Olympiad problems

2018 Romania National Olympiad, 4

Find the natural number $n$ for which $$\sqrt{\frac{20^n- 18^n}{19}}$$ is a rational number.

1984 Czech And Slovak Olympiad IIIA, 4

Tags: algebra , irrational number , rational , number theory

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

2001 Junior Balkan Team Selection Tests - Moldova, 5

Tags: number theory , rational

Determine if there is a non-natural natural number $n$ with the property that $\sqrt{n + 1} + \sqrt{n - 1}$ is rational.

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?

1995 Yugoslav Team Selection Test, Problem 1

Tags: number theory , rational , rational number

Determine all triples $(x,y,z)$ of positive rational numbers with $x\le y\le z$ such that $x+y+z,\frac1x+\frac1y+\frac1z$, and xyz are natural numbers.

2018 Junior Balkan Team Selection Tests - Romania, 1

Tags: algebra , rational

Prove that the equation $x^2+y^2+z^2 = x+y+z+1$ has no rational solutions.

1999 IMO Shortlist, 4

Tags: number theory , decimal representation , periodic function , rational

Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write \[\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , \] where $r=r(p)$; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by \[ f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}\] a) Prove that $S$ is infinite. b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$

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.

1998 Romania National Olympiad, 3

Tags: algebra , rational , number theory

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

1974 Yugoslav Team Selection Test, Problem 1

Tags: rational , floor function , number theory , algebra , function

Assume that $a$ is a given irrational number. (a) Prove that for each positive real number $\epsilon$ there exists at least one integer $q\ge0$ such that $aq-\lfloor aq\rfloor<\epsilon$. (b) Prove that for given $\epsilon>0$ there exist infinitely many rational numbers $\frac pq$ such that $q>0$ and $\left|a-\frac pq\right|<\frac\epsilon q$.

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.

2014 Saudi Arabia GMO TST, 4

Tags: rational , set , algebra

Let $X$ be a set of rational numbers satisfying the following two conditions: (a) The set $X$ contains at least two elements, (b) For any $x, y$ in $X$, if $x \ne y$ then there exists $z$ in $X$ such that either $\left| \frac{x- z}{y - z} \right|= 2$ or $\left| \frac{y -z}{x - z} \right|= 2$ . Prove that $X$ contains infinitely many elements.

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

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.

2013 Hanoi Open Mathematics Competitions, 15

Tags: algebra , rational

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

1969 Czech and Slovak Olympiad III A, 1

Tags: algebra , equation , quadratic polynomial , rational

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

1988 All Soviet Union Mathematical Olympiad, 469

Tags: rational , square , number theory

If rationals $x, y$ satisfy $x^5 + y^5 = 2 x^2 y^2$, show that $1-x y$ is the square of a rational.

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.

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.

2019 Czech-Polish-Slovak Junior Match, 1

Tags: number theory , rational , integer

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

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.

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.

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.

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?

2006 Abels Math Contest (Norwegian MO), 3

Tags: number theory , diophantine , rational , integer

(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.