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We have all had the experience of being stumped by a math problem. The way a person responds when this happens varies from person to person, but for some it can be a deflating experience. It is precisely when a person is stuck, however, that the greatest potential for learning and growth reveals itself. Instead of resigning oneself to looking up someone else's solution to the problem, there are opportunities to challenge oneself. This chapter is intended to provide the stuck problem solver with an alternative to reading a complete solution. It is a source of hints, some small and some large, to help readers to persevere in finding their own solutions before reading the full solutions that appear in Chapter 11.
For every exercise in this book, we provide two types of hints. First, there are “Hints to Get Started”, designed to get the reader thinking about how to begin solving the problem. Then, if that is not sufficient or if the reader simply wants further guidance, “More Extensive Hints” provide more detailed strategies for solving the problems, with less work for the reader to fill in to complete the solution. In reading the more extensive hints for a problem, it is assumed that the reader has already read the hints to get started for it.
Many of the AIME problems have several good solutions, and in the full solutions in the final chapter, multiple solutions are sometimes given. However, the hints are usually oriented towards one solution, the one that we feel is most likely to be easiest for the reader to discover.
To reiterate, readers who are not sure how to begin working on a problem are encouraged to consult the hints and use them to take a fresh look at the problem on which they are stuck. This will be a more effective way to learn and practice than reading a full solution provided in Chapter 11.
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