Mathematics Problem Archive

Graham: For every non-equilateral triangle T, show that it is possible to color the plane with three colors so that there is no monochromatic (congrue...

Straus: Is every polygonal room in the plane illuminable from some point? See this....

/Clark : What is the (homogeneous adjacency) spectrum of the Fano plane?...

Is the weak order on S_(n) (the "inversion" poset) Sperner?...

If the critical set of $u(x)=\sum_{k=1}^n a_k/|x-x_k|$ for positive point charges in $\mathbb R^3$ is finite, how many points can it contain? In parti...

For a monic polynomial $p$ of degree $d$, determine the maximum length of the lemniscate $E(p)=\{z:|p(z)|=1\}$. Is the extremal asymptotically $p(z)=z...

Let $f$ be holomorphic in the unit disk with a simple zero at $0$, exactly two simple $1$-points at $b$ and $-b$, and no other zeros or $1$-points. De...

Do there exist seven planar points in general position—no three collinear and no four concyclic—such that every pairwise distance is an integer?...

For the classical polynomial Liénard system $$\dot x=y-F(x),\qquad \dot y=-x,$$ let $\operatorname{Lie}(n)$ be the maximum number of limit cycles when...

Are there smooth vector fields in $\mathbb{R}^3$ satisfying the hypotheses of the Markus–Yamabe conjecture and having periodic orbits?...

For $n\in\mathbb{N}$, let $\Pi(n)$ be the product of its decimal digits, and let $\operatorname{Pm}(n)$ be the least positive integer such that $\Pi^{...

Determine all single sided (resp. double sided) sequences that are pattern Sturmian....

Given a C∞ metric in a neighborhood of a point in a 2-dimensional Riemannian manifold, does there exist an isometric embedding of some neighborhood of...

Do non-singular real plane projective curves of an odd degree consisting of a single connected component form a connected set (i.e., are they rigid is...

Is it true that any component of the complement of the discriminant variety of a versal deformation contains a Morsification, whose all $\mu(f)$ criti...

Give a similar universal estimate of the radius of convergence for multidimensional Newton’s method of [Shub and Smale1993]....