Problem #1038 is ostensibly a problem about polynomials, but it can instead be phrased as a question about discrete probability measures $\mu = \sum_i p_i \delta_{a_i}$ on the interval $[-1,1]$. For any such measure, form the logarithmic potential
$$ U_\mu(x) = \int \log |x-y|\ d\mu(y) = \sum_i p_i \log |x_i - a_i|$$
and consider the quantity
$$ |{ x: U_\mu(x) < 0 }|.$$
What is the infimum and supremum of this quantity?
The supremum conjecturally is attained at $2 \sqrt{2}$, taking the $(a_i,p_i)$ to be $(+1,0.5)$ and $(-1,0.5)$ (i.e., $\mu$ is the uniform distribution on ${-1,+1}$). This may be hard to prove completely. But the infimum has more scope for room of improvement. Currently the best upper and lower bounds on the infimum are
$$1 \leq \inf |\{ x: U_\mu(x) < 0 \}| \leq 1.84 $$
where the upper bound was found by the AI-powered tool AlphaEvolve. Details to both bounds can be found in the in the comments to the problem. Perhaps one can do better on either side?
Problem #1038 is ostensibly a problem about polynomials, but it can instead be phrased as a question about discrete probability measures$\mu = \sum_i p_i \delta_{a_i}$ on the interval $[-1,1]$ . For any such measure, form the logarithmic potential
$$ U_\mu(x) = \int \log |x-y|\ d\mu(y) = \sum_i p_i \log |x_i - a_i|$$
and consider the quantity
$$ |{ x: U_\mu(x) < 0 }|.$$
What is the infimum and supremum of this quantity?
The supremum conjecturally is attained at$2 \sqrt{2}$ , taking the $(a_i,p_i)$ to be $(+1,0.5)$ and $(-1,0.5)$ (i.e., $\mu$ is the uniform distribution on ${-1,+1}$ ). This may be hard to prove completely. But the infimum has more scope for room of improvement. Currently the best upper and lower bounds on the infimum are
$$1 \leq \inf |\{ x: U_\mu(x) < 0 \}| \leq 1.84 $$
where the upper bound was found by the AI-powered tool AlphaEvolve. Details to both bounds can be found in the in the comments to the problem. Perhaps one can do better on either side?