Question 47
Let $a_1,\ldots,a_n;\ b_1,\ldots,b_n;\ c_1,\ldots,c_n$ be real numbers. Prove that
$$
\begin{aligned}
&\sqrt{\sum_{i=1}^n(3a_i-b_i-c_i)^2}+\sqrt{\sum_{i=1}^n(3b_i-a_i-c_i)^2}+\sqrt{\sum_{i=1}^n(3c_i-a_i-b_i)^2}\\
&\qquad\ge\sqrt{\sum_{i=1}^na_i^2}+\sqrt{\sum_{i=1}^nb_i^2}+\sqrt{\sum_{i=1}^nc_i^2}.
\end{aligned}
$$
Source page cross-reference: page 21.
$$
\begin{aligned}
&\sqrt{\sum_{i=1}^n(3a_i-b_i-c_i)^2}+\sqrt{\sum_{i=1}^n(3b_i-a_i-c_i)^2}+\sqrt{\sum_{i=1}^n(3c_i-a_i-b_i)^2}\\
&\qquad\ge\sqrt{\sum_{i=1}^na_i^2}+\sqrt{\sum_{i=1}^nb_i^2}+\sqrt{\sum_{i=1}^nc_i^2}.
\end{aligned}
$$
Source page cross-reference: page 21.