151 problems

Strengthened truncated-mean inequality. For every $-1<t<1$,

Let $\beta(L)$ denote the beta-value associated with a list $L$, and let list concatenation be written by juxtaposition. Let $m,n$ be non-negative integers with $n>m$, and let $L$…

Let $(m_1,n_1)$ and $(m_2,n_2)$ be pairs of non-negative integers, where $(m_1,n_1)$ is strictly better balanced than $(m_2,n_2)$, meaning that the first pair is strictly closer to…

Let $\varphi(n)$, $\psi(n)$ and $\sigma(n)$ denote the Euler totient, Dedekind psi and sum-of-divisors functions, respectively. For integers $k\geq1$ and $n\geq2$, form the three q…

Let $\varphi(n)$, $\psi(n)$ and $\sigma(n)$ denote the Euler totient, Dedekind psi and sum-of-divisors functions, respectively. For integers $k\geq1$ and $n\geq2$, use…

Nonnegative-coefficient conjecture. If $r\in\mathbb N$ is arbitrary but fixed, then $h(t,a;r)$, viewed as a polynomial in $(t,a)$, has nonnegative coefficients. This would consider…

Let $X$, $Y$, and $Z$ be upward closed subsets of $Q_n$. It is conjectured that … Notation: the poset $Q_n$ is the power set of a set with $n$ elements, ordered by subset relation.…

For every integer $n\ge 0$ and every nonzero polynomial $p$ of degree $n$ whose zeros all lie on the unit circle $\mathbb{T}$, define $Q_n(z)=1+z^n$. For all $0\le s\le t\le\infty$…

Let $T$ be a nondegenerate planar triangle, and let $0<\lambda_1^N(T)\leq\lambda_2^N(T)$ be its first two positive Neumann Laplace eigenvalues. Define the harmonic and geometric me…

For $\nu>0$, let $J_\nu$ denote the Bessel function of the first kind, let $j_{\nu,n}$ be its $n$th positive zero, and define the normalized Turán expression by…

For every integer $n\ge 1$ and every real symmetric positive definite matrix $G\in\mathbb{R}^{n\times n}$, the matrix $\left(G\circ G^{-1}\right)^{-1}$ is entrywise nonnegative, wh…

For every integer $n\ge 1$, every real number $\rho>1$, and all complex numbers $z_1,\ldots,z_n$ satisfying $|z_j|\le \rho$ for $1\le j\le n$, prove that…

Let $B_1,\ldots,B_n$ be independent Bernoulli random variables with parameters $p_1,\ldots,p_n\in[0,1]$, let $S=\sum_{i=1}^n B_i$, and write $P_k(p)=\mathbb{P}_p(S=k)$. Determine t…

For $0\leq\mu_1\leq\cdots\leq\mu_n$ and $\lambda>\sum_{i=1}^n\mu_i$, let $X_1,\ldots,X_n$ range over independent nonnegative random variables satisfying $\mathbb{E}X_i=\mu_i$. Defi…

For every integer $n\ge 1$, every $-1<\alpha<\beta$, and every $0<c\le 1$, let $\pi_n$ be the monic degree-$n$ polynomial orthogonal on $[-c,c]$ with respect to the Jacobi weight…

Rump's 100 Euro Conjecture. There exists $x\in\mathbb{R}^n\setminus\{0\}$ such that

For $g\geq2$, $n\geq1$, and $d_1,\ldots,d_n\geq1$ satisfying $|\mathbf d|=3g-3+n$, let $C(\mathbf d)$ denote the primitive normalized Witten intersection number, and write…

Let $n \in \mathbb{Z}_{\ge 1}$, let $\mu_1 \ge \dotsc \ge \mu_n > 0$ satisfy … and let $X_1,\dotsc,X_n$ be independent non-negative random variables with $EX_i=1$ for every $i$. De…

Phelps–Rodrigues extremal conjecture. If $p>1$, equality in Borcea's variance inequality occurs if and only if

Monotonicity conjecture. For any $n\in\mathbf{N}$, we have

Let $K\subset\mathbb{R}^n$ be a convex body, and let $V$ denote mixed volume, with $K[k]$ denoting $k$ copies of $K$. Godbersen's conjecture. For each convex body…

Monotonicity conjecture. Let $t\geq3$ be an integer. Then

Wishart Gaussian product inequality conjecture.

Gaunt–Merkle median bounds conjecture. For $r>0$,

The quartic-invariant conjecture for $q(n)$. The following inequalities hold: