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Parte intreaga
Posted: Sat Jun 13, 2009 9:36 pm
by Claudiu Mindrila
\( n\in\mathbb{N},\ a\ge1+\sqrt{3}\Longrightarrow\left[\frac{\left[\frac{na^{2}+2}{a}\right]+2}{a}\right]=n \)
Posted: Sat Jun 13, 2009 11:15 pm
by alex2008
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 \( 0<\frac{2}{a}\le \frac{2}{1+\sqrt{3}}<1\Rightarrow \left[\frac{2}{a}\right]=0 \)
Posted: Sat Jun 13, 2009 11:16 pm
by alex2008
Posted: Sun Jun 14, 2009 4:03 pm
by mihai++
Prima solutie nu prea e corecta. Ai scos \( na \) din partea intreaga fara sa stii ca e intreg. Cred ca nu m-am pacalit.
Posted: Sun Jun 14, 2009 4:14 pm
by alex2008
Eu de fapt am vrut sa folosesc \( \left[\frac{[x]}{n}\right]=\left[\frac{x}{n}\right] \) , da nu cred ca merge nici asa .