ABSTRACT
This chapter introduces 10 Monthly problems related to the mean value theorem, eigenvalues and determinants of some special matrices, weighted trigonometric sums and Dirichlet series, infinite sum-product identity, and polynomial zero identities. In addition to developing valuable techniques for solving these problems, it also includes some interesting historical accounts of ideas and contexts. Among them, we will encounter various extensions of the mean value theorem, Chebyshev polynomials, ( 0 , 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003168065/00c14962-ad03-4e9f-bf7b-805c548b14be/content/math5_3388.tif"/> -matrices, the Cauchy matrix, Dodgson condensation, Dirichlet convolution, the Woods-Robbbins identity and the alternating sign matrix conjecture. It is intended to facilitate a natural transition that bridges problem-solving to independent exploration of new results.
