ALIBABA 2021 Problem 5

Suppose that $A$ is a finite subset of $\mathbb{R}^d$ such that

(a) every three distinct points in $A$ contain two points that are exactly at unit distance apart, and

(b) the Euclidean norm of every point $v$ in $A$ satisfies $$\sqrt{\frac{1}{2}-\frac{1}{2\vert A\vert}} \le \|v\| \le \sqrt{\frac{1}{2}+\frac{1}{2\vert A\vert}}.$$ Prove that the cardinality of $A$ is at most $2d+4$ .