Olympiad problems

2021 AMC 12/AHSME Fall, 5

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 Kyiv City MO Round 1, Problem 1

Tags: fraction , construction , algebra

Represent $\frac{1}{2021}$ as a difference of two irreducible fractions with smaller denominators. [i](Proposed by Bogdan Rublov)[/i]

1992 Chile National Olympiad, 3

Tags: fraction , combinatorics , algebra

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

1996 Greece Junior Math Olympiad, 4a

Tags: fraction , number theory , 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).

2010 Regional Olympiad of Mexico Northeast, 2

Tags: fraction , algebra , number theory

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.

2006 Singapore Junior Math Olympiad, 2

Tags: number theory , sum , fraction

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.

2009 Postal Coaching, 2

Tags: fraction , prime , number theory

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

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} $.

2020 Malaysia IMONST 2, 2

Tags: algebra , fraction

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

1999 Czech And Slovak Olympiad IIIA, 1

Tags: min , integer polynomial , algebra , 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.

2022 AMC 12/AHSME, 1

Tags: fraction

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

2017 Israel Oral Olympiad, 2

Tags: fraction , algebra , simplify

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

2017 South Africa National Olympiad, 1

Tags: algebra , fraction

Together, the two positive integers $a$ and $b$ have $9$ digits and contain each of the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once. For which possible values of $a$ and $b$ is the fraction $a/b$ closest to $1$?

2015 BAMO, 3

Tags: algebra , fraction , compare

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.

2023 Romania EGMO TST, P4

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 $b \le \sqrt{n}$.

1984 Dutch Mathematical Olympiad, 4

Tags: fraction , combinatorics , algebra

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?

2008 Postal Coaching, 1

Tags: floor function , number theory , natural , algebra , irrational , fraction

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.

2021 AMC 12/AHSME Fall, 5

2022 Kyiv City MO Round 1, Problem 1

1992 Chile National Olympiad, 3

1996 Greece Junior Math Olympiad, 4a

2010 Regional Olympiad of Mexico Northeast, 2

2006 Singapore Junior Math Olympiad, 2

2009 Postal Coaching, 2

III Soros Olympiad 1996 - 97 (Russia), 9.6

2020 Malaysia IMONST 2, 2

1999 Czech And Slovak Olympiad IIIA, 1

2022 AMC 12/AHSME, 1

2017 Israel Oral Olympiad, 2

2017 South Africa National Olympiad, 1

2015 BAMO, 3

2023 Romania EGMO TST, P4

1984 Dutch Mathematical Olympiad, 4

2008 Postal Coaching, 1

2005 Korea Junior Math Olympiad, 1

2019 India PRMO, 9

1987 Mexico National Olympiad, 1

2024 ITAMO, 6