Question

Daca [x]=2010, aflati [ 1/x ], [ (1+x)/x]

Answer 1

Partea intreaga are proprietatea  $[x] \leq x <[x]+1$.

De aici, deducem:  
$2010 \leq x <2011\\ \\ \\ \dfrac{1}{2010} \geq \dfrac{1}{x}>\dfrac{1}{2011}.$

Tot din prima proprietate, se deduce:

 $x-1<[x] \leq x \\ \\ \\ \dfrac{1}{x}-1<\left[\dfrac{1}{x}\right] \leq \dfrac{1}{x}\\ \\ \\ \dfrac{1}{2010}-1<\left[\dfrac{1}{x}\right] \leq \dfrac{1}{2011}$

Stim ca x este pozitiv si ca $\dfrac{1}{2011}<1$, de unde deducem:

$\left[\dfrac{1}{x}\right]=0$

$\left[\dfrac{1+x}{x}\right]=\left[\dfrac{1}{x}+1\right]=\left[\dfrac{1}{x}\right]+1=1.$