This is the English translation of the book originally published in Russian. It contains 20 essays, each dealing with a separate mathematical topic. The topics range from brilliant mathematical statements with interesting proofs, to simple and effective methods of problem-solving, to interesting properties of polynomials, to exceptional points of the triangle. Many of the topics have a long and interesting history. The author has lectured on them to students worldwide. The essays are independent of one another for the most part, and each presents a vivid mathematical result that led to current research in number theory, geometry, polynomial algebra, or topology.
Conjugate numbers Rational parametrizations of the circle Sums of squares of polynomials Representing numbers as the sum of two squares Can any knot be unraveled? Construction of a regular 17-gon The Markov equation Integer-valued polynomials Chebyshev polynomials Vectors in geometry The averaging method and geometric inequalities Intersection points of the diagonals of regular polygons The chromatic polynomial of a graph Brocard points Diophantine equations for polynomials The Pascal lines One butterfly and two butterflies theorems The Van der Waerden theorem on arithmetical progressions Isogonal conjugate points Cubic curves related to the triangle.