Found problems: 106
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?
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?
Determine the integers $x$ and $y$ for which $\sqrt{4^x + 5^y}$ is rational.
Find the rational roots (if any) of the equation $$abx^2 + (a^2 + b^2 )x +1 = 0 , \,\,\,\, (a, b \in Z).$$
Let $x, y$ and $z$ be rational numbers satisfying $$x^3 + 3y^3 + 9z^3 - 9xyz = 0.$$
Prove that $x = y = z = 0$.
Given a real number $\alpha$ satisfying $\alpha^3 = \alpha + 1$. Determine all $4$-tuples of rational numbers $(a, b, c, d)$ satisfying: $a\alpha^2 + b\alpha+ c = \sqrt{d}.$
Determine the values of the positive integer $n$ for which $$A =\sqrt{\frac{9n - 1}{n + 7}}$$ is rational.
Get rid of irrationality in the denominator of a fraction $$\frac{1}{\sqrt[3]{4}+\sqrt[3]{2}+2}$$.
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.
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?
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.
Show that $\sqrt[3]{\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5}$ is 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?
On the board are written $100$ mutually different positive real numbers, such that for any three different numbers $a, b, c$ is $a^2 + bc$ is an integer. Prove that for any two numbers $x, y$ from the board , number $\frac{x}{y}$ is rational.
For a given positive integer $n$, let $f(x) =x^{n}$. Is it possible for the decimal number
$$0.f(1)f(2)f(3)\ldots$$
to be rational? (Example: for $n=2$, we are considering $0.1491625\ldots$)
$a$ 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.
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.
Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.
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 $p$.
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.
Let $n$ be a natural number and let $P (t) = 1 + t + t^2 + ... + t^{2n}$.
If $x \in R$ such that $P (x)$ and $P (x^2)$ are rational numbers, prove that $x$ is rational number.
Find all pairs $(a, b)$ of positive rational numbers such that $\sqrt[b]{a}= ab$
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.
Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.
Determine whether the equation $x^2 +xy+y^2 = 2$ has a solution $(x,y)$ in rational numbers.