Olympiad problems

2018 China Western Mathematical Olympiad, 2

Let $n \geq 2$ be an integer. Positive reals $x_1, x_2, \cdots, x_n$ satisfy $x_1x_2 \cdots x_n = 1$. Show: $$\{x_1\} + \{x_2\} + \cdots + \{x_n\} < \frac{2n-1}{2}$$ Where $\{x\}$ denotes the fractional part of $x$.

2023 CIIM, 1

Tags: fractional part , integral , infinite series , integration , calculus

Determine all the pairs of positive real numbers $(a, b)$ with $a < b$ such that the following series $$\sum_{k=1}^{\infty} \int_a^b\{x\}^k dx =\int_a^b\{x\} dx + \int_a^b\{x\}^2 dx + \int_a^b\{x\}^3 dx + \cdots$$ is convergent and determine its value in function of $a$ and $b$. [b]Note: [/b] $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.

2020 Iran Team Selection Test, 5

Tags: divisibility , algebra , fractional part

For every positive integer $k>1$ prove that there exist a real number $x$ so that for every positive integer $n<1398$: $$\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.$$ [i]Proposed by Mohammad Amin Sharifi[/i]

2017 Balkan MO Shortlist, N5

Tags: quadratic residue , number theory , fractional part , arithmetic mean , odd , prime

Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .

1999 USAMO, 3

Tags: geometry , fractional part , modular arithmetic , algebra , polynomial

Let $p > 2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$, such that \[ \left\{ \dfrac{ra}{p} \right\} + \left\{ \dfrac{rb}{p} \right\} + \left\{ \dfrac{rc}{p} \right\} + \left\{ \dfrac{rd}{p} \right\} = 2 \] for any integer $r$ not divisible by $p$. Prove that at least two of the numbers $a+b$, $a+c$, $a+d$, $b+c$, $b+d$, $c+d$ are divisible by $p$. (Note: $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.)

2011 District Olympiad, 4

Tags: algebra , fractional part

Let be a nonzero real number $ a, $ and a natural number $ n. $ Prove the implication: $$ \{ a \} +\left\{\frac{1}{a}\right\} =1 \implies \{ a^n \} +\left\{\frac{1}{a^n}\right\} =1 , $$ where $ \{\} $ is the fractional part.

2006 Bosnia and Herzegovina Team Selection Test, 3

Tags: positive integer , fractional part , inequality , inequalities , number theory

Prove that for every positive integer $n$ holds inequality $\{n\sqrt{7}\}>\frac{3\sqrt{7}}{14n}$, where $\{x\}$ is fractional part of $x$.

2017 Romania National Olympiad, 1

Tags: fractional part , integer part , floor , equation , logarithm , algebra

Solve in the set of real numbers the equation $ a^{[ x ]} +\log_a\{ x \} =x , $ where $ a $ is a real number from the interval $ (0,1). $ $ [] $ and $ \{\} $ [i]denote the floor, respectively, the fractional part.[/i]

2024 Israel National Olympiad (Gillis), P2

Tags: fractional part , remainder , number theory

A positive integer $x$ satisfies the following: \[\{\frac{x}{3}\}+\{\frac{x}{5}\}+\{\frac{x}{7}\}+\{\frac{x}{11}\}=\frac{248}{165}\] Find all possible values of \[\{\frac{2x}{3}\}+\{\frac{2x}{5}\}+\{\frac{2x}{7}\}+\{\frac{2x}{11}\}\] where $\{y\}$ denotes the fractional part of $y$.

1997 Romania National Olympiad, 2

Tags: fractional part , algebra

Let $n\geq 3$ be a natural number and $x\in \mathbb{R}$, for which $\{ x\} =\{ x^2\} =\{ x^n\} $ (with $\{ x\} $ we denote the fractional part of $x$). Prove that $x$ is an integer.

1990 Greece National Olympiad, 4

Tags: integer and fractional part , fractional part , integer part , algebra , floor function

Froa nay real $x$, we denote $[x]$, the integer part of $x$ and with $\{x\}$ the fractional part of $x$, such that $x=[x]+\{x\}$. a) Find at least one real $x$ such that$\{x\}+\left\{\frac{1}{x}\right\}=1$ b) Find all rationals $x$ such that $\{x\}+\left\{\frac{1}{x}\right\}=1$

II Soros Olympiad 1995 - 96 (Russia), 9.8

Tags: fractional part , algebra

Let ${a}$ be the fractional part of the number $a$, that is, $\{a\} = a - [a]$, where$ [a]$ is the integer part of $ a$. (For example, $\{1.7\} = 1.7 -1 = 0.7$,$\{-\sqrt2 \}= -\sqrt2 -(-3) = 3-\sqrt2$.) a) How many solutions does the equation have $$ \{5\{4\{3\{2\{x\}\}\}\}\}=1\,\, ?$$ b) Find its greatest solution.

