Challenging problems from analysis, combinatorics, algebra and probability

Our goal is to have a joint discussion on well-chosen specific problems involving topics such as combinatorial and algebraic identities, convergence of sequences of numbers, convergence of integrals, inequalities, casino numbers, approximation of functions, finding values of the Riemann zeta function, etc.
We start with slightly non-standard and attractive problems which are easy to formulate. The emphasis, however, will be on the diversity of ideas and techniques used to find the solutions and the possibility to create any kind of extensions. This is a good way to arrive at very general problems whose solutions could be a serious task.
There will be something unexpected and challenging in each of the problems. It is quite interesting, and it will be well demonstrated, that some problems from Analysis, Algebra and Combinatorics can be solved very elegantly by using only ideas and techniques from Probability. It is more than curious that sometime these are, yet, the only available solutions.
This lecture will be available to a wide audience: undergraduate and graduate students in mathematical sciences, PhD students, and even professionals.
Questions and suggestions from the audience during the lecture will be very welcome.