Olympiad problems

2023 CIIM, 1

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 $x$.

2003 Gheorghe Vranceanu, 1

Tags: fractional part , base 10 , number theory

Find all nonnegative numbers $ n $ which have the property that $ a_{2}\neq 9, $ where $ \sum_{i=1}^{\infty } a_i10^{-i} $ is the decimal representation of the fractional part of $ \sqrt{n(n+1)} . $

III Soros Olympiad 1996 - 97 (Russia), 11.1

Tags: trigonometry , algebra , fractional part

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 $a$.)

2020 Iran Team Selection Test, 5

Tags: fractional part , algebra , divisibility

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]

2018 China National Olympiad, 3

Tags: approximation , inequality , fractional part , 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 $x$.

2015 Costa Rica - Final Round, 3

Tags: algebra , fractional part , functional equation , functional

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 $\{-1.75\} = 0.25$.

2017 Balkan MO Shortlist, N5

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

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 $p$ .

2012 District Olympiad, 1

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

Solve in $ \mathbb{R} $ the equation $ [x]^5+\{ x\}^5 =x^5, $ where $ [],\{\} $ are the integer part, respectively, the fractional part.

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.

2017 Romania Team Selection Test, P4

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

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 $p$ .

2024 Junior Balkan Team Selection Tests - Romania, P2

Tags: fractional part , number theory

For any positive integer $n{}$ define $a_n=\{n/s(n)\}$ where $s(\cdot)$ denotes the sum of the digits and $\{\cdot\}$ denotes the fractional part.[list=a] [*]Prove that there exist infinitely many positive integers $n$ such that $a_n=1/2.$ [*]Determine the smallest positive integer $n$ such that $a_n=1/6.$ [/list] [i]Marius Burtea[/i]

2007 Nicolae Coculescu, 1

Tags: algebra , floor , fractional part , equation

Let be two real numbers $ x,y, $ and a natural number $ n_0 $ such that $ \{ n_0x \} = \{ n_0y \} $ and $ \{ (n_0+1)x \} = \{ (n_0+1)y \} ,$ where $ \{\} $ denotes the fractional part. Show that $ \{ nx \} =\{ ny \} , $ for any natural number $ n. $ [i]Ovidiu Pop[/i]

2000 IMO Shortlist, 2

Tags: floor function , algebra , fractional part

Let $ a, b, c$ be positive integers satisfying the conditions $ b > 2a$ and $ c > 2b.$ Show that there exists a real number $ \lambda$ with the property that all the three numbers $ \lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $ \left(\frac {1}{3}, \frac {2}{3} \right].$

2006 Cezar Ivănescu, 2

Tags: number theory , albegra , fractional part , algebra

[b]a)[/b] Prove that $ \{ a \} +\{ 1/a \} <3/2, $ for any positive real number $ a. $ [b]b)[/b] Give an example of a number $ b $ satisfying $ \{ b \} +\{ 1/b \} =1. $ [i]{} means fractional part[/i]

1997 Romania National Olympiad, 2

Tags: algebra , fractional part

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.

2000 USA Team Selection Test, 3

Tags: inequalities , number theory , fractional part

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 $r, s$, \[ \left\lceil \frac{p-1}{3} \right\rceil \le f(r, s) \le \left\lfloor \frac{2(p-1)}{3} \right\rfloor. \]

2013 Romania National Olympiad, 3

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

Find all real $x > 0$ and integer $n > 0$ so that $$ \lfloor x \rfloor+\left\{ \frac{1}{x}\right\}= 1.005 \cdot n.$$

1986 Traian Lălescu, 1.3

Tags: continuity , real analysis , fractional part

Prove that the application $ \mathbb{R}\ni x\mapsto 2x+ \{ x\} $ and its inverse are bijective and continuous.

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.

1998 All-Russian Olympiad Regional Round, 10.5

Tags: fractional part , algebra

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

2023 Romania National Olympiad, 1

Tags: fractional part , algebra , inequality

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 $x.$)

2006 Bosnia and Herzegovina Team Selection Test, 3

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

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

2010 Mathcenter Contest, 2

Tags: fractional part , number theory , algebra

A positive rational number $x$ is called [i]banzai [/i] if the following conditions are met: $\bullet$ $x=\frac{p}{q}>1$ where $p,q$ are comprime natural numbers $\bullet$ exist constants $\alpha,N$ such that for all integers $n\geq N$,$$\mid \left\{\,x^n\right\} -\alpha\mid \leq \dfrac{1}{2(p+q)}.$$ Find the total number of banzai numbers. Note:$\left\{\,x\right\}$ means fractional part of $x$ [i](tatari/nightmare)[/i]

2016 District Olympiad, 2

Tags: perfect square , number theory , fractional part , equation , algebra

If $ a,n $ are two natural numbers corelated by the equation $ \left\{ \sqrt{n+\sqrt n}\right\} =\left\{ \sqrt a\right\} , $ then $ 1+4a $ is a perfect square. Justify this statement. Here, $ \{\} $ is the usual fractionary part.

2006 Mathematics for Its Sake, 1

Tags: fractional part , algebra , floor , equation

Solve in the set of real numbers the equation $$ 16\{ x \}^2-8x=-1, $$ where $ \{\} $ denotes the fractional part.