Various elementary techniques for solving problems in algebra, geometry, and combinatorics are explored in this second edition of Mathematics as Problem Solving. Each new chapter builds on the previous one, allowing the reader to uncover new methods for using logic to solve problems. Topics are presented in self-contained chapters, with classical solutions as well as Soifer's own discoveries. With roughly 200 different problems, the reader is challenged to approach problems from different angles.

Mathematics as Problem Solving is aimed at students from high school through undergraduate levels and beyond, educators, and the general reader interested in the methods of mathematical problem solving.

y = −p x y = q. If we can find p and q , then by the converse of the Vi`ete theorem, {x, y} will be the solution set of the equation z 2 + pz + q = 0. Let us therefore rewrite the given system in terms of p and q . Because x 3 + y 3 = (x + y)3 − 3x y(x + y), it is easy to do: (− p)3 − 3q(− p) = −7 q = −2, that is, p 3 − 3 pq = 7 q = −2. By substituting −2 for q in the first equation, we get p3 + 6 p − 7 = 0 that is, ( p − 1)( p 2 + p + 7) = 0 p−1=0 p=1 We have z2 p=1 q=−2 , p2 + p + 7 =

< 0, will have to be checked.) x 2 − 5x + 6 = 0 x1 = 2; x2 = 3 ln |x| = 0 x3 = 1; x4 = −1. The check shows that x 4 = −1 is a solution of this equation, as well, of course, as x = 1, 2, 3. 3.18. Solve the following equation: √ x +1− √ x − 1 = 1. Solution. By multiplying both sides of the given equation by √ x − 1, we get √ √ x + 1 + x − 1 = 2. Now we add this equation to the original one to get √ 2 x + 1 = 3. Therefore, x +1= 9 4 or x = 54 . 3.19. Solve the following equation: 3 1+

will follow soon, as will new expanded editions of the books [9, 1, 10]. All my books will be published by Springer. Write back to me; your solutions, problems, and ideas are always welcome! Alexander Soifer Colorado Springs, Colorado May 8, 2008 Preface to the First Edition Remember but him, who being demanded, to what purpose he toiled so much about an Art, which could by no means come to the knowledge of many. Few are enough for me; one will suffice, yea, less than one will content me,

knowledge is required to understand many of his problems. Moreover, many problems allow young mathematicians to advance and find partial solutions. The book after Erd˝os will be either The Art on the Frontier of Cultures: The Fang People of West Equatorial Africa and Their Neighbors or Memory in Flashback. The former could be a result of my ongoing study of African art and culture, inspired by the great anthropologist James W. Fernandez. The latter will be a collection of humorous and noteworthy

Some Celebrated Ideas 1.14. Given a prime p and a positive integer n . Prove that if n 2 is divisible by p , then n is divisible by p . 1.15. Prove that √ 6 is an irrational number. 1.16. A known theorem states that any point C of the perpendicular bisector of a segment AB is equidistant from A and B . Prove the converse. 1.17. A known theorem states that if a convex quadrilateral is inscribed in a circle, then the sums of its opposite angles are equal. Prove the converse. 1.18. A known