On December 5th I have taken the Mother of all Math Exams. Established in 1938 by William Lowell Putnam to encourage healthful rivalry between colleges and institutions, it is a notoriously difficult and challenging exam featuring problems from many topics of Pure and Theoretical Math.
Every year about 2000 students take the exam. Some do it for fun, some for bragging rights, and some are good enough to win prizes or to use their score to get into a prestigious graduate school.
Exam is taken in two sessions, each one lasting 3 hours, with a break for lunch in between. There are 6 problems to solve during each session. Each question features a problem from one or more branches of Mathematics. Solving a problem means solving it correctly and showing all relevant work, usually by constructing a rigorous proof of your answer. A correct solution of a problem yields 10 points. Partial credit may be given if a significant progress has been done towards a solution, but such credit is rare. Out of all the students who take the exam, roughly half of them receive a score of 0.
I have a good feeling that I have solved one of the problems. I will know the actual results in March, when the exam will be graded.
Here is the problem:
Can any positive rational number be expressed as a quotient of products of factorials?
For example, 10/9 = (2!5!)/(3!3!3!)