Found problems: 30
If $a$ is a rational number and $b$ is an irrational number such that $ab$ is rational, then which of the following is false?
(A)$ab^2$ is irrational
(B)$a^2b$ is rational
(C)$\sqrt{ab}$ is rational
(D)$a+b$ is irrational
An $n$-tuple $(x_1, x_2,\dots, x_n)$ is called unearthly if $q_1x_1 +q_2x_2 +\dots+q_nx_n$ is irrational for any non-negative rational coefficients $q_1, q_2, \dots, q_n$ where $q_i$’s are not all zero. Prove that it is possible to select an unearthly $n$-tuple from any $2n-1$ distinct irrational numbers.
Let be given an irrational number $a$ in the interval $(0,1)$ and a positive integer $N$.
Prove that there exist positive integers $p,q,r,s$ such that $\frac{p}{q} < a <\frac{r}{s}, \frac{r}{s} -\frac{p}{q}<\frac{1}{N}$, and $rq- ps = 1$.
The function $f : (0,1) \to R$ is defined by
$f(x) = x$ if $x$ is irrational,
$f(x) = \frac{p+1}{q}$ if $x =\frac{p}{q}$ , where $(p,q) = 1$.
Find the maximum value of $f$ on the interval $(7/8,8/9)$.
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.
Find (with proof) all monic polynomials $f(x)$ with integer coefficients that satisfy the following two conditions.
1. $f (0) = 2004$.
2. If $x$ is irrational, then $f (x)$ is also irrational.
(Notes: Apolynomial is monic if its highest degree term has coefficient $1$. Thus, $f (x) = x^4-5x^3-4x+7$ is an example of a monic polynomial with integer coefficients.
A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2/5$ and $-9$ are rational, while $\sqrt2$ and $\pi$ are well known to be irrational.)
For each positive $ x \in \mathbb{R}$, define
$ E(x)=\{[nx]: n\in \mathbb{N}\}$
Find all irrational $ \alpha >1$ with the following property:
If a positive real $ \beta$ satisfies $ E(\beta) \subset E(\alpha)$. then $ \frac{\beta}{\alpha}$ is a natural number.
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 $n > 1$?
A real number $a$ satisfies the equality $\frac{1}{a} = a - [a]$. Prove that $a$ is irrational.
Prove that if $ \cos \pi x =\frac{1}{3} $ then $ x $ is an irrational number.
The $n$-th digit of number $a = 0.12457...$ equals the first digit of the integer part of the number $n\sqrt2$. Prove that $a$ is irrational number. (6)
Non-zero numbers $a$ and $b$ satisfy the equality $$a^2b^2(a^2b^2 + 4) = 2(a^6 + b^6).$$ Prove that at least one of them is irrational.
Prove that among any $6$ irrational numbers you can pick three numbers $a$, $b$ and $c$ such that numbers $a+b$, $b+c$ and $c+a$ are irrational
In a triangle $ABC$, the bisector of the largest angle $\angle A$ meets $BC$ at point $D$. Let $E$ and $F$ be the feet of perpendiculars from $D$ to $AC$ and $AB$, respectively. Let $R$ denote the ratio between the areas of triangles $DEB$ and $DFC$.
(a) Prove that, for every real number $r > 0$, one can construct a triangle ABC for which $R$ is equal to $r$.
(b) Prove that if $R$ is irrational, then at least one side length of $\vartriangle ABC$ is irrational.
(c) Give an example of a triangle $ABC$ with exactly two sides of irrational length, but with rational $R$.
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.
For the positive integer $k$ we denote by the $a_n$ , the $k$ from the left digit in the decimal notation of the number $2^n$ ($a_n = 0$ if in the notation of the number $2^n$ less than the digits). Consider the infinite decimal fraction $a = \overline{0, a_1a_2a_3...}$. Prove that the number $a$ is irrational.
Show that $\tan \frac{\pi}{3n}$ is irrational for all positive integers $n$.
Let $p$ be a prime. Show that $\sqrt[3]{p} +\sqrt[3]{p^5} $ is irrational.
Show that $\log_{10} 2$ is irrational.
The number $x$ from $[0,1]$ is written as an infinite decimal fraction. Having rearranged its first five digits after the point we can obtain another fraction that corresponds to the number $x_1$. Having rearranged five digits of $x_k$ from $(k+1)$-th till $(k+5)$-th after the point we obtain the number $x_{k+1}$.
a) Prove that the sequence $x_i$ has limit.
b) Can this limit be irrational if we have started with the rational number?
c) Invent such a number, that always produces irrational numbers, no matter what digits were transposed.
Let $x_0=2024^{2024}$ and $x_{n+1}=|x_n-\pi|$ for $n \ge 0$. Show that there exists a value of $n$ such that $x_{n+2}=x_n$.
Let $p$ and $q$ be two positive integers such that $1 \le q \le p$. Also let $a = \left( p +\sqrt{p^2 + q} \right)^2$.
a) Prove that the number $a$ is irrational.
b) Show that $\{a\} > 0.75$.
Let $a,b,c,d$ be positive irrational numbers with $a+b = 1$.
Show that $c+d = 1$ if and only if $[na]+[nb] = [nc]+[nd]$ for all positive integers $n$.
Prove that the number $\tan \frac{\pi}{3^n}$ is irrational for any natural $n$.
As shown in figure , a circle of radius $1$ passes through vertex $A$ of $\vartriangle ABC$ and is tangent to the side $BC$ at the point $D$ , intersect sides $AB$ and $AC$ at points $E$ and $F$ respectively . Also$ EF$ bisects $\angle AFD$, and $\angle ADC = 80^o$ , Is there a triangle that satisfies the condition, so that $\frac{AB+BC+CA}{AD^2}$ is an irrational number, and the irrational number is the root of a quadratic equation with integral coefficients? If it does not exist, please prove it; if it exists, find the quadratic equation that satisfies the condition.
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