Question 161
a) Prove that $\frac{1}{2}(a+b)^{2}+\frac{1}{4}(a+b) \geq a \sqrt{b}+b \sqrt{a}$, $a, b \geq 0$
b) Show that for all $x, y, z \in\left[0, \frac{\pi}{2}\right)$ the following inequality holds:
$$
\begin{aligned}
& (\operatorname{tg} x+\operatorname{tg} y)^{2}+(\operatorname{tg} y+\operatorname{tg} z)^{2}+(\operatorname{tg} z+\operatorname{tg} x)^{2}+\operatorname{tg} x+\operatorname{tg} y+\operatorname{tg} z \geq \\
& \geq 2(\operatorname{tg} x \sqrt{\operatorname{tg} y}+\operatorname{tg} y \sqrt{\operatorname{tg} x}+\operatorname{tg} y \sqrt{\operatorname{tg} z}+\operatorname{tg} z \sqrt{\operatorname{tg} y}+\operatorname{tg} z \sqrt{\operatorname{tg} x}+\operatorname{tg} x \sqrt{\operatorname{tg} z})
\end{aligned}
$$
b) Show that for all $x, y, z \in\left[0, \frac{\pi}{2}\right)$ the following inequality holds:
$$
\begin{aligned}
& (\operatorname{tg} x+\operatorname{tg} y)^{2}+(\operatorname{tg} y+\operatorname{tg} z)^{2}+(\operatorname{tg} z+\operatorname{tg} x)^{2}+\operatorname{tg} x+\operatorname{tg} y+\operatorname{tg} z \geq \\
& \geq 2(\operatorname{tg} x \sqrt{\operatorname{tg} y}+\operatorname{tg} y \sqrt{\operatorname{tg} x}+\operatorname{tg} y \sqrt{\operatorname{tg} z}+\operatorname{tg} z \sqrt{\operatorname{tg} y}+\operatorname{tg} z \sqrt{\operatorname{tg} x}+\operatorname{tg} x \sqrt{\operatorname{tg} z})
\end{aligned}
$$