*“An integer is called formidable if it can be written as a sum of distinct powers of 4, and successful if it can be written as a sum of distinct powers of 6. Can 2005 be written as a sum of a formidable number and a successful number? Prove your answer.”*

This problem, from the 2005 Bay Area Mathematical Olympiad, is intended primarily for bright and math-focused high schoolers. Imagine my delight to find four of my students — three in middle school, no less — not only fearlessly diving in, but actually **solving** it.

Congratulations to Cameron R., Sloan D., Seika N., and Jack D.!

2008 is going to be a very good year.

I have to admit I am somewhat older than a middle-schooler. I am getting the answer “no”. Is that correct?

Yes, but without a proof, that wouldn’t get much credit. ๐

Fair enough. To be the sum of a formidable number and a successful number, 2005 will have to be the sum of a subset of the following terms — each chosen 0 or 1 time:

1 4 16 64 256 1024

1 6 36 216 1296

We cannot choose both 1024 and 1296, because that exceeds 2005.

Hence the maximum we can achieve is the sum of 1 4 16 64 256 1 6 36 216 1296

But that maximum is less than 2005.

QED (quite easily done).

Complete, direct, concise: I don’t know how I could improve on that! Well done.

…and thanks for the inspiration and reminder to start posting again now that I’m back from vacation.