Question 105
Let $a, b, c \in \mathbb{R}$ with $a, b, c \geq 1$. Show that
$$
\min \left(\frac{10 a^{2}-5 a+1}{b^{2}-5 b+10}, \frac{10 b^{2}-5 b+1}{c^{2}-5 c+10}, \frac{10 c^{2}-5 c+1}{a^{2}-5 a+10}\right) \leq a b c
$$
$$
\min \left(\frac{10 a^{2}-5 a+1}{b^{2}-5 b+10}, \frac{10 b^{2}-5 b+1}{c^{2}-5 c+10}, \frac{10 c^{2}-5 c+1}{a^{2}-5 a+10}\right) \leq a b c
$$