By Titu Andreescu
This signficantly revised and extended moment version of Mathematical Olympiad Challenges is a wealthy number of difficulties prepare through skilled and recognized professors and coaches of the U.S. overseas Mathematical Olympiad crew. 1000's of gorgeous, demanding, and instructive difficulties from algebra, geometry, trigonometry, combinatorics, and quantity conception from quite a few mathematical competitions and journals were chosen and up-to-date. the issues are clustered through subject into self-contained sections with options supplied individually. old insights and asides are awarded to stimulate additional inquiry. The emphasis all through is on artistic suggestions to open-ended problems.
New to the second one edition:
* thoroughly rewritten discussions precede all the 30 devices, adopting a extra uncomplicated variety with extra obtainable and welcoming examples
* Many new or improved examples, difficulties, and solutions
* extra references and reader feedback were incorporated
Featuring better motivation for complicated highschool and starting students, in addition to teachers and Olympiad coaches, this article can be utilized for artistic problem-solving classes, specialist instructor improvement seminars and workshops, self-study, or as a coaching source for mathematical competitions.
This [book] is…much greater than simply one other number of attention-grabbing, not easy difficulties, yet is as a substitute geared up in particular for studying. The publication expertly weaves jointly similar difficulties, in order that insights progressively turn into innovations, tips slowly turn into equipment, and techniques ultimately evolve into mastery…. The publication is geared toward inspired highschool and starting students and instructors...I strongly suggest this booklet for somebody drawn to inventive problem-solving in mathematics…. It has already taken up a prized place in my own library, and is certain to supply me with many hours of highbrow pleasure.
—The Mathematical Gazette (Review of the 1st Edition)
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Extra info for Mathematical Olympiad Challenges
2 Look at the Endpoints This section is about inequalities that are proved by using the fact that certain real functions reach their extrema at the endpoints of the interval of definition. Two kinds of functions are considered: • linear functions, which have both extrema at the endpoints of their domain, and • convex functions, whose maximum is attained on the boundary of the domain. The main idea is to view an expression as a linear or convex function in each of the variables separately and use this to bound the expression from above or below.
Prove that among the numbers 1 1 1 1 1 1 + , + , + x 4−y y 4−z z 4−x there is at least one that is greater than or equal to 1. 4. Find all real solutions to the system of equations √ x + y = 4z − 1, √ y + z = 4x − 1, z+x = 4y − 1. 5. Let x, y be numbers in the interval (0, 1) with the property that there exists a positive number a different from 1 such that logx a + logy a = 4 logxy a. Prove that x = y. 6. Find all real triples (x, y, z) that satisfy x4 + y4 + z4 − 4xyz = −1. 7. Find all triples of real numbers x, y, z satisfying 2xy − z2 ≥ 1, z − |x + y| ≥ −1.
Prove that there exists at least one rectangle in the dissection. Find the sum of the areas of all rectangles from the dissection. 16. Find with proof all possible values of the largest angle of a triangle that can be dissected into five disjoint triangles similar to it. 5 Regular Polygons This section discusses two methods for solving problems about regular polygons. The first method consists in the use of symmetries of these polygons. We illustrate it with the following fascinating fact about the construction of the regular pentagon.