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I am a student who is just starting high school and am very interested in taking part in the IMO (International Mathematical Olympiad). I am currently reading about inequalities. Having studied many of the classical inequalities, I have found that a good knowledge of functions and functional equations would be very beneficial. However, the books I have looked up seem very complicated and I have not found any book offering a friendly introduction into the subject for beginners. None of what I have looked up also relate to olympiad problems. Can anyone please recommend me a book that could help me get acquainted with the topic. Thanks.

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  • $\begingroup$ You might find imomath.com/index.php?options=338 useful, but I almost always turn to eqworld.ipmnet.ru/en/solutions/fe.htm and especially eqworld.ipmnet.ru/en/solutions/fe/fe-toc3.htm $\endgroup$ – Henry Aug 31 '16 at 21:28
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    $\begingroup$ Related: math.stackexchange.com/questions/272591/… and math.stackexchange.com/questions/241445/… $\endgroup$ – Henry Aug 31 '16 at 21:31
  • $\begingroup$ @Henry I have checked the links, IMO math is providing problem-solving techniques (assuming you have prior knowledge). Also the other two links are giving more of a handout, listing and summing up the important parts. But I would like a good book to really understand the topic, and maybe also through exercises. I.e. I don't really know where all the equations in the last two links came from. I really need to understand the topic so that I could eventually solve problems, Olympiad level. I really appreciate your help but I just wanted to make it clear for anyone else who might want to answer. $\endgroup$ – Basem Fouda Aug 31 '16 at 21:35
  • $\begingroup$ Sorry, I didn't see the last comment. I think it would be helpful. $\endgroup$ – Basem Fouda Aug 31 '16 at 21:36
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    $\begingroup$ Functional Equations and how to solve them - Christopher G Small, is totally awesome. $\endgroup$ – gambler101 Aug 31 '16 at 23:05
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Hint: Since you put the focus on inequalities I'd like to point to The Cauchy-Schwarz Master Class by J. Michael Steele. This book gives a comprehensive treatment of inequalities from many different perspectives. It is both, instructive and a pleasure to read. If you don't know it, it might provide you with a lot of nice gems.

A valuable, systematical primer into functional equations is Introduction to Functional Equations by Costas Efthimiou. It also provides IMO examples. Maybe it is even online available.

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