HMMT 2022 U34

Estimate $A$, the number of unordered triples of integers $(a, b, c)$ so that there exists a nondegenerate triangle with side lengths $a, b$, and $c$ fitting inside a $100 \times 100$ square. An estimate of $E$ earns $\max (0,\lfloor 20-|A-E| / 1000\rfloor)$ points.