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Claim. E. Altman, On a problem of P. Erdős, Amer. Math. Monthly 70 (1963), no. 2, 148–157, proves that the vertices of every convex $n$-gon in the plane determine at least $\lfloor n/2\rfloor$ distinct distances. Points in strictly convex position have no three on a line, so the first question of Problem 1082 holds for every such set. The theorem is the whole of Problem 93, of which the first question is a stronger form.

Covers. The first question for sets in convex position, for every $n$. Nothing for other sets. Nothing on the second question, whose convex case is the open Problem 982.

Depends on. Altman's claim page on Problem 93.

Acceptance. Refereed: the paper appeared in the American Mathematical Monthly. Not reviewed: the site labels Problem 1082 FALSIFIABLE and says only that it is a stronger form of Problem 93, so there is no curator credit for this case; the curator's acceptance of the theorem as the solution of Problem 93 is recorded on that problem's claim page.