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.
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.