Parte intreaga

Claudiu Mindrila · Post by **Claudiu Mindrila** » Sat Jun 13, 2009 9:36 pm

\( n\in\mathbb{N},\ a\ge1+\sqrt{3}\Longrightarrow\left[\frac{\left[\frac{na^{2}+2}{a}\right]+2}{a}\right]=n \)

alex2008 · Post by **alex2008** » Sat Jun 13, 2009 11:15 pm

Solutia 1. \( \left[\frac{\left[\frac{na^2+2}{a}\right]+2}{a}\right]=\left[\frac{na+\left[\frac{2}{a}\right]+2}{a}\right]=n+\left[\frac{\left[\frac{2}{a}\right]+2}{a}\right]=n+\left[\frac{2}{a}\right]=n \) , deoarece \( 0<\frac{2}{a}\le \frac{2}{1+\sqrt{3}}<1\Rightarrow \left[\frac{2}{a}\right]=0 \)

alex2008 · Post by **alex2008** » Sat Jun 13, 2009 11:16 pm

Solutia 2 aici

mihai++ · Post by **mihai++** » Sun Jun 14, 2009 4:03 pm

Prima solutie nu prea e corecta. Ai scos \( na \) din partea intreaga fara sa stii ca e intreg. Cred ca nu m-am pacalit.

alex2008 · Post by **alex2008** » Sun Jun 14, 2009 4:14 pm

Eu de fapt am vrut sa folosesc \( \left[\frac{[x]}{n}\right]=\left[\frac{x}{n}\right] \) , da nu cred ca merge nici asa .