American Mathematical Monthly Problem

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 ...

Associativity of the Secant Method

1. INTRODUCTION. This paper has its genesis in a problem the author first came upon while in college. Although the areas covered here are well travelled and nothing here is guaranteed original, it covers a pleasant nexus of many mathematical strands. Furthermore, we show the value of good notation and of reading an old master for solving a problem. Consider iterates of the function m (x) = 1 I ...