Olympiad problems

1981 Austrian-Polish Competition, 2

The sequence $a_0, a_1, a_2, ...$ is defined by $a_{n+1} = a^2_n + (a_n - 1)^2$ for $n \ge 0$. Find all rational numbers $a_0$ for which there exist four distinct indices $k, m, p, q$ such that $a_q - a_p = a_m - a_k$.

1980 Bundeswettbewerb Mathematik, 1

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.

1976 Swedish Mathematical Competition, 3

Tags: algebra , perfect square , rational

If $a$, $b$, $c$ are rational, show that \[ \frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}+\frac{1}{(a-b)^2} \] is the square of a rational.

1993 ITAMO, 2

Tags: quadratic trinomial , trinomial , number theory , rational roots , rational , prime , quadratic

Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.

1989 All Soviet Union Mathematical Olympiad, 499

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

Do there exist two reals whose sum is rational, but the sum of their $n$ th powers is irrational for all $n > 1$? Do there exist two reals whose sum is irrational, but the sum of whose $n$ th powers is rational for all $n > 1$?

2017 Latvia Baltic Way TST, 13

Tags: algebra , rational , radical

Prove that the number $$\sqrt{1 + \frac{1}{n^2} + \frac{1}{(n+1)^2}}$$ is rational for all natural $n$.

1996 Tuymaada Olympiad, 7

Tags: algebra , equation , exponential , rational , exponential equation

In the set of all positive real numbers define the operation $a * b = a^b$ . Find all positive rational numbers for which $a * b = b * a$.

1969 Polish MO Finals, 3

Tags: geometry , octagon , rational , symmetry

Prove that an octagon, whose all angles are equal and all sides have rational length, has a center of symmetry.

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?

1989 Bundeswettbewerb Mathematik, 1

Tags: rational , decimal representation , periodic , number theory

For a given positive integer $n$, let $f(x) =x^{n}$. Is it possible for the decimal number $$0.f(1)f(2)f(3)\ldots$$ to be rational? (Example: for $n=2$, we are considering $0.1491625\ldots$)

2006 MOP Homework, 1

Tags: set , rational , number theory , strong induction , rational number

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 $(0, 1)$.

2015 Poland - Second Round, 1

Tags: polynomial , algebra , sum , rational , irrational number

Real numbers $x_1, x_2, x_3, x_4$ are roots of the fourth degree polynomial $W (x)$ with integer coefficients. Prove that if $x_3 + x_4$ is a rational number and $x_3x_4$ is a irrational number, then $x_1 + x_2 = x_3 + x_4$.

2011 Junior Balkan Team Selection Tests - Romania, 4

Tags: rational , sum , algebra

Let $k$ and $n$ be integer numbers with $2 \le k \le n - 1$. Consider a set $A$ of $n$ real numbers such that the sum of any $k$ distinct elements of $A$ is a rational number. Prove that all elements of the set $A$ are rational numbers.

1940 Putnam, B5

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

2018 Danube Mathematical Competition, 3

Tags: number theory , sum , product , rational , positive integer

Find all the positive integers $n$ with the property: there exists an integer $k > 2$ and the positive rational numbers $a_1, a_2, ..., a_k$ such that $a_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n$.

1986 Swedish Mathematical Competition, 3

Tags: algebra , rational

Let $N \ge 3$ be a positive integer. For every pair $(a,b)$ of integers with $1 \le a <b \le N$ consider the quotient $q = b/a$. Show that the pairs with $q < 2$ are equally numbered as those with $q > 2$.

2001 Swedish Mathematical Competition, 2

Tags: algebra , rational , radical

Show that $\sqrt[3]{\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5}$ is rational.

1995 Portugal MO, 6

Tags: rational , algebra , geometric progression

Prove that a real number $x$ is rational if and only if the sequence $x, x+1, x+2, x+3, ..., x+n, ...$ contains, at least least three terms in geometric progression.

1981 Austrian-Polish Competition, 5

Tags: polynomial , rational , algebra

Let $P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4$ be a polynomial with rational coefficients. Show that if $P(x)$ has exactly one real root $\xi$, then $\xi$ is a rational number.

1995 Singapore MO Open, 1

Tags: polynomial , rational , root , algebra

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

1986 Spain Mathematical Olympiad, 5

Tags: rational , triangle , coordinates , analytic geometry , algebra

Consider the curve $\Gamma$ defined by the equation $y^2 = x^3 +bx+b^2$, where $b$ is a nonzero rational constant. Inscribe in the curve $\Gamma$ a triangle whose vertices have rational coordinates.

1985 Poland - Second Round, 4

Tags: algebra , rational , radical , perfect cube , number theory

Prove that if for natural numbers $ a, b $ the number $ \sqrt[3]{a} + \sqrt[3]{b} $ is rational, then $ a, b $ are cubes of natural numbers.

2013 Hanoi Open Mathematics Competitions, 15

Tags: rational , algebra

Denote by $Q$ and $N^*$ the set of all rational and positive integer numbers, respectively. Suppose that $\frac{ax + b}{x} \in Q$ for every $x \in N^*$: Prove that there exist integers $A,B,C$ such that $\frac{ax + b}{x}= \frac{Ax + B}{Cx}$ for all $x \in N^* $

2021 Poland - Second Round, 4

Tags: rational , algebra

There are real numbers $x, y$ such that $x \ne 0$, $y \ne 0$, $xy + 1 \ne 0$ and $x + y \ne 0$. Suppose the numbers $x + \frac{1}{x} + y + \frac{1}{y}$ and $x^3+\frac{1}{x^3} + y^3 + \frac{1}{y^3}$ are rational. Prove that then the number $x^2+\frac{1}{x^2} + y^2 + \frac{1}{y^2}$ is also rational.

2020 India National Olympiad, 5

Tags: tiling , descent , trigonometry , niven theorem , rational , galois theory , number 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]