Phlembac Adib Hasan wrote:Problem 5

A Hydra has 2019 heads and is immune to damage from conventional weapons. However, with one blow of a magical sword, Hercules can cut off its 9, 10, 11 or 12 heads. In each of these cases, 5, 18, 7 and 0 heads grow on its shoulder. The Hydra will die only if all the heads are cut off. Can Hercules kill the Hydra with his sword?

**Problem 6**

An isosceles trapezoid has equal sides of length $a$, parallel sides of lengths $b$ and $c$, and diagonals of length $d$. Without using Ptolemy's theorem, show that $d^2 = a^2 + bc$.