Found problems: 71
Find the common fraction with the smallest positive denominator lying between the fractions $\frac{96}{35}$ and $\frac{97}{36} $.
Does there exist a positive integer $N$ which is divisible by at least $2024$ distinct primes and whose positive divisors $1 = d_1 < d_2 < \ldots < d_k = N$ are such that the number
\[
\frac{d_2}{d_1}+\frac{d_3}{d_2}+\ldots+\frac{d_k}{d_{k-1}}
\]
is an integer?
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.
(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.
The fraction $\frac23$ can be eypressed as a sum of two distinct unit fractions: $\frac12 + \frac16$ .
Show that the fraction $\frac{p-1}{p}$ where $p\ge 5$ is a prime cannot be expressed as a sum of two distinct unit fractions.
Let $p> 3$ be a prime number. Determine the number of all ordered sixes $(a, b, c, d, e, f)$ of positive integers whose sum is $3p$ and all fractions $\frac{a + b}{c + d},\frac{b + c}{d + e},\frac{c + d}{e + f},\frac{d + e}{f + a},\frac{e + f}{a + b}$ have integer values.
Prove that if the sum of two irreducible fractions is an integer then the two fractions have the same denominator.
Find all positive integers $n$ that can be subtracted from both the numerator and denominator of the fraction $\frac{1234}{6789}$, to get, after the reduction, the fraction of form $\frac{a}{b}$, where $a, b$ are single digit numbers.
[i]Proposed by Bogdan Rublov[/i]
Write $2017$ following numbers on the blackboard: $-\frac{1008}{1008}, -\frac{1007}{1008}, ..., -\frac{1}{1008}, 0,\frac{1}{1008},\frac{2}{1008}, ... ,\frac{1007}{1008},\frac{1008}{1008}$ .
One processes some steps as: erase two arbitrary numbers $x, y$ on the blackboard and then write on it the number $x + 7xy + y$. After $2016$ steps, there is only one number. The last one on the blackboard is
(A): $-\frac{1}{1008}$ (B): $0$ (C): $\frac{1}{1008}$ (D): $-\frac{144}{1008}$ (E): None of the above
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.
(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.
Let the set $M =\{\frac{1998}{1999},\frac{1999}{2000} \}$.
The set $M$ is completed with new fractions according to the rule:
take two distinct fractions$ \frac{p_1}{q_1}$ and $\frac{p_2}{q_2}$ from $M$ thus there are no other numbers in $M$ located between them and a new fraction is formed, $\frac{p_1+p_2}{q_1+q_2}$ which is included in $M$, etc.
Show that, after each procedure, the newly obtained fraction is irreducible and is different from the fractions previously included in $M$.
What is the value of
$$3 + \frac{1}{3+\frac{1}{3+\frac{1}{3}}}?$$
$\textbf{(A) } \frac{31}{10} \qquad \textbf{(B) } \frac{49}{15} \qquad \textbf{(C) } \frac{33}{10} \qquad \textbf{(D) } \frac{109}{33} \qquad \textbf{(E) } \frac{15}{4}$
Find this:
$ (1+\frac{1}{5})(1+\frac{1}{6})...(1+\frac{1}{2023})(1+\frac{1}{2024}) $
Find a irreducible fraction with denominator not greater than 2005, that is closest to $\frac{9}{25}$ but is not $\frac{9}{25}$
Find all fractions which can be written simultaneously in the forms $\frac{7k- 5}{5k - 3}$ and $\frac{6l - 1}{4l - 3}$
, for some integers $k, l$.
Which number we need to substract from numerator and add to denominator of $\frac{\overline{28a3}}{7276}$ such that we get fraction equal to $\frac{2}{7}$
Find all triples of positive integers \( a, b, c \) that satisfy the equation:
\[
a + \frac{1}{b + \frac{1}{c}} = 20.25.
\]
Prove that for all positive real numbers $a$, $b$, $c$ such that $abc=1$ the following inequality holds:
$$\frac{a}{2b+c^2}+\frac{b}{2c+a^2}+\frac{c}{2a+b^2}\le \frac{a^2+b^2+c^2}3.$$
Simplify the fraction: $\frac{(1^4+4)\cdot (5^4+4)\cdot (9^4+4)\cdot ... (69^4+4)\cdot(73^4+4)}{(3^4+4)\cdot (7^4+4)\cdot (11^4+4)\cdot ... (71^4+4)\cdot(75^4+4)}$.
Let $x_1,...,x_n$ be positive real numbers, satisfying $x_1+\dots+x_n=n$. Prove that
$\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\leq\frac{4}{x_1\cdot x_2\cdot\dots\cdot x_n}+n-4$.
Given an integer $h > 1$. Let's call a positive common fraction (not necessarily irreducible) [i]good[/i] if the sum of its numerator and denominator is equal to $h$. Let's say that a number $h$ is [i]remarkable[/i] if every positive common fraction whose denominator is less than $h$ can be expressed in terms of good fractions (not necessarily various) using the operations of addition and subtraction.
Prove that $h$ is remarkable if and only if it is prime.
(Recall that an common fraction has an integer numerator and a natural denominator.)
Let $ r$ be a positive rational. Prove that $\frac{8r + 21}{3r + 8}$ is a better approximation to $\sqrt7$ that $ r$.
If $ p$ is a positive integer, then $ \frac {3p \plus{} 25}{2p \minus{} 5}$ can be a positive integer, if and only if $ p$ is:
$ \textbf{(A)}\ \text{at least }3 \qquad\textbf{(B)}\ \text{at least }3\text{ and no more than }35 \qquad\textbf{(C)}\ \text{no more than }35$
$ \textbf{(D)}\ \text{equal to }35 \qquad\textbf{(E)}\ \text{equal to }3\text{ or }35$
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.
(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.
(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.
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.