Solve in real numbers $\frac{(x+2)^4}{x^3}-\frac{(x+2)^2}{2x}\ge - \frac{x}{16}$

Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $D$ be the midpoint of side $BC$ and $BE,CZ$ be the altitudes of the triangle $ABC$. Line $ZE$ intersects line $BC$ at point $O$. (i) Find all the angles of the triangle $ZDE$ in terms of angle $\angle A$ of the triangle $ABC$ (ii) Find the angle $\angle BOZ$ in terms of angles $\angle B$ and $\angle C$ of the triangle $ABC$

Find all positive integers $x$, for which the equation $$a+b+c=xabc$$has solution in positive integers. Solve the equation for these values of $x$

We are having  99 equal circles in a row and in the interior, we write inside them all the numbers from 1 up to 99 (one number in each circle).We color each of the circles with one of the two colors available: red and green. A coloring is good if it has the ability: Red circles lying in the interval of the numbers from 1 up to 50  are more than the red circles lying in the interval of the numbers from 51  up to 99 . a) Find how many different colorings can be constructed. b) Find how many different good colorings can be constructed. (Note: Two colorings are different, if they have different color in at least one of their circles.)