Olympiad problems

2017 F = ma, 1

Tags: fraction

A motorcycle rides on the vertical walls around the perimeter of a large circular room. The friction coefficient between the motorcycle tires and the walls is $\mu$. How does the minimum $\mu$ needed to prevent the motorcycle from slipping downwards change with the motorcycle's speed, $s$? $\textbf{(A)}\mu \propto s^{0} \qquad \textbf{(B)}\mu \propto s^{-\frac{1}{2}}\qquad \textbf{(C)}\mu \propto s^{-1}\qquad \textbf{(D)}\mu \propto s^{-2}\qquad \textbf{(E)}\text{none of these}$

1969 Czech and Slovak Olympiad III A, 3

Tags: fraction , sequence , prime , number theory

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

2016 IMO Shortlist, A5

Tags: fraction , algebra

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 $b \le \sqrt{n}$.

2008 Postal Coaching, 1

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

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.

2015 Junior Regional Olympiad - FBH, 4

Tags: digit , fraction

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

2025 Kyiv City MO Round 1, Problem 1

Tags: fraction , number theory , continued fraction , algebra

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

2022 AMC 10, 1

Tags: fraction

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

2006 Singapore Junior Math Olympiad, 2

Tags: sum , fraction , number theory

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.

2017 Romania Team Selection Test, P1

Tags: fraction , algebra

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 Vietnamese Southern Summer School contest, Problem 2

Tags: cyclic inequality , fraction , inequalities

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

2017 F = ma, 19

Tags: fraction

19) A puck is kicked up a ramp, which makes an angle of $30^{\circ}$ with the horizontal. The graph below depicts the speed of the puck versus time. What is the coefficient of friction between the puck and the ramp? A) 0.07 B) 0.15 C) 0.22 D) 0.29 E) 0.37

2024 Durer Math Competition Finals, 1

Tags: fraction , algebra

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 \[\frac{a+2b+3c}{c+d}=\frac{b+2c+3d}{d+a}=\frac{c+2d+3a}{a+b}=\frac{d+2a+3b}{b+c}.\]

2021 Kyiv City MO Round 1, 8.1

Tags: number theory , fraction , 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]

2022/2023 Tournament of Towns, P5

Tags: fraction , number theory

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.)

Kvant 2023, M2742

Tags: fraction , number theory

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.)

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? $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

2019 Peru Cono Sur TST, P1

Tags: number theory , fraction , inequalities

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

2020 Brazil National Olympiad, 1

Tags: number theory , fraction , algebra

Prove that there are positive integers $a_1, a_2,\dots, a_{2020}$ such that $$\dfrac{1}{a_1}+\dfrac{1}{2a_2}+\dfrac{1}{3a_3}+\dots+\dfrac{1}{2020a_{2020}}=1.$$

III Soros Olympiad 1996 - 97 (Russia), 9.6

Tags: fraction , algebra

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

1991 Mexico National Olympiad, 1

Tags: fraction , irreducible , number theory

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

2017 Azerbaijan Team Selection Test, 3

Tags: fraction , algebra

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 Hanoi Open Mathematics Competitions, 5

Tags: fraction , combinatorics , algebra

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

2021 AMC 10 Fall, 7

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? $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

1987 Mexico National Olympiad, 1

Tags: fraction , number theory

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

2014 Greece National Olympiad, 2

Tags: fraction , number theory

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