\[\boxed{\text{1062\ (1062).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[\frac{1}{3} \leq \frac{a^{2} - a + 1}{a^{2} + a + 1} \leq 3\]
\[1)\frac{1}{3} \leq \frac{a^{2} - a + 1}{a^{2} + a + 1}\]
\[3a^{2} - 3a + 3 \geq a^{2} + a + 1\]
\[2a² - 4a + 2 \geq 0\]
\[(a - 1)^{2} \geq 0 \Longrightarrow при\ любом\ \text{a.}\]
\[2)\frac{a^{2} - a + 1}{a^{2} + a + 1} \leq 3\]
\[a^{2} - a + 1 \leq 3a^{2} + 3a + 3\]
\[2a^{2} + 4a + 2 \geq 0\]
\[(a + 1)^{2} \geq 0 \Longrightarrow при\ любом\ \text{a.}\]
\[Ответ:при\ любом\ a.\]