ are rational.

1995 Tournament Of Towns, (456) 1

Tags: geometry , combinatorics , analytic geometry , rational , combinatorial geometry

Does there exist a sphere passing through only one rational point? (A rational point is a point whose Cartesian coordinates are all rational numbers.) (A Rubin)

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]