Juan Carlos Ortiz Rhoton with his "tasty" math books. (Image posted with permission from the Ortiz Rhoton family.)
Juan Carlos Ortiz Rhoton from Guadalajara, Mexico loves math as much as he loves food! He got in touch with Science House Foundation to thank us for supporting the Art of Problem Solving, Worldwide Online Olympiad Training, known as WOOT. With the gentleman’s gracious permission, excerpts of his letter have been posted below along with the story of his new skills and magnificent experience.
Art of Problem Solving is a natural fit for Juan Carlos. The organization works to help develop math skills through online coursework, games and curriculum development. WOOT is their preparation and testing program that brings together many of the best students from around the world to learn their “Olympiad problem solving skills”.
After completing WOOT, grantee Juan Carlos (pictured above) wrote us an eloquent and intriguing letter to share the math equations he solved and to thank the Science House Foundation for supporting his in participation.
Wrote Juan Carlos:
I realized how much I had to learn, that the process was long, life lasting I could say. But aside from Math, I also learned to find the process very “tasty”, it was like not only “eating” Math to get full, but to enjoy its delicious flavor while eating it. …The joy of learning became meaningful and therefore, my level of resilience increased.
I enjoyed all the problems in WOOT, whether I solved them or not, because they taught me something. I think my favorite problem from all those that I solved in WOOT, though not by far, is problem number 5. It is challenging, though not very much. I didn’t solve it in the test it came in (called Practice Olympiad), but I found it wonderful when I solved it afterwards. (EDITOR’S NOTE: If you want to take a chance at solving problem 5, see below for the problem and a special note from Juan Carlos).
Thank you all, Science House Foundation Team, for helping students achieve a deeper level of understanding how the world works.
Juan Carlos, thank YOU! You (and the many young people around the world with whom you share an interest in math) are the reason why Science House Foundation was created. The effort that you are willing to make to create a future for this planet that includes a fine citizen like yourself is why we work together at Science House Foundation each day. It makes us happy to see you succeed.
And for those of you who have patiently waited for problem 5…pencils ready!
Without further delay:
“Let P be the set of all primes, and M be a subset of P containing at least three elements. For any proper subset A of M, all of the prime factors of the number (∏p€Ap) – 1 (the product of all elements of A minus 1) are in M. Prove that M=P.”
A note from Juan Carlos:
I find this problem interesting because it is fun. I used all the techniques I like and find useful. Seeing small cases (like proving that 2, 3, 5, 7 and 11 are in M) is useful and gives hindsight of the problem.
Proving that M is infinite is a natural step and is also useful. That was a difficult part. The proof seemed to come naturally, although it took some time. The last part, proving M=P, is the one I didn’t prove in the test (3 hrs. for 3 problems). It needs the Pigeonhole Principle to be completely solved, but I found out after the time limit.
I am attaching both, the official solution which is given to us 2 weeks after the submission deadline (first 2 pages), and my own (pages 3 and 4).