Sendov's Conjecture

Claim: if a degree-n ≥ 2 polynomial with distinct roots has all roots in |z| ≤ 1, then each root a lies within distance 1 of at least one root of the derivative (critical point).

Posed 1958; proven for n ≤ 9 (with heroic effort at the tail), for roots near the unit circle (Brown–Xiang 1990s), for real-rooted polynomials, and — Terence Tao 2020 — for all sufficiently large n. What remains is the finite middle range, where the interplay between Gauss–Lucas geometry and root separation gets genuinely delicate. Bulgarian mathematicians consider it a national mathematical monument; the world's complex analysts keep chipping.