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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