Olympiad problems

2015 India Regional MathematicaI Olympiad, 3

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 $k, l$.

2024 ITAMO, 6

Tags: inequalities , integer , extermal , fraction , maximal

For each integer $n$, determine the smallest real number $M_n$ such that \[\frac{1}{a_1}+\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots+\frac{a_{n-1}}{a_n} \le M_n\] for any $n$-tuple $(a_1,a_2,\dots,a_n)$ of integers such that $1<a_1<a_2<\dots<a_n$.

2000 Poland - Second Round, 1

Tags: algebra , fraction , rational number

Decide, whether every positive rational number can present in the form $\frac{a^2 + b^3}{c^5 + d^7}$, where $a, b, c, d$ are positive integers.

2019 India PRMO, 9

Tags: fraction

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 $p - 3q$?

2024 Korea Junior Math Olympiad (First Round), 1.

Tags: fraction , calculating

Find this: $ (1+\frac{1}{5})(1+\frac{1}{6})...(1+\frac{1}{2023})(1+\frac{1}{2024}) $

2023 Romania National Olympiad, 1

Tags: algebra , fraction

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.

1970 Czech and Slovak Olympiad III A, 1

Tags: number theory , prime , fraction , harmonic , number , prime divisor

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 $a.$

2016 Junior Regional Olympiad - FBH, 2

Tags: number theory , fraction

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

2014 Contests, 2

Tags: fraction , number theory

Find all the integers $n$ for which $\frac{8n-25}{n+5}$ is cube of a rational number.

2015 Middle European Mathematical Olympiad, 1

Tags: inequalities , fraction , muirhed inequality

Prove that for all positive real numbers $a$, $b$, $c$ such that $abc=1$ the following inequality holds: $$\frac{a}{2b+c^2}+\frac{b}{2c+a^2}+\frac{c}{2a+b^2}\le \frac{a^2+b^2+c^2}3.$$

2005 Korea Junior Math Olympiad, 1

Tags: algebra , irreducible , fraction

Find a irreducible fraction with denominator not greater than 2005, that is closest to $\frac{9}{25}$ but is not $\frac{9}{25}$

2009 Postal Coaching, 2

Tags: number theory , prime , fraction

Find all pairs $(x, y)$ of natural numbers $x$ and $y$ such that $\frac{xy^2}{x+y}$ is a prime

2013 Israel National Olympiad, 6

Tags: algebra , inequalities , inequality , fraction

Let $x_1,...,x_n$ be positive real numbers, satisfying $x_1+\dots+x_n=n$. Prove that $\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\leq\frac{4}{x_1\cdot x_2\cdot\dots\cdot x_n}+n-4$.

1993 Chile National Olympiad, 3

Tags: fraction , algebra

Let $ r$ be a positive rational. Prove that $\frac{8r + 21}{3r + 8}$ is a better approximation to $\sqrt7$ that $ r$.

1999 Czech And Slovak Olympiad IIIA, 1

Tags: algebra , min , integer polynomial , fraction

We are allowed to put several brackets in the expression $$\frac{29 : 28 : 27 : 26 :... : 17 : 16}{15 : 14 : 13 : 12 : ... : 3 : 2}$$ always in the same places below each other. (a) Find the smallest possible integer value we can obtain in that way. (b) Find all possible integer values that can be obtained. Remark: in this problem, $$\frac{(29 : 28) : 27 : ... : 16}{(15 : 14) : 13 : ... : 2},$$ is valid position of parenthesis, on the other hand $$\frac{(29 : 28) : 27 : ... : 16}{15 : (14 : 13) : ... : 2}$$ is forbidden.

2015 BAMO, 3

Tags: fraction , compare , algebra

Which number is larger, $A$ or $B$, where $$A = \dfrac{1}{2015} (1 + \dfrac12 + \dfrac13 + \cdots + \dfrac{1}{2015})$$ and $$B = \dfrac{1}{2016} (1 + \dfrac12 + \dfrac13 + \cdots + \dfrac{1}{2016}) \text{ ?}$$ Prove your answer is correct.

2015 Bundeswettbewerb Mathematik Germany, 2

Tags: number theory , fraction , decimal representation

In the decimal expansion of a fraction $\frac{m}{n}$ with positive integers $m$ and $n$ you can find a string of numbers $7143$ after the comma. Show $n>1250$. [i]Example:[/i] I mean something like $0.7143$.

2022 AMC 10, 5