\[\ \boxed{\text{1011.\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]
\[x + y + z = 1\]
\[\sqrt{4x + 1} + \sqrt{4y + 1} +\]
\[+ \sqrt{4z + 1} \leq 5\]
\[\left. \ \begin{matrix} \sqrt{4x + 1} = \frac{(4x + 1) + 1}{2} \\ \sqrt{4y + 1} = \frac{(4y + 1) + 1}{2} \\ \sqrt{4z + 1} = \frac{(4z + 1) + 1}{2} \\ \end{matrix} \right\} \Longrightarrow\]
\[\Longrightarrow \frac{(4x + 1) + 1}{2} +\]
\[+ \frac{(4y + 1) + 1}{2} +\]
\[+ \frac{(4z + 1) + 1}{2} =\]
\[= \frac{4x + 2}{2} + \frac{4y + 2}{2} + \frac{4z + 2}{2} =\]
\[= 2x + 1 + 2y + 1 + 2z + 1 =\]
\[= 2 \cdot (x + y + z) + 3 \Longrightarrow при\ x +\]
\[+ y + z = 1 \Longrightarrow\]
\[2 \cdot 1 + 3 = 5,\ \ 5 = 5 \Longrightarrow ч.т.д.\]