Olympiad problems

Kvant 2023, M2742

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.)

2004 Bosnia and Herzegovina Junior BMO TST, 3

Tags: algebra , Sum , Fractions

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 $w =\frac{a}{b}+\frac{c}{d}$ .

2016 AMC 12/AHSME, 1

Tags: factorial , Fractions , AMC , AMC 10 , AMC 10 A

What is the value of $\dfrac{11!-10!}{9!}$? $\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$

2016 Czech And Slovak Olympiad III A, 1

Tags: number theory , Integers , Sum , 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.

2019 India PRMO, 9

Tags: Fractions

Let the rational number $p/q$ be closest to but not equal to $22/7$ among all rational numbers with denominator $< 100$. What is the value of $p - 3q$?

1970 Czech and Slovak Olympiad III A, 1

Tags: number theory , prime , prime divisors , Fractions , harmonic , numbers

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 $a.$

2022 AMC 12/AHSME, 1

Tags: 2022 AMC , 2022 AMC 10a , 2022 AMC 12A , AMC 10 , AMC 12 , AMC , Fractions

What is the value of $$3 + \frac{1}{3+\frac{1}{3+\frac{1}{3}}}?$$ $\textbf{(A) } \frac{31}{10} \qquad \textbf{(B) } \frac{49}{15} \qquad \textbf{(C) } \frac{33}{10} \qquad \textbf{(D) } \frac{109}{33} \qquad \textbf{(E) } \frac{15}{4}$

2021 AMC 10 Fall, 7

Tags: SFFT , Fractions , AMC , AMC 10 , number theory

Call a fraction $\frac{a}{b}$, not necessarily in the simplest form [i]special[/i] if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions? $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

2022 AMC 10, 5

Tags: AMC , AMC 10 , 2022 AMC , 2022 AMC 10B , Fractions

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

2013 Israel National Olympiad, 6

Tags: algebra , Inequality , Fractions , 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 IMO Shortlist, A5

Tags: IMO Shortlist , algebra , Fractions , Hi

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 $b \le \sqrt{n}$.

1969 Czech and Slovak Olympiad III A, 3

Tags: number theory , prime , Sequences , Fractions

Let $p$ be a prime. How many different (infinite) sequences $\left(a_k\right)_{k\ge0}$ exist such that for every positive integer $n$ \[\frac{a_0}{a_1}+\frac{a_0}{a_2}+\cdots+\frac{a_0}{a_n}+\frac{p}{a_{n+1}}=1?\]

2025 Kyiv City MO Round 1, Problem 1

Tags: algebra , number theory , Fractions , continued fraction

Find all triples of positive integers $ a, b, c $ that satisfy the equation: \[ a + \frac{1}{b + \frac{1}{c}} = 20.25. \]

2016 Junior Regional Olympiad - FBH, 2

Tags: number theory , Fractions

Which fraction is bigger: $\frac{5553}{5557}$ or $\frac{6664}{6669}$ ?

2021 AMC 12/AHSME Fall, 5

Tags: SFFT , Fractions , AMC , AMC 10 , number theory

Call a fraction $\frac{a}{b}$, not necessarily in the simplest form [i]special[/i] if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions? $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

1987 Mexico National Olympiad, 1

Tags: number theory , Fractions

Prove that if the sum of two irreducible fractions is an integer then the two fractions have the same denominator.

2023 Romania EGMO TST, P4

Tags: IMO Shortlist , algebra , Fractions , Hi

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 Peru IMO TST, 15

Tags: IMO Shortlist , algebra , Fractions , Hi

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 $b \le \sqrt{n}$.

2006 Singapore Junior Math Olympiad, 2

Tags: number theory , Sum , Fractions

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.

2024 CAPS Match, 1

Tags: number theory , Intersection , Fractions , decimal representation

Determine whether there exist 2024 distinct positive integers satisfying the following: if we consider every possible ratio between two distinct numbers (we include both $a/b$ and $b/a$), we will obtain numbers with finite decimal expansions (after the decimal point) of mutually distinct non-zero lengths.

2020 Malaysia IMONST 2, 2

Tags: algebra , Fractions

Prove that \[1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots +\frac{1}{2019}-\frac{1}{2020}=\frac{1}{1011}+\frac{1}{1012}+\cdots +\frac{1}{2020}\]

2017 Brazil Team Selection Test, 1

Tags: IMO Shortlist , algebra , Fractions , Hi

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 $b \le \sqrt{n}$.

1991 Mexico National Olympiad, 1

Tags: number theory , Fractions , Irreducible

Evaluate the sum of all positive irreducible fractions less than $1$ and having the denominator $1991$.

2014 Junior Regional Olympiad - FBH, 2

Tags: Fractions

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?

2024 Durer Math Competition Finals, 1

Tags: algebra , Fractions

Describe all ordered sets of four real numbers $(a, b, c, d)$ for which the values $a + b, b + c, c + d, d + a$ are all non-zero and \[\frac{a+2b+3c}{c+d}=\frac{b+2c+3d}{d+a}=\frac{c+2d+3a}{a+b}=\frac{d+2a+3b}{b+c}.\]