Olympiad problems

2023 Ukraine National Mathematical Olympiad, 8.3

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]

2018 District Olympiad, 1

Tags: number theory , fractional part

Prove that $\left\{ \frac{m}{n}\right\}+\left\{ \frac{n}{m}\right\} \ne 1$ , for any positive integers $m, n$.

2023 Brazil Cono Sur TST, 4

Tags: fractional part , inequality

Let $n$ be a positive integer. Prove that $n\sqrt{19}\{n\sqrt{19}\} > 1$, where $\{x\}$ denotes the fractional part of $x$.

2002 District Olympiad, 1

Tags: algebra , fractional part

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]

2024 District Olympiad, P2

Tags: sequence , fractional part , algebra

Consider the sequence $(a_n)_{n\geqslant 1}$ defined by $a_1=1/2$ and $2n\cdot a_{n+1}=(n+1)a_n.$[list=a] [*]Determine the general formula for $a_n.$ [*]Let $b_n=a_1+a_2+\cdots+a_n.$ Prove that $\{b_n\}-\{b_{n+1}\}\neq \{b_{n+1}\}-\{b_{n+2}\}.$ [/list]

2015 India Regional MathematicaI Olympiad, 6

Tags: number theory , fractional part , integer

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 $a$.)