Open-problem statements, with their sources attached.

Determine whether positive integers $a,b,c$ exist such that $a^2+b^2$, $a^2+c^2$, $b^2+c^2$, and $a^2+b^2+c^2$ are all perfect squares.

Every Jordan curve $\gamma:S^1\to\mathbb{R}^2$ contains four points that are the vertices of a nondegenerate square.

Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no three on a line. Does $A$ determine at least $\lfloor n/2\rfloor$ distinct distances?

Is there some $c > 0$ such that every measurable $A \subseteq \mathbb{R}^2$ of measure $\geq c$ contains the vertices of a triangle of area 1?

What is the size of the largest $A \subseteq \mathbb{R}^n$ such that every three points from $A$ determine an isosceles triangle? That is, for any three points $x$, $y$, $z$ from $A$, at least two of the distances $|x - y|$, $|y - z|$, $|x - z|$ are equal.

Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$. Estimate $\alpha(n)$.

Estimate a lower bound for$\alpha(n)$.

Estimate an upper bound for$\alpha(n)$.

How many rotated (about the origin) copies of the 'pyjama set' $\\{(x, y) \in \mathbb{R}^2 : \text{dist}(x, \mathbb{Z}) \leq \varepsilon\\}$ are needed to cover $\mathbb{R}^2$?

In particular, can one find a better bound than the best-known bound from [KrLe25]?

Is there a better bound than the best-known bound from [KrLe25]? This is an existential version of the main problem that does not require providing the bound explicitly.

Is $\varepsilon^{-C}$ rotations enough?

Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?

Can a unit square be covered by rectangles of width 1 / (n + 1) and height 1 / (n + 2)?

Equivalently, can a unit square be packed with rectangles of width 1 / (n + 1) and height 1 / (n + 2)?

Inscribed square problem* Does every Jordan curve admit an inscribed square?

Inscribed rectangle problem* Does every Jordan curve admit inscribed rectangles of any given aspect ratio?

What is the smallest square that can contain 11 unit squares?

Reference: Wikipedia

What is the smallest square that can contain 17 unit squares?

Reference: Wikipedia

What is the smallest circle that can contain 3 unit squares?

Reference: Wikipedia

What is the smallest square that can contain 21 unit circles?