Olympiad problems

1976 Bundeswettbewerb Mathematik, 3

A set $S$ of rational numbers is ordered in a tree-diagram in such a way that each rational number $\frac{a}{b}$ (where $a$ and $b$ are coprime integers) has exactly two successors: $\frac{a}{a+b}$ and $\frac{b}{a+b}$. How should the initial element be selected such that this tree contains the set of all rationals $r$ with $0 < r < 1$? Give a procedure for determining the level of a rational number $\frac{p}{q}$ in this tree.

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.

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 $0.1491625\ldots$)

1994 Bundeswettbewerb Mathematik, 4

Tags: arithmetic sequence , geometric sequence , rational , algebra

Let $a,b$ be real numbers ($b\ne 0$) and consider the infinite arithmetic sequence $a, a+b ,a +2b , \ldots.$ Show that this sequence contains an infinite geometric subsequence if and only if $\frac{a}{b}$ is rational.

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.

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.

1981 Austrian-Polish Competition, 2

Tags: sequence , recurrence relation , algebra , rational

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 $a_q - a_p = a_m - a_k$.

1989 Chile National Olympiad, 5

Tags: geometry , rational , altitude

The lengths of the three sides of a $ \triangle ABC $ are rational. The altitude $ CD $ determines on the side $AB$ two segments $ AD $ and $ DB $. Prove that $ AD, DB $ are rational.

2015 Latvia Baltic Way TST, 8

Tags: algebra , rational , number theory

Given a fixed rational number $q$. Let's call a number $x$ [i]charismatic [/i] if we can find a natural number $n$ and integers $a_1, a_2,.., a_n$ such that $$x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} \cdot ... \cdot(q + n)^{a_n} .$$ i) Prove that one can find a $q$ such that all positive rational numbers are charismatic. ii) Is it true that for all $q$, if the number $x$ is charismatic, then $x + 1$ is also charismatic?

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^* $

2019 Czech-Polish-Slovak Junior Match, 1

Tags: integer , number theory , rational

Rational numbers $a, b$ are such that $a+b$ and $a^2+b^2$ are integers. Prove that $a, b$ are integers.

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 $(0, 1)$.

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.

1990 All Soviet Union Mathematical Olympiad, 516

Tags: quadratic , root , rational , algebra , trinomial

Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots.

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.

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.

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

1984 Czech And Slovak Olympiad IIIA, 4

Tags: number theory , irrational number , rational , algebra

Let $r$ be a natural number greater than $1$. Then there exist positive irrational numbers $x, y$ such that $x^y = r$ . Prove it.

Estonia Open Senior - geometry, 2011.2.3

Tags: ratio , geometry , rational , area

Let $ABC$ be a triangle with integral side lengths. The angle bisector drawn from $B$ and the altitude drawn from $C$ meet at point $P$ inside the triangle. Prove that the ratio of areas of triangles $APB$ and $APC$ is a rational number.

1995 Portugal MO, 6

Tags: rational , geometric progression , algebra

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.

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 $y$.

1978 Kurschak Competition, 1

Tags: algebra , number theory , rational

$a$ and $b$ are rationals. Show that if $ax^2 + by^2 = 1$ has a rational solution (in $x$ and $y$), then it must have infinitely many.

2013 Nordic, 3

Tags: sequence , rational , number theory

Define a sequence ${(n_k)_{k\ge 0}}$ by ${n_{0 }= n_{1} = 1}$, and ${n_{2k} = n_k + n_{k-1} }$ and ${n_{2k+1} = n_k}$ for ${k \ge 1}$. Let further ${q_k = n_k }$ / ${ n_{k-1} }$ for each ${k \ge 1}$. Show that every positive rational number is present exactly once in the sequence ${(q_k)_{k\ge 1}}$

1975 IMO Shortlist, 15

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?

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.