Let $A\subset {\mathbb{R}}^2$ be a set of $n$ points with no three on a line. Does $A$ determine at least $\lfloor n/2\rfloor$ distinct distances? In fact, must there exist a single point from which there are at least $\lfloor n/2\rfloor$ distinct distances?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

* (D(A)={,|x-y|:x,y\in A,\ x\neq y,}) be the **set of pairwise distances**, and * for a “pinned” point (p\in A), (D_p(A)={,|p-q|:q\in A\setminus{p},}).

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.