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### Re: MPMS Problem Solving Marathon

Posted: Tue Jan 08, 2019 3:59 pm
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?
You made that problem up, didn't you?

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\$.

### Re: MPMS Problem Solving Marathon

Posted: Tue Jan 08, 2019 4:19 pm
Produce \$DA\$ upto \$E\$ such that \$AE=BC\$.It follows that \$AEBC\$ is a parallelogram.So,\$BE=AC=BD=d\$.By Stewart's theorem on \$\bigtriangleup BED,d^2b+d^2c=m(a^2+bc)\$ where \$m=b+c\$.Cancelling \$b+c=m\$ from both sides we finally have \$d^2=a^2+bc\$.