Tonight our math circle worked on the 50th anniversary American High School Math Exams (AHSME) problems, which contained one problem from each of the first five decades of the AHSME. (For those who don't know, the AHSME was the predecessor of today's AMC12 contest. And it all started in New York State in 1950--yay for New York pride!)
Here is one of the problems a number of tonight's students especially enjoyed from the 1991 AHSME contest given 20 years ago, before any of the students eligible to take this year's AMC contests were born.
To clarify these instructions, let's be clear that you can use the four allowed procedures as many times as you like, in any order that you like.
It looks very daunting at first--there are so many possibilities to consider.
If you want a hint, check the comments.