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Found problems: 2

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 $\{x\}+\left\{\frac{1}{x}\right\}=1$

2023 Taiwan TST Round 3, A
Tags: integer and fractional part , algebra
Show that there exists a positive constant $C$ such that, for all positive reals $a$ and $b$ with $a + b$ being an integer, we have $$\left\{a^3\right\} + \left\{b^3\right\} + \frac{C}{(a+b)^6} \le 2. $$ Here $\{x\} = x - \lfloor x\rfloor$ is the fractional part of $x$. [i]Proposed by Li4 and Untro368.[/i]