Found problems: 106
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.
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.
Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots.
Find rational $x$ in $(3,4)$ such that $\sqrt{x-3}$ and $\sqrt{x+1}$ are 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 $a_q - a_p = a_m - a_k$.
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 $n > 1$?
Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.
Does there exist a triangle $ABC$ whose sides are rational numbers and $BC$ equals to the altitude from $A$?
Let $a$ and $b$ be distinct positive real numbers, such that $a -\sqrt{ab}$ and $b -\sqrt{ab}$ are both rational numbers. Prove that $a$ and $b$ are rational numbers.
Prove that the number
$$\sqrt{1 + \frac{1}{n^2} + \frac{1}{(n+1)^2}}$$
is rational for all natural $n$.
Prove that an octagon, whose all angles are equal and all sides have rational length, has a center of symmetry.
Let $(a_n)_{n\ge 0}$ be the sequence of rational numbers with $a_0 = 2016$ and $a_{n+1} = a_n + \frac{2}{a_n}$ for all $n \ge 0$.
Show that the sequence does not contain a square of a rational number.
Proposed by Theresia Eisenkölbl
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 ${(q_k)_{k\ge 1}}$
Prove that if $ \cos \pi x =\frac{1}{3} $ then $ x $ is an irrational number.
Given are the non zero natural numbers $a,b,c$ such that the number $\frac{a\sqrt2+b\sqrt3}{b\sqrt2+c\sqrt3}$ is rational.
Prove that the number $\frac{a^2+b^2+c^2}{a+b+c}$ is an integer .
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.
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.
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.
Find the natural number $n$ for which $$\sqrt{\frac{20^n- 18^n}{19}}$$ is a 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.
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^* $
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.
Show that for any rational number $a$ the equation $y =\sqrt{x^2 +a}$ has infinitely many solutions in rational numbers $x$ and $y$.
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.
Does there exist a sphere passing through only one rational point? (A rational point is a point whose Cartesian coordinates are all rational numbers.)
(A Rubin)