This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 71

2008 Postal Coaching, 5
Tags: fraction , combinatorics , irreducible
Let $n \in N$. Find the maximum number of irreducible fractions a/b (i.e., $gcd(a, b) = 1$) which lie in the interval $(0,1/n)$.

2022 AMC 10, 5
Tags: fraction
What is the value of $\frac{(1+\frac{1}{3})(1+\frac{1}{5})(1+\frac{1}{7})}{\sqrt{(1-\frac{1}{3^2})(1-\frac{1}{5^2})(1-\frac{1}{7^2})}}?$ $\textbf{(A) }\sqrt{3} \qquad \textbf{(B) }2 \qquad \textbf{(C) }\sqrt{15} \qquad \textbf{(D) }4 \qquad \textbf{(E) }\sqrt{105}$

2020 Brazil National Olympiad, 1
Tags: fraction , algebra , number theory
Prove that there are positive integers $a_1, a_2,\dots, a_{2020}$ such that $$\dfrac{1}{a_1}+\dfrac{1}{2a_2}+\dfrac{1}{3a_3}+\dots+\dfrac{1}{2020a_{2020}}=1.$$

1999 Junior Balkan Team Selection Tests - Moldova, 5
Tags: algebra , set , irreducible , fraction
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 $M$.

2019 Peru Cono Sur TST, P1
Tags: fraction , number theory , inequalities
Find all a positive integers $a$ and $b$, such that $$\frac{a^b+b^a}{a^a-b^b}$$ is an integer

2016 AMC 10, 1
Tags: factorial , fraction
What is the value of $\dfrac{11!-10!}{9!}$? $\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$

2014 Junior Regional Olympiad - FBH, 2
Tags: fraction
In one class in the school, number of abscent students is $\frac{1}{6}$ of number of students who were present. When teacher sent one student to bring chalk, number of abscent students was $\frac{1}{5}$ of number of students who were present. How many students are in that class?

2013 Israel National Olympiad, 6
Tags: algebra , inequality , fraction , inequalities
Let $x_1,...,x_n$ be positive real numbers, satisfying $x_1+\dots+x_n=n$. Prove that $\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\leq\frac{4}{x_1\cdot x_2\cdot\dots\cdot x_n}+n-4$.

2016 Junior Regional Olympiad - FBH, 2
Tags: number theory , fraction
Which fraction is bigger: $\frac{5553}{5557}$ or $\frac{6664}{6669}$ ?

2017 Romania Team Selection Test, P1
Tags: algebra , fraction
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 $b \le \sqrt{n}$.

2017 Azerbaijan Team Selection Test, 3
Tags: algebra , fraction
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 $b \le \sqrt{n}$.

2021 Kyiv City MO Round 1, 8.1
Tags: fraction , number theory , algebra
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]

2016 AMC 12/AHSME, 1
Tags: factorial , fraction
What is the value of $\dfrac{11!-10!}{9!}$? $\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$

2023 Romania National Olympiad, 1
Tags: fraction , algebra
Determine all sequences of equal ratios of the form \[ \frac{a_1}{a_2} = \frac{a_3}{a_4} = \frac{a_5}{a_6} = \frac{a_7}{a_8} \] which simultaneously satisfy the following conditions: $\bullet$ The set $\{ a_1, a_2, \ldots , a_8 \}$ represents all positive divisors of $24$. $\bullet$ The common value of the ratios is a natural number.

2010 Singapore Junior Math Olympiad, 4
Tags: number theory , fraction
A student divides an integer $m$ by a positive integer $n$, where $n \le 100$, and claims that $\frac{m}{n}=0.167a_1a_2...$ . Show the student must be wrong.

1970 Czech and Slovak Olympiad III A, 1
Tags: number theory , prime , fraction , harmonic , prime divisor , number
Let $p>2$ be a prime and $a,b$ positive integers such that \[\frac ab=1+\frac12+\frac13+\cdots+\frac{1}{p-1}.\] Show that $p$ is a divisor of $a.$

2024 Baltic Way, 19
Tags: construction , fraction , number theory , divisor
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?

2015 Middle European Mathematical Olympiad, 1
Tags: fraction , inequalities , muirhed inequality
Prove that for all positive real numbers $a$, $b$, $c$ such that $abc=1$ the following inequality holds: $$\frac{a}{2b+c^2}+\frac{b}{2c+a^2}+\frac{c}{2a+b^2}\le \frac{a^2+b^2+c^2}3.$$

1993 Chile National Olympiad, 3
Tags: algebra , fraction
Let $ r$ be a positive rational. Prove that $\frac{8r + 21}{3r + 8}$ is a better approximation to $\sqrt7$ that $ r$.

2015 Junior Regional Olympiad - FBH, 4
Tags: digit , fraction
Which number we need to substract from numerator and add to denominator of $\frac{\overline{28a3}}{7276}$ such that we get fraction equal to $\frac{2}{7}$

2017 F = ma, 1
Tags: fraction
A motorcycle rides on the vertical walls around the perimeter of a large circular room. The friction coefficient between the motorcycle tires and the walls is $\mu$. How does the minimum $\mu$ needed to prevent the motorcycle from slipping downwards change with the motorcycle's speed, $s$? $\textbf{(A)}\mu \propto s^{0} \qquad \textbf{(B)}\mu \propto s^{-\frac{1}{2}}\qquad \textbf{(C)}\mu \propto s^{-1}\qquad \textbf{(D)}\mu \propto s^{-2}\qquad \textbf{(E)}\text{none of these}$

2006 Chile National Olympiad, 1
Tags: fraction , algebra
Juana and Juan have to write each one an ordered list of fractions so that the two lists have the same number of fractions and that the difference between the sum of all the fractions from Juana's list and the sum of all fractions from Juan's list is greater than $123$. The fractions in Juana's list are $$\frac{1^2}{1}, \frac{2^2}{3},\frac{3^2}{5},\frac{4^2}{7},\frac{5^2}{9},...$$ And the fractions in John's list are $$\frac{1^2}{3}, \frac{2^2}{5},\frac{3^2}{7},\frac{4^2}{9},\frac{5^2}{11},...$$ Find the least amount of fractions that each one must write to achieve the objective.

2004 Bosnia and Herzegovina Junior BMO TST, 3
Tags: algebra , sum , fraction
Let $a, b, c, d$ be reals such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 7$ and $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}= 12$. Find the value of $w =\frac{a}{b}+\frac{c}{d}$ .

2022 Junior Balkan Team Selection Tests - Moldova, 4
Tags: fraction , number theory , rational
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 $p$.

2015 India Regional MathematicaI Olympiad, 3
Tags: number theory , integer , fraction
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 $k, l$.