Olympiad problems

2023 Ukraine National Mathematical Olympiad, 9.4

Find the smallest real number $C$, such that for any positive integers $x \neq y$ holds the following: $$\min(\{\sqrt{x^2 + 2y}\}, \{\sqrt{y^2 + 2x}\})<C$$ Here $\{x\}$ denotes the fractional part of $x$. For example, $\{3.14\} = 0.14$. [i]Proposed by Anton Trygub[/i]

2004 Alexandru Myller, 4

Tags: number theory , floor function , n-th root , integer part , fractional part

Find the real numbers $ x>1 $ having the property that $ \sqrt[n]{\lfloor x^n \rfloor } $ is an integer for any natural number $ n\ge 2. $ [i]Mihai Piticari[/i] and [i]Dan Popescu[/i]

2013 Romania National Olympiad, 3

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

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

2004 District Olympiad, 3

Tags: algebra , fractional part

[b]a)[/b] Show that there are infinitely many rational numbers $ x>0 $ such that $ \left\{ x^2 \right\} +\{ x \} =0.99. $ [b]b)[/b] Show that there are no rational numbers $ x>0 $ such that $ \left\{ x^2 \right\} +\{ x \} =1. $ $ \{\} $ denotes the usual fractional part.

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.

2003 Gheorghe Vranceanu, 1

Tags: algebra , fractional part , integer part

For a real number $ k\ge 2, $ solve the equation $ \frac{\{x\}[x]}{x} =k. $