I can not resist quoting from one of the AOPS articles (also included in the pdf above):

What is Problem Solving?
by Richard Rusczyk
http://www.artofproblemsolving.com/Resources/articles.php?page=problemsolving

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... true mathematics is not a process of memorizing formulas and applying them to problems tailor-made for those formulas. Instead, the successful mathematician possesses fewer tools, but knows how to apply them to a much broader range of problems. We use the term “problem solving” to distinguish this approach to mathematics from the ‘memorize-use-forget’ approach.

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After MOP I relearned math throughout high school. I was unaware that I was learning much more. ... The skills the problem solvers developed in math transferred, and these students flourished.

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We use math to teach problem solving because it is the most fundamental logical discipline. ... There are many paths to strong problem solving skills. Mathematics is the shortest.