Related papers: Crux's Crux's crux

An technically interesting proof of a known theorem.

These notes represent an extended version of a talk I gave for the participants of the IMO 2009 and other interested people. We introduce diophantine equations and show evidence that it can be hard to solve them. Then we demonstrate how one…

We consider a recent The Vee's fair soup division problem, provide its partial solution, and pose a related open problem.

The paper studies the rewriting problem, that is, the decision problem whether, for a given conjunctive query $Q$ and a set $\mathcal{V}$ of views, there is a conjunctive query $Q'$ over $\mathcal{V}$ that is equivalent to $Q$, for cases…

We give a new solution to the famous Gion shrine geometry problem from eighteenth-century Japan. Like the classical Japanese solution, ours is given in the form of a degree ten equation. However, our polynomial has the advantage of being…

This is a survey of old and new problems and results in additive number theory.

A recently claimed resolution to the lattice Gribov problem in the context of chiral lattice gauge theories is examined. Unfortunately, I find that the old problem remains.

Expository article on the problem of determining the maximum number of equiangular lines with a fixed angle, and the associated problem of second eigenvalue multiplicity in graphs.

Given two points in the plane, and a set of "obstacles" given as curves through the plane with assigned weights, we consider the point-separation problem, which asks for the minimum-weight subset of the obstacles separating the two points.…

We present a solution of Exercise 1.2.1 of [2] which yields a short new proof of a key step in one of proofs of Brouwer's fixed point theorem, 1910. A few people asked the author about the details of the solution and they might be…

The present paper aims to survey known results and to point out the wealth of rather important open problems that are out there.

The problem of finding the convex hull of an IFS fractal is relevant in both theoretical and computational settings. Various methods exist that approximate it, but our aim is its exact determination. The finiteness of extremal points is…

This is an update of my problem list.

We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.

We consider an obstacle problem for elastic curves with fixed ends. We attempt to extend the graph approach provided in [8]. More precisely, we investigate nonexistence of graph solutions for special obstacles and extend the class of…

In this note, we establish the validity of a conjecture recently proposed in Mathematics Magazine and connect it to the existing interesting results

Based on Lyndon words, a new Sudoku-like puzzle is presented and some relative theoretical questions are proposed.

This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…

In this paper, we consider the planar forced N-pendulum equation. Multiple rotational solutions are obtained.

A square trisection is a problem of assembling three identical squares from a larger square, using a minimal number of pieces. This paper presents an historical overview of the square trisection problem starting with its origins in the…