This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 106

1997 Bundeswettbewerb Mathematik, 2
Tags: rational , analytic geometry , number theory
Show that for any rational number $a$ the equation $y =\sqrt{x^2 +a}$ has infinitely many solutions in rational numbers $x$ and $y$.

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?

1981 Austrian-Polish Competition, 2
Tags: recurrence relation , sequence , rational , algebra
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 $a_q - a_p = a_m - a_k$.

2009 Dutch IMO TST, 5
Tags: equal segments , rational , polygon , analytic geometry , 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.

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.

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]

2008 Mathcenter Contest, 4
Tags: number theory , algebra , integer , rational
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]

Estonia Open Senior - geometry, 2011.2.3
Tags: ratio , area , rational , geometry
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.

1976 Poland - Second Round, 5
Tags: irrational , rational , trigonometry , algebra
Prove that if $ \cos \pi x =\frac{1}{3} $ then $ x $ is an irrational number.

1978 Kurschak Competition, 1
Tags: rational , number theory , algebra
$a$ 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.

1975 IMO, 5
Tags: point set , rational , roots of unity , trigonometry , algebra
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?

2013 Balkan MO Shortlist, N9
Tags: rational , number theory
Let $n\ge 2$ be a given integer. Determine all sequences $x_1,...,x_n$ of positive rational numbers such that $x_1^{x_2}=x_2^{x_3}=...=x_{n-1}^{x_n}=x_n^{x_1}$

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^* $

2022 Junior Balkan Team Selection Tests - Moldova, 4
Tags: rational , fraction , number theory
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 $p$.

2002 District Olympiad, 2
Tags: irrational number , rational , algebra
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.

1975 IMO Shortlist, 15
Tags: trigonometry , point set , roots of unity , rational , algebra
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?

2018 Junior Balkan Team Selection Tests - Romania, 1
Tags: rational , algebra
Prove that the equation $x^2+y^2+z^2 = x+y+z+1$ has no rational solutions.

2002 All-Russian Olympiad Regional Round, 11.1
Tags: rational , irrational , number theory , algebra
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.

1993 ITAMO, 2
Tags: quadratic trinomial , rational roots , rational , prime , quadratic , trinomial , number theory
Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.

2006 Abels Math Contest (Norwegian MO), 3
Tags: rational , number theory , diophantine , 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.

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.

1990 All Soviet Union Mathematical Olympiad, 516
Tags: rational , algebra , quadratic , root , trinomial
Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots.

2006 MOP Homework, 1
Tags: rational , rational number , set , strong induction , number theory
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 $(0, 1)$.

1985 Poland - Second Round, 4
Tags: perfect cube , rational , radical , algebra , 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.

2015 Hanoi Open Mathematics Competitions, 13
Tags: equation , rational , algebra
Give rational numbers $x, y$ such that $(x^2 + y^2 - 2) (x + y)^2 + (xy + 1)^2 = 0 $ Prove that $\sqrt{1 + xy}$ is a rational number.