Found problems: 106
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.
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?
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
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?
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.
Determine if there is a non-natural natural number $n$ with the property that $\sqrt{n + 1} + \sqrt{n - 1}$ is rational.
An ant is moving on the cooridnate plane, starting form point $(0,-1)$ along a straight line until it reaches the $x$- axis at point $(x,0)$ where $x$ is a real number. After it turns $90^o$ to the left and moves again along a straight line until it reaches the $y$-axis . Then it again turns left and moves along a straight line until it reaches the $x$-axis, where it once more turns left by $90^o$ and moves along a straight line until it finally reached the $y$-axis.
Can both the length of the ant's journey and distance between it's initial and final point be:
(a) rational numbers ?
(b) integers?
Justify your answers
PS. Collected [url=https://artofproblemsolving.com/community/c949609_2019_nigerian_senior_mo_round_4]here[/url]
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?
In the set of all positive real numbers define the operation $a * b = a^b$ .
Find all positive rational numbers for which $a * b = b * a$.
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^* $
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.
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.
Let $r$ be a natural number greater than $1$. Then there exist positive irrational numbers $x, y$ such that $x^y = r$ . Prove it.
Let $P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4$ be a polynomial with rational coefficients. Show that if $P(x)$ has exactly one real root $\xi$, then $\xi$ is a rational number.
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
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]
We consider the sequence $\{u_n\}$ defined by recursion $u_{n+1} =\frac{u_n(u_n + 1)}{n}$ for $n \ge 1$.
(a) Determine the terms of the sequence for $u_1 = 1$.
(b) Show that if a member of the sequence is rational, then all subsequent members are also rational numbers.
(c) Show that for every natural number $K$ there is a $u_1 > 1$ such that the first $K$ terms of the sequence are natural numbers.
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 $\left|a-\frac pq\right|<\frac\epsilon q$.
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.
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
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 $q > 2$.
Let ${a_n}$ be a sequence defined by $a_1=2$, $a_{n+1}=\left[ \frac {3a_n}{2}\right]$ $\forall n \in \mathbb N$
$0.a_1a_2...$ rational or irrational?
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.
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.
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]