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[quote=pettifogger][quote=Anonymous][quote=Anonymous]Would these be good for a strong math student, but not one who is into competitions?
Are they fun or rote? [/quote]
As other posters have said, it depends.

But I take a bit of an issue with your question.  First off, "rote" is typically used as a derogatory term by those wanting to push their snakeoil "critical thinking" humbug. But let's apply the principle of charity here and assume by "rote" you mean that something is being repeated in order to learn it by heart.

Yes, actually, successful competitors end up memorizing things, including but not limited to:
- all squares until at least 27^2
- all cubes until at least 11^3
- all powers of two until at least 2^16
- Pascal's triangle until at least 6 choose k
- triangular numbers until at least 55
- all primitive Pythagorean triples until their sum exceeds 90
- Heronian triangles with small areas
- fraction to decimal conversion for powers of 2, for 3, 7, 9, and 11 (at least).

So if you think that, for instance, being asked to answer questions like "what's the hundredth digit after the period of 13/9" or "compute 32*38 in your head" is rote and you should use a calculator then Mathcounts is not for you.

A quick recall of math facts and a high level of fluency in mental arithmetic is however a characteristic of most famous mathematicians from Gauss to von Neumann. [/quote]
Your description suggesting that one should memorize a lot of things in order to be successful in Mathcounts, while not inaccurate, is a bit overblown and I would not want others reading this to take your advice to heart. While *some* of the above listed items are useful mainly because they can appear again and again in problems, one can do quite well in Mathcounts through states and even nationals without needing to memorize the above. Below are three fine points to consider for those interested in doing well in Mathcounts:

1) Practicing doing lots of challenging problems (i.e from past math contests) is the number one way to become really good. This is true for any math contest, not just Mathcounts (and generally true for many skills in life). By practicing problem solving, students tend to internalize and/or pick up the patterns related to a subset of the above mentioned "facts" without having to explicitly memorize them. Problem solving ideas and techniques, (i.e reasoning through the mathematics), is much more important than the above list of facts.

2) As a corollary, many difficult Mathcounts problems are not made difficult due to the above facts you stated. They are difficult because they require creative ideas, which can usually only be acquired via lots of practice working on problems (point #1 above) and careful reasoning.

3) Mathcounts as a contest does have a weakness in that it tends to turn off many students who are not very quick at computations but nevertheless are excellent problem solvers. These type of students may feel (and rightfully so) that they cannot compete at the highest levels simply because of the speed of the rounds (30 questions in 40 minute Mathcounts Sprint round, Target/Countdown rounds, etc). For example, there are significant numbers of students who could solve some AIME problems but not actually be able to finish Mathcounts rounds because of the very short allotted time. It is sad that they cannot have a chance to display their deep problem solving skills simply because they cannot make it far enough in the competition (i.e to states and beyond where Mathcounts problems are more interesting/difficult and they can shine). Not all is lost though, as many of these students are aware and looking forward to many high school math contests that are not quite so focused on speed.[/quote]

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