Olympiad problems

2002 District Olympiad, 1

Prove the identity $ \left[ \frac{3+x}{6} \right] -\left[ \frac{4+x}{6} \right] +\left[ \frac{5+x}{6} \right] =\left[ \frac{1+x}{2} \right] -\left[ \frac{1+x}{3} \right] ,\quad\forall x\in\mathbb{R} , $ where $ [] $ is the integer part. [i]C. Mortici[/i]

2008 Gheorghe Vranceanu, 1

At what index the harmonic series has a fractional part of $ 1/12? $

integer part number theory fractional part floor function floor and ceiling functions fractional parts

2001 Tuymaada Olympiad, 7

Several rational numbers were written on the blackboard. Dima wrote off their fractional parts on paper. Then all the numbers on the board squared, and Dima wrote off another paper with fractional parts of the resulting numbers. It turned out that on Dima's papers were written the same sets of numbers (maybe in different order). Prove that the original numbers on the board were integers. (The fractional part of a number $x$ is such a number $\{x\}, 0 \le \{x\} <1$, that $x-\{x\}$ is an integer.)

integer fractional part algebra fractional parts

2003 Gheorghe Vranceanu, 1

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)} . $

base 10 number theory fractional part fractional parts

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 $x$.

calculus infinite series integral integration fractional part algebra fractional parts

2007 Nicolae Coculescu, 1

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]

floor equation fractional part algebra fractional parts