Challenging and Fun Putnam Problems: A Journey Through Advanced Mathematics

The William Lowell Putnam Mathematical Competition is one of the most prestigious academic problem-solving competitions for undergraduate college students. Every year, this competition features a series of challenging and intriguing problems that test the participants' knowledge of advanced mathematics. In this article, we delve into some of the most notable and interesting Putnam problems from recent years, showcasing the diverse and sophisticated nature of these challenges.

The Experiment of Solving Putnam Problems

I've had the privilege of engaging with various Putnam problems over the years, and I must admit, some of them were particularly intriguing and demanding. The struggle to solve these problems, often lasting for a year or more, has been both a challenge and a rewarding experience. Here are some problems I've managed to solve and some that continue to elude me:

Putnam 2015 Problems

Problem A2

Let (a_0 1), (a_1 2), and (a_n 4a_{n-1} - a_{n-2}) for (n geq 2). Find an odd prime factor of (a_{2015}).

Problem A3

Compute (log_2 left( prod_{a1}^{2015} prod_{b1}^{2015} e^{2pi i a b /2015} right)) Here, (i) is the imaginary unit such that (i^2 -1).

Problem A4

For each real number (x), let (f(x) sum_{n in S_x} frac{1}{2^n}), where (S_x) is the set of positive integers (n) for which (lfloor nx rfloor) is even. What is the largest real number (L) such that (f(x) geq L) for all (x in [0,1])?

Problem B1

Let (f) be a three times differentiable function defined on (mathbb{R}) and real-valued such that (f) has at least five distinct real zeros. Prove that (f'(6)f'(12)f'(8)) has at least two distinct real zeros.

Other Interesting Putnam Problems

Here are a couple of additional problems I found particularly challenging but enjoyable to think about:

Problem A4 (2014)

Suppose (X) is a random variable that takes on only nonnegative integer values with (mathbb{E}[X] 1), (mathbb{E}[X^2] 2), and (mathbb{E}[X^3] 5). Determine the smallest possible value of the probability of the event (X 0).

Problem B–6 (2012)

Let (p) be an odd prime number such that (p equiv 2 pmod{3}). Define a permutation (pi) of the residue classes modulo (p) by (pi(x) x^3 pmod{p}). Show that (pi) is an even permutation if and only if (p equiv 3 pmod{4}).

Reflections and Insights

Putnam problems are not just exercises in mathematics but are mini-explorations into the depths of mathematical thinking and problem-solving techniques. While some of these problems may seem daunting at first, the journey to uncovering their solutions can be both enlightening and rewarding. These problems often require a blend of theoretical knowledge and creative problem-solving skills, making the process a continuous learning experience.

Although the Putnam competition is geared towards students with a deep interest in mathematics, the problems can provide valuable insights and inspiration for anyone interested in mathematical puzzles and challenges.

Conclusion

The problems discussed in this article represent just a fraction of the vast and challenging landscape of Putnam problems. They serve as a reminder of the beauty and complexity of advanced mathematics and the intellectual stimulation provided by such problems. Whether one finds success in solving these problems or not, the journey of exploration and learning they offer is invaluable.

Keywords

*Putnam Problems

*Advanced Mathematics

*Mathematical Challenges