Reddit reviews 100 Great Problems of Elementary Mathematics (Dover Books on Mathematics)

I tried to read Arnold's "Mathematical problems from 5 to 15" when I was 22, there are two possibilities...

• I was actually 10 years old, judging by the amount of questions I could answer.
  
  
• As I had 22 and 22∉{5,6,7,...,15}, this forbid me to answer all of it.
  
  To understand the joke, see the last problem of this list, it's easy to find a copy of it with Google.
  
  But seriously, not being able to answer problems from past grades is not a big deal, we forget stuff all the time, just try to study it again. When you try to solve problems from past grades without success, it's probably because you don't remember the pragmatic way to answer it and when trying to answer it, you try to "invent" a solution. I think that this is a great opportunity to contemplate how superb mathematics is: These pragmatic solutions we learn at school, even from very basic mathematics took centuries to be developed in the form it is presented to you. Newton's "Standing on the shoulders of giants" couldn't be more fit.
  
  You should take a look at Dörrie's "100 great problems of elementary mathematics: Their history and solution"

A nice pair of books you might like is the two volume set Challenging Mathematical Problems with Elementary Solutions, by Yaglom and Yaglom. Here's the table of contents from volume 1 and the table of contents from volume 2, so you can see what areas the problems are drawn from.

I haven't looked at this book, but noticed it on the site while looking up the details on the above two books: The USSR Olympiad Problem Book. It's from the height of the cold war, when Russia was generally kicking ass in math, so I'd expect it to be pretty good.

Actually, there's about a bazillion books of problems at Dover, most quite affordable, so I'd recommend browsing around that site a bit.

Plenty of problem books have been put together from the International Mathematical Olympiad problems, such as this one. I think you can find most of these problems online if you prefer.

A book less of the "here are problems you can solve!" variety and more of the "here are some neat problems and their solutions!" is 100 Great Problems of Elementary Mathematics: Their History and Solution by Heinrich Dorrie. It covers everything from the Hermite-Lindemann theorem to calculating how to tack into a north wind as quickly as possible. It's just all over the place, which might be something you'd like given your desire for variety. Here's the table of contents.