11 problems

For every integer $m\ge 1$, let $X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbb{P}^5$ be the Fermat fourfold. The Hodge conjecture asserts that every rational Hodge…

For each integer $n\geq 1$, determine … where … In particular, determine the exact value of $M_{\mathrm{bal}}(8)$; the supplied report claims $M_{\mathrm{bal}}(8)=116$, excluding…

For each integer $k\ge 1$, determine the smallest integer $n\ge k$ such that $n(n-1)(n-2)\cdots(n-k)$ divides the central binomial coefficient $\binom{2n}{n}$; equivalently, determ…

Consider the separable constrained optimization problem $\min_{x_1,x_2,x_3}\ f_1(x_1)+f_2(x_2)+f_3(x_3)$ subject to $A_1x_1+A_2x_2+Ix_3=b$, where the first two objective functions…

A position in $4 \times n$ Chomp is a quadruple $(a,b,c,d)$ of non-negative integers with $a\geq b\geq c\geq d\geq 0$; a P-position is a position from which the previous player has…

Let $p$ and $q$ be coprime integers with $q$ odd. For the periodic problem $\Sigma(p/q)$, let $\gamma_k(p/q)$ denote the length of the gap with label $k$. Gap-decay and extremal-fr…

Let $m$ be an odd integer with $m\geq 5$, and let $W\in\mathbb{Z}_2*\mathbb{Z}_4$. Let $\tau_W(\lambda)$ denote the trace polynomial associated with $W$. The trace-polynomial nonex…

Let $B_3$ be the three-strand braid group, let $\Sigma_3^{\scriptstyle *}$ denote its dual generating set, and let $\mathcal{S}(B_3,\Sigma_3^{\scriptstyle *})$ and…

Let $t_{2}(2,q)$ be the smallest size of a complete arc in $\mathrm{PG}(2,q)$, and let the bounds denoted in the source by … be the explicit upper bounds obtained from the computed…

Let $a_\rho^{(3)}$ be the coefficients in the monomial expansion of $g_3$. Kerov's numerical conjecture. For $k=3$, the values of $500\cdot5!\cdot7!\,a_\rho^{(3)}$ are given by the…

Let $c_\rho^{(3)}$ be the coefficients in the monomial expansion of $f_3$. Goulden–Rattan's numerical conjecture. For $k=3$, the values of $2\cdot 6!\cdot 8!\,c_\rho^{(3)}$ are giv…