III Soros Olympiad 1996 - 97 (Russia), 11.1

Tags: algebra , fractional part , trigonometry

Find the smallest positive root of the equation $$\{tg x\}=\sin x. $$ ($\{a\}$ is the fractional part of $a$, $\{a\}$ is equal to the difference between $ a$ and the largest integer not exceeding $a$.)

2015 India Regional MathematicaI Olympiad, 6

Tags: number theory , integer , fractional part

Find all real numbers $a$ such that $3 < a < 4$ and $a(a-3\{a\})$ is an integer. (Here $\{a\}$ denotes the fractional part of $a$.)

2015 Costa Rica - Final Round, 3

Tags: algebra , fractional part , functional , functional equation

Indicate (justifying your answer) if there exists a function $f: R \to R$ such that for all $x \in R$ fulfills that i) $\{f(x))\} \sin^2 x + \{x\} cos (f(x)) cosx =f (x)$ ii) $f (f(x)) = f(x)$ where $\{m\}$ denotes the fractional part of $m$. That is, $\{2.657\} = 0.657$, and $\{-1.75\} = 0.25$.

1985 Tournament Of Towns, (086) 2

Tags: fractional part , integer part , floor function

The integer part $I (A)$ of a number $A$ is the greatest integer which is not greater than $A$ , while the fractional part $F(A)$ is defined as $A - I(A)$ . (a) Give an example of a positive number $A$ such that $F(A) + F( 1/A) = 1$ . (b) Can such an $A$ be a rational number? (I. Varge, Romania)

2008 Gheorghe Vranceanu, 2

Tags: kronecker , fractional part , inequalities , number theory

Show that there is a natural number $ n $ that satisfies the following inequalities: $$ \sqrt{3} -\frac{1}{10}<\{ n\sqrt 3\} +\{ (n+1)\sqrt 3 \} <\sqrt 3. $$

2000 USA Team Selection Test, 3

Tags: fractional part , number theory , inequalities

Let $p$ be a prime number. For integers $r, s$ such that $rs(r^2 - s^2)$ is not divisible by $p$, let $f(r, s)$ denote the number of integers $n \in \{1, 2, \ldots, p - 1\}$ such that $\{rn/p\}$ and $\{sn/p\}$ are either both less than $1/2$ or both greater than $1/2$. Prove that there exists $N > 0$ such that for $p \geq N$ and all $r, s$, \[ \left\lceil \frac{p-1}{3} \right\rceil \le f(r, s) \le \left\lfloor \frac{2(p-1)}{3} \right\rfloor. \]

2024 CIIM, 6

Tags: real analysis , series , fractional part , convergence , ana

Given a real number $x$, define the series \[ S(x) = \sum_{n=1}^{\infty} \{n! \cdot x\}, \] where $\{s\} = s - \lfloor s \rfloor$ is the fractional part of the number $s$. Determine if there exists an irrational number $x$ for which the series $S(x)$ converges.

2018 China National Olympiad, 3

Tags: fractional part , approximation , inequality , diophantine approximation , number theory

Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$, one has $$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$$ where $\{ x \}$ denotes the fractional part of $x$.

2023 Ukraine National Mathematical Olympiad, 8.3

Tags: algebra , fractional part , inequalities

Positive integers $x, y$ satisfy the following conditions: $$\{\sqrt{x^2 + 2y}\}> \frac{2}{3}; \hspace{10mm} \{\sqrt{y^2 + 2x}\}> \frac{2}{3}$$ Show that $x = y$. Here $\{x\}$ denotes the fractional part of $x$. For example, $\{3.14\} = 0.14$. [i]Proposed by Anton Trygub[/i]

1998 All-Russian Olympiad Regional Round, 10.5

Tags: algebra , fractional part

Solve the equation $\{(x + 1)^3\} = x^3$, where $\{z\}$ is the fractional part of the number z, i.e. $\{z\} = z - [z]$.

VI Soros Olympiad 1999 - 2000 (Russia), 9.2

Tags: algebra , fractional part , floor function , number theory

Solve the equation $[x]\{x\} = 1999x$, where $[x]$ denotes the largest integer less than or equal to $x$, and $\{x\} = x -[x] $

2023 Romania National Olympiad, 1

Tags: fractional part , inequality , algebra

For natural number $n$ we define \[ a_n = \{ \sqrt{n} \} - \{ \sqrt{n + 1} \} + \{ \sqrt{n + 2} \} - \{ \sqrt{n + 3} \}. \] a) Show that $a_1 > 0,2$. b) Show that $a_n < 0$ for infinity many values of $n$ and $a_n > 0$ for infinity values of natural numbers of $n$ as well. ( We denote by $\{ x \} $ the fractional part of $x.$)

V Soros Olympiad 1998 - 99 (Russia), 9.6

Tags: fractional part , algebra

How many solutions satisfying the condition $1 < x < 5$ does the equation $\{x[x]\} = 0.5$ have? (Here $[x]$ is the integer part of the number $x$, $\{x\} = x - [x]$ is the fractional part of the number $x$.)