n variabile intr-un interval

[alex2008]

Fie \( x_1,x_2,...,x_n\in \mathbb{R}\ ,\ n\ge 2 \) si fie \( a=x_1+x_2+...+x_n \). Daca \( a\ge 0 \) si \( x_1^2+x_2^2+...+x_n^2\le \frac{a^2}{n-1} \) sa se demonstreze ca oricare ar fi \( i\in\{1,2,...,n\} \) avem \( x_i\in \left[0,\frac{2a}{n}\right] \).

[Mateescu Constantin]

Folosind inegalitatea Cauchy-Schwarz avem:

\( (a-x_1)^{2}\leq(n-1)(x_2^{2}+x_3^{2}+\dots+x_n^{2})\leq(n-1)\left(\frac{a^{2}}{n-1}-x_1^{2}\right). \)

Astfel, \( a^{2}-2ax_1+x_1^{2}\leq a^{2}-(n-1)x_1^{2} \)

\( \Longleftrightarrow x_1\left(x_1-\frac{2a}{n}\right)\leq 0 \), de unde rezulta concluzia.