TITLE: An Inequality problem 123
QUESTION [5 upvotes]: Let $x_1,x_2,\ldots,x_n\in \left [ 0,1 \right ]$, prove that $(1+x_{1}+x_2+\cdots+x_n)^2\geq 4(x_1^2+x_2^2+\cdots+x_n^2)$

REPLY [2 votes]: Since $x_i \in [0,1]$, we have: $1\geq x_i\geq x_i^2\geq 0$, thus: 
$x_1+x_2+...+x_n \geq x_1^2+x_2^2+...+x_n^2\geq 0 \longrightarrow (1+x_1+...+x_n)^2\geq (1+x_1^2+...+x_n^2)^2 \geq 4(x_1^2+...+x_n^2)$ by the well-known inequality: $(1+a)^2 - 4a = (1-a)^2 \geq 0$ is true for $a = x_1^2+...+x_n^2$.