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.

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 well-prepared students aspiring to a top-level 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 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