Find a Hadamard matrix of order 668

Prove a tight lower bound on Ramsey numbers for a class of off-diagonal book graphs.

Construct hypergraphs as large as possible that do not have a certain easy-to-check, difficult-to-find property.

Prove that certain “small” Diophantine equations have infinitely many solutions.

Improve best-known upper bounds by constructing specific combinatorial objects.

Improve the exponent in the upper bound that degree has over sensitivity.

Present a KLT del Pezzo surface in characteristic 3 with more than 7 singular points.

Construct an $(n,q,r)$-Steiner system with $n > q > r > 5$, $r < 10$, and $n < 200$.

Give a presentation for the absolute Galois group of the field of $2$-adic numbers as a profinite group.

Find a polynomial whose Galois group is the Mathieu group $M_{23}$.

Find partitions whose stretched LR-coefficients, when expressed as a polynomial, have a negative coefficient.

Find explicit embeddings of symplectic balls into a single target ball, taking up all but $\epsilon$ of the target ball's volume.

Adapt Apéry's proof of the irrationality of $\zeta(3)$ to other constants.

Improve the constant factor in the exponent of GNFS.

Devise an algorithm that decides whether a knot has unknotting number equal to 1.