Learn English – What does “taking mock offence” mean

"To throw down" means to fight someone (usually physical). It's a colloquial expression.

Interestingly, one entry on Urban Dictionary attributes its origin to "throwing down the gauntlet" an old tradition of throwing down ones gloves to challenge someone. The person challenging the other would throw down his glove or gauntlet, and the person accepting the challenge 'would take it up'.

You also see it used in popular culture like the TV show Throwdown with Bobby Flay, which is a show where Bobby Flay challenges people to cooking competitions.

"Hard sums" is used to mean "difficult mathematics" in a self-deprecating way, depending on the speaker.

If someone says

I can do hard sums.

and they are:

• A four year-old: It means they've started doing addition.
• An eight year-old: It means they've mastered multiplication.
• A high-school student: It means they can do some simple calculus.
• An undergraduate: It means they can do more complex calculus / analysis / algebra.
• A post-graduate: It means they're Einstein. :-)

And each group is more self-deprecating than the last.