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 Problems Problem of the month (elementary group theory): Let G be a finite group with n elements and let m be a positive integer. If m and n are relatively prime, then every element of G has a unique mth root. That is, for each g in G, prove there is a unique h with hm = g. Problem of the month (elementary geometry): Find a polytope (such as a tetrahedron) in 3 dimensions with an odd number of vertices, edges, and faces or prove one does not exist. For example, two tetrahedrons stuck together (along a triangle) is a polytope with 5 vertices, 9 edges, but 6 faces, so is not such an example. (Thanks Mike at Noisebridge!) Problem of the month (polynomials): Given a positive integer n, prove that the nth cyclotomic polynomial has linear x term -µ(n), where µ is the mobius function on the positive integers. (Thanks Lionel Levine!) More Problems of the Month Problem Proposals: American Mathematical Monthly [11446] Matrices whose products are all different | pdf [11422] Normality of a matrix given iterated symmetric commutators | problem | solution [11377] A Determinant Generated by a Polynomial | pdf [11346] An Unusual GCD/LCM Relationship | pdf [11321] Characteristic polynomials of rational symmetric matrices | pdf [11288] A polynomial product identity| pdf [11231] A problem involving word equations in groups | pdf [11204] A trace formula for sums of products of matrices | pdf [11123] Snapshots of points moving on a line | pdf [11098] Asymptotic behavior of a certain combinatorial sum | pdf [10928] Powers sums of a convergent sum | pdf | ps [10723] A sum congruence modulo a prime | pdf | ps Mathematics Magazine [1775] Graphs with a path connectivity property| pdf [1770] Convex linear recurrences| pdf [1750] Arithmetical progressions modulo a prime | pdf [1684] Counting certain equivalence classes of words | pdf | ps (* denotes to appear in an issue) Selected Solutions: [11226] pdf [11190] pdf [11096] pdf [11028] pdf | ps [11085] pdf [11077] pdf | ps[10873] pdf | ps [10851] pdf | ps [Put05] pdf

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