Olympiad problems

1976 Swedish Mathematical Competition, 3

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.

1976 Bundeswettbewerb Mathematik, 3

Tags: algebra , tree , rational

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.

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.

2013 Dutch IMO TST, 2

Tags: perfect square , number theory , rational

Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.

1992 ITAMO, 6

Tags: number theory , perfect cube , rational

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.

1979 Swedish Mathematical Competition, 2

Tags: algebra , radical , rational

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

1969 Czech and Slovak Olympiad III A, 1

Tags: algebra , equation , quadratic polynomial , rational

Find all rational numbers $x,y$ such that \[\left(x+y\sqrt5\right)^2=7+3\sqrt5.\]

1996 Tuymaada Olympiad, 7

Tags: equation , algebra , 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 $a * b = b * a$.

2011 QEDMO 8th, 3

Tags: rational , number theory

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.

1969 Polish MO Finals, 3

Tags: geometry , symmetry , octagon , rational

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

1959 Polish MO Finals, 4

Tags: polynomial , algebra , rational

Prove that if a quadratic equation $$ ax^2 + bx + c = 0$$ with integer coefficients has a rational root, then at least one of the numbers $ a $, $ b $, $ c $ is even.

1940 Putnam, B5

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

2015 Latvia Baltic Way TST, 8

Tags: number theory , algebra , rational

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?

2011 Junior Balkan Team Selection Tests - Romania, 4

Tags: algebra , rational , sum

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.

2019 Ecuador NMO (OMEC), 6

Tags: geometry , rational , area

Let $n\ge 3$ be a positive integer. Danielle draws a math flower on the plane Cartesian as follows: first draw a unit circle centered on the origin, then draw a polygon of $n$ vertices with both rational coordinates on the circumference so that it has two diametrically opposite vertices, on each side draw a circumference that has the diameter of that side, and finally paints the area inside the $n$ small circles but outside the unit circle. If it is known that the painted area is rational, find all possible polygons drawn by Danielle.

1998 Romania National Olympiad, 3

Tags: number theory , algebra , rational

Find the rational roots (if any) of the equation $$abx^2 + (a^2 + b^2 )x +1 = 0 , \,\,\,\, (a, b \in Z).$$

2019 Argentina National Olympiad, 4

Tags: algebra , rational

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.

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 $q > 2$.

2012 Junior Balkan Team Selection Tests - Romania, 2

Tags: algebra , equation , rational

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 $x = y$.

2007 Germany Team Selection Test, 3

Tags: rational , decimal representation , number theory

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]

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)

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: $a\alpha^2 + b\alpha+ c = \sqrt{d}.$

2001 Swedish Mathematical Competition, 2

Tags: algebra , rational , radical

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

2013 Romania National Olympiad, 4

Tags: algebra , rational

A set $M$ of real numbers will be called [i]special [/i] if it has the properties: (i) for each $x, y \in M, x\ne y$, the numbers $x + y$ and $xy$ are not zero and exactly one of them is rational; (ii) for each $x \in M, x^2$ is irrational. Find the maximum number of elements of a [i]special [/i] set.

1986 Spain Mathematical Olympiad, 5

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

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.