Olympiad problems

2010 Regional Olympiad of Mexico Northeast, 2

Of all the fractions $\frac{x}{y}$ that satisfy $$\frac{41}{2010}<\frac{x}{y}<\frac{1}{49}$$ find the one with the smallest denominator.

III Soros Olympiad 1996 - 97 (Russia), 9.6

Tags: algebra , Fraction

Find the common fraction with the smallest positive denominator lying between the fractions $\frac{96}{35}$ and $\frac{97}{36} $.

2017 Vietnamese Southern Summer School contest, Problem 2

Tags: inequalities , cyclic inequality , Fraction

Let $a,b,c$ be the positive real numbers satisfying $a^2+b^2+c^2=3$. Prove that: $$\frac{a}{b(a+c)}+\frac{b}{c(b+a)}+\frac{c}{a(c+b)}\geq \frac{3}{2}.$$

2015 Junior Regional Olympiad - FBH, 4

Tags: Fraction , Digits

Which number we need to substract from numerator and add to denominator of $\frac{\overline{28a3}}{7276}$ such that we get fraction equal to $\frac{2}{7}$

2006 Chile National Olympiad, 1

Tags: algebra , Fraction

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.

2017 Israel Oral Olympiad, 2

Tags: algebra , Fraction , simplify

Simplify the fraction: $\frac{(1^4+4)\cdot (5^4+4)\cdot (9^4+4)\cdot ... (69^4+4)\cdot(73^4+4)}{(3^4+4)\cdot (7^4+4)\cdot (11^4+4)\cdot ... (71^4+4)\cdot(75^4+4)}$.

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

Tags: calculating , Fraction

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

2008 Postal Coaching, 1

Tags: floor function , number theory , Fraction , natural , algebra , 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.

1996 Greece Junior Math Olympiad, 4a

Tags: number theory , Fraction , simplify , Even

If the fraction $\frac{an + b}{cn + d}$ may be simplified using $2$ (as a common divisor ), show that the number $ad - bc$ is even. ($a, b, c, d, n$ are natural numbers and the $cn + d$ different from zero).

2000 Poland - Second Round, 1

Tags: algebra , rational number , Fraction , Poland

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.

1984 Dutch Mathematical Olympiad, 4

Tags: combinatorics , algebra , Fraction

By placing parentheses in the expression $1:2:3$ we can get two different number values: $(1 : 2) : 3 = \frac16$ and $1 : (2 : 3) = \frac32$. Now brackets are placed in the expression $1:2:3:4:5:6:7:8$. Multiple bracket pairs are allowed, whether or not in nest form. (a) What is the largest numerical value we can get, and what is the smallest? (b) How many different number values can be obtained?

2014 Greece National Olympiad, 2

Tags: number theory , Fraction

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

2014 Contests, 2

Tags: number theory , Fraction

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

2009 Postal Coaching, 2

Tags: prime , number theory , Fraction

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

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 $\frac{9}{25}$

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 $(0,1/n)$.

1999 Czech And Slovak Olympiad IIIA, 1

Tags: min , Fraction , Integer Polynomial , algebra

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.

2017 Hanoi Open Mathematics Competitions, 5

Tags: algebra , Fraction , combinatorics

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

2019 Peru Cono Sur TST, P1

Tags: inequalities , Fraction , Numberteory

Find all a positive integers $a$ and $b$, such that $$\frac{a^b+b^a}{a^a-b^b}$$ is an integer

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: $$\frac{a}{2b+c^2}+\frac{b}{2c+a^2}+\frac{c}{2a+b^2}\le \frac{a^2+b^2+c^2}3.$$

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$.

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.

1992 Chile National Olympiad, 3

Tags: combinatorics , algebra , Fraction

Determine the number of times and the positions in which it appears $\frac12$ in the following sequence of fractions: $$ \frac11, \frac21, \frac12 , \frac31 , \frac22 , \frac13 , \frac41,\frac32,\frac23,\frac14,..., \frac{1}{1992}$$

2013 Tournament of Towns, 6

Tags: prime , Fraction , divides , divisor , number theory , prime numbers

The number $1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n}$ is represented as an irreducible fraction. If $3n+1$ is a prime number, prove that the numerator of this fraction is a multiple of $3n + 1$.

1999 Junior Balkan Team Selection Tests - Moldova, 5

Tags: algebra , Fraction , set , Irreducible

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 $M$.