Olympiad problems

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}) $

2022 AMC 12/AHSME, 1

Tags: fraction

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

2024 CAPS Match, 1

Tags: number theory , intersection , fraction , 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.

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} $.

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]

2021 AMC 12/AHSME Fall, 5

Tags: sfft , fraction , 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? $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

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 $1$?

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.

2017 Peru IMO TST, 15

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 $b \le \sqrt{n}$.

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 $k, l$.

1956 AMC 12/AHSME, 48

Tags: fraction

If $ p$ is a positive integer, then $ \frac {3p \plus{} 25}{2p \minus{} 5}$ can be a positive integer, if and only if $ p$ is: $ \textbf{(A)}\ \text{at least }3 \qquad\textbf{(B)}\ \text{at least }3\text{ and no more than }35 \qquad\textbf{(C)}\ \text{no more than }35$ $ \textbf{(D)}\ \text{equal to }35 \qquad\textbf{(E)}\ \text{equal to }3\text{ or }35$

1996 Greece Junior Math Olympiad, 4a

Tags: number theory , simplify , even , fraction

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).

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 $p - 3q$?

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.

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.

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?

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.

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 $b \le \sqrt{n}$.

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.

2011 IFYM, Sozopol, 8

Tags: algebra , fraction

The fraction $\frac{1}{p}$, where $p$ is a prime number coprime with 10, is presented as an infinite periodic fraction. Prove that, if the number of digits in the period is even, then the arithmetic mean of the digits in the period is equal to $\frac{9}{2}$.

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 $\frac{9}{25}$