BG-JBMO-TST 2018

In the quadrilateral $ABCD$ , we have $\measuredangle BAD = 100^{\circ}$ , $\measuredangle BCD = 130^{\circ}$ , and $AB=AD=1$ centimeter. Find the length of diagonal $AC$ .

For all positive reals $a$ and $b$ , show that $$\frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab > 18.$$

Find all positive integers $n$ such that the number $$n^6 + 5n^3 + 4n + 116$$ is the product of two or more consecutive numbers.