Has anyone competed in or been part of an institution that participated in this competition? My school doesn't appear to have been involved in it past 2001 which is a shame because I really would like to pit myself against people from other schools or heck just challenge myself. Anyway I came across a problems archive and was doing a few practice problems so really this post is less about the competition and more about whether I am ready. This was question 1 on the 1995 test and I think I have a solution but I'm not sure it is "rigorous": (solution to problem A-1 located at http://www.unl.edu/amc/a-activities/a7-problems/putnam/-pdf/1995.pdf [Broken] )(adsbygoogle = window.adsbygoogle || []).push({});

Let [tex]a, b, c \in T[/tex]

[tex]abc \in T[/tex]

[tex](ab)c \in T[/tex]

Let [tex]d, e, f \in U[/tex]

[tex]def \in U[/tex]

[tex]d(ef) \in U[/tex]

Assume [tex](ab) \in U[/tex]

then [tex](ab)ef \in U[/tex]

[tex]ab(ef) \in U[/tex]

For this to be true [tex] a, b \in U[/tex] but since U and T are disjoint this is a contradiction so [tex] ab \in T[/tex]

[tex]Let g = ab \in T[/tex]

[tex]gc \in T[/tex]

Is it proven? Please pardon the bad LaTex I will edit this post if it doesn't come out right. Well come to think of it I don't need that g = ab part right?

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# William Lowell Putnam Competition

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