American Mathematical Monthly Problem 11403

The Problem of Squarable Lunes

An Elementary Problem Equivalent to the Riemann Hypothesis

The problem is: Let Hn = n ∑ j=1 1 j be the n-th harmonic number. Show, for each n ≥ 1, that

A Binomial-like Matrix Equation

We show that a pair of matrices satisfying a certain algebraic identity, reminiscent of the binomial theorem, must have the same characteristic polynomial. This is a generalization of Problem 4 (11th grade) from the Romanian National Mathematical Olympiad 2011.

The Three-Body Problem and the Shape Sphere

The three-body problem defines dynamics on the space of triangles in the plane. The shape sphere is the moduli space of oriented similarity classes of planar triangles and lies inside shape space, a Euclidean three-space parameterizing oriented congruence classes of triangles. We derive and investigate the geometry and dynamics induced on these spaces by the three-body problem. We present two t...

Problem 11066

Problem # 11066 American Mathematical Monthly 111 (2004) Let R be a ring such that for any x, y ∈ R there exist nonnegative integersm = m(x, y) and n = n(x, y) such that xy = xyxy Prove that R is commutative. Proof. First we prove the following lemmas. Lemma 1 Let R be a ring, x, y ∈ R and m,n nonnegative integers such that xy = (x+ 1)y = 0 Then y = 0. Proof of Lemma1. Let k denote the minimal ...