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Just because a given item is never explicitly referenced in mathematics tournament problems does not mean that knowing it will not help you in tournaments. To take a trivial example, you will probably never be asked in a tournament to give a mnemonic for the first 8 digits of pi, but if you have memorized the sentence “May I have a large container of coffee?”, you will have that (based on the number of letters in each word), and therefore have the first 8 digits of pi at reliable and immediate recall, a situation that could help you in a tournament. To take a deeper example, if you know the topological definition of convexity, the you can apply it to the geometrical figures you encounter in tournaments. Just as ‘angle’ can refer to the opening, or to the measure of the opening, so also reference to a given polygon can refer strictly to the union of the line segments making up the figure, or to the entire region defined by that figure. When the polygon is said to be convex, what is meant is the topological definition applied to the latter case.
Another focused consideration is this: If it is well-known that tournament problems are often constructed with a certain theorem in mind, such as is indeed the case for the angle-bisector theorem, then it would be a good idea to be well-versed in the PROOF of that theorem, even though in a tournament you will never be asked to state the theorem, much less to prove the theorem. Something you have only a tenuous grasp of can forsake you in a noisy / hostile environment – but if you have actually gone through the PROOF of the concerned theorem from time to time, then it will be reliably at your command in a noisy / hostile environment – and the stress of a tournament – if not the very status of being a tournament – means that a tournament is a noisy / hostile environment. Perhaps this quote from Bruce Lee will cement the point that I’m trying to make: “I do not fear the man who has practiced ten thousand kicks one time. I fear the man who has practiced one kick ten thousand times.”
So, in a word, knowing background material can help you in a highly foregrounded regime such as a tournament.
keywords: Mathematics, Logic, advantage, preparation