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AwesomeMath Admission Tests

Titu Andreescu, Navid Safaei. and Alessandro Ventullo
XYZ Press
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Henry Ricardo
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Note: this is a review of two volumes in this series covering 2006-2014 and 2015-2021.  
These books present problems that have constituted part of the admissions requirements for the Awesome Math Summer Program (AMSP) for middle school and high school students, cofounded and directed by Titu Andreescu, who has created the ”vast majority” of the problems. Professor Andreescu is well-known to problemists. At various times, he has served as head coach and leader of the US IMO team, director of the MAA AMC program, cofounder of the Purple Math! competition, and founder/editor-in-chief of the free online journal Mathematical Reflections.  In addition, he is the author/co-author of over 50 problem-oriented books.
The problems (over 500 in total) are at a precalculus level and cover the ”four traditional areas of competition math”: algebra, geometry, number theory, and combinatorics. Complete solutions (often multiple solutions) to all problems are provided. If you are familiar with any of Andreescu’s competition math books, you have come to expect clever problems, insightful commentary, and an occasional reference to the mathematical literature. Each book contains an identical Glossary of definitions,theorems, and formulas applied in the solutions—Angle Bisector Theorem, Bertrand’s Postulate,..., Trigonometric Identities, Vieta’s Formulas.
Here’s a small sample of problems, one from each of the main areas listed above:
  • Let \(a_{n}=2-\frac{1}{n^{2}+\sqrt{n^{4}+\frac{1}{4}}}\), \(n=1, 2, \ldots \).  Prove that \(\sqrt{a_{1}}+\sqrt{a_{2}}+ \cdots + \sqrt{a_{119}}\) is an integer.
  • Let \(ABCD\) be a quadrilateral inscribed in a semicircle of diameter \(AD=x\).  If \(AB=a\), \(BC=b\), \(CD=c\), prove that \(x^{3}-(a^{2}+b^{2}+c^{2})x-2abc=0\).
  • Find the greatest \(n\) for which \(n!\) has fewer than 2017 positive divisors.
  • Consider the word COUSINS and all different "words" that can be obtained by permuting its digits.  Arrange them in a list of alphabetical order.  What is the position of the word COSINUS on the list.
These are books with many uses. You may be searching for ideas for your own math circle or math camp, trying to spice up a course in one of the four main area covered by these books, or just looking for personal mathematical challenges. Anyone who appreciates a good problem will enjoy these books. 


Henry Ricardo ( retired from Medgar Evers College (CUNY) as Professor of Mathematics and is currently affiliated with the Westchester Area Math Circle. He is the author of A Modern Introduction to Differential Equations (Third Edition) and A Modern Introduction to Linear Algebra. He has a special interest in mathematical problem solving