HMMT February Guts Round, 2024

Problem:

Let $p$ denote the proportion of teams, out of all participating teams, who submitted a negative response to problem 5 of the Team round (e.g. "there are no such integers"). Estimate $P=\lfloor 10000 p\rfloor$. An estimate of $E$ earns $\max (0,\lfloor 20-|P-E| / 20\rfloor)$ points.

If you have forgotten, problem 5 of the Team round was the following: "Determine, with proof, whether there exist positive integers $x$ and $y$ such that $x+y, x^{2}+y^{2}$, and $x^{3}+y^{3}$ are all perfect squares."