Subject 2

Solve in $\mathbb{R}$ the equation: $\frac{1}{\{x\}}=\frac{1}{x}+\frac{1}{[x]}$, where $\{x\}$ represents the fractional part of $x$ and $[x]$ represents the integer part of $x$.
Subject 2

Solve in $\mathbb{R}$ the equation: $\frac{1}{\{x\}}=\frac{1}{x}+\frac{1}{[x]}$,

where $\{x\}$ represents the fractional part of $x$ and $[x]$ represents the integer part of $x$.

Solution

If $x>0$ and $\frac{1}{x}+\frac{1}{[x]}<2 \Rightarrow [x] \geq 2 \Rightarrow \frac{1}{x}+\frac{1}{[x]}<1$

2p.

$x \in(1 ; 2) \Rightarrow [x]=1 \Rightarrow \frac{1}{x-1}=\frac{1}{x}+1$

2p.

$x^{2}-x-1=0, x \in(1 ; 2) \Rightarrow x=\frac{1+\sqrt{5}}{2}$

1p.