“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. 😉
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.