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Circumstance

Hello,

I am looking for books that encourage creative problem-solving at the high school level. I attended secondary school in the United States and I have noticed that there are techniques that I was not taught thoroughly.

Request

Please suggest books that address the following:

  • Substitution/change of variable for single variable equations
  • Factoring non-algebraic expressions
  • "smart" manipulation of expressions

I have listed YouTube videos depicting the kinds of problems I would like to be able to solve. The videos are in Russian, but they all involve solving equations.

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

I am aware of The Art of Problem Solving but I feel it addresses contest math.

Thank you.

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  • $\begingroup$ I feel it addresses contest math --- To me the videos (I only saw the very beginnings to see what the equations are) seem to be highly focused on the types of problems that arise in contest math, so I don't understand your concern about AOPS. That said, some of the older advanced school algebra texts I listed in this answer are probably worth looking at. $\endgroup$ Mar 22 at 14:25
  • $\begingroup$ Not a book, but these "variants of the Entrance Examinations" for Moscow State University (department of Mechanics and Mathematics) should be of interest: 1995 and 1996 and 1997 and 1998. I came across these (continued) $\endgroup$ Mar 22 at 14:31
  • $\begingroup$ on the internet early 1998. At this time I had a super strong but small class of 5 or 6 students in a multivariable calculus class (taught it several times in the late 1990s at LSMSA) who were very interested in these types of problems when I brought photocopies of the 1995 problems to class sometime in February 1998. Since one of my students was fluent in Russian, she was able to tell us what the problems said when it wasn't obvious. I wound up devoting the last 20 minutes of a class once a week for us to talk about the problems and present solutions to the others. $\endgroup$ Mar 22 at 14:37
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I like The Art and Craft of Problem Solving, by Paul Zeitz. But you probably already know of it, since it's mentioned on the aops site.

I think it addresses some of what you list above. But it goes so much deeper too.

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