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 problemsolving, 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. Readership Advanced high school and undergraduate students, mathematics educators at secondary and university level; general mathematical audience. Reviews "This volume is quite delightful ... Each essay ... engages the reader's interest in an immediate and lively manner ... can be highly recommended for wellprepared students aspiring to a toplevel mathematics degree and will prove a marvelous source for teachers ... a good bedtime read for Putnam competitors ... a pleasure indeed for the general lover of mathematics."  Mathematical Reviews "A worthwhile resource for mathematics departments to have on hand ... this collection serves as a nice resource for accessible topics beyond the standard curriculum around the level of calculus."  MAA Online Table of Contents  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 17gon
 The Markov equation
 Integervalued 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
