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Claim. For $k\ge2$ put $A_k=\{r(k-r):1\le r\le k-1\}$; since $r(k-r)$ is unchanged by $r\mapsto k-r$, this is the set $\{r(k-r):1\le r\le k/2\}$ of Problem 443. Hegyvári proves (Theorem 1.1) that for $m>n$

$$|A_n\cap A_m|\le\tau_{m,n} =\#\{(M,N):MN=m^2-n^2,\ M+N<2m,\ 0<M\le N<M+2n\},$$

with equality when $m$ is even and $n$ is odd; (Corollary 1.2) that for every $\epsilon>0$ there is $n_0$ with $|A_n\cap A_m|<m^{(2+\epsilon)\log2/\log\log m}$ whenever $m>n>n_0$, so, by the symmetry of the intersection in $m$ and $n$, it has size $(mn)^{o(1)}$ for all sufficiently large $m\ne n$; and (Theorem 1.3) that for every $s$ there are infinitely many pairs $(n,m)$ with $|A_n\cap A_m|=s$, so the size is unbounded. Together these answer both questions of the corrected Statement, which takes $m\ne n$. The method is elementary: $k(n-k)=r(m-r)$ is rewritten as $(m-2r+n-2k)(m-2r-n+2k)=m^2-n^2$ and the divisors of $m^2-n^2$ are counted. The source card lists the paper's results.

Other proofs and formalizations. The site credits an independent, unpublished solution by Cambie with the same two conclusions; no manuscript is posted, so it has no claim page. Boris Alexeev announced on the problem's forum thread on 4 February 2026 a Lean proof, produced with the system Aristotle, whose header names Hegyvári and Cambie as the informal authors and says it proves Theorem 1.1, Corollary 1.2, Theorem 1.3 and the paper's conditional sum-product application; the linked file is pinned to a later revision of the repository, of 30 June 2026, the commit that the formal-conjectures statement file for the problem cites. That development is a formalization of this result, not a separate claim. This repository has not built or audited it, so it is not listed as formalized evidence here; the formal-conjectures file states the two questions and is not itself a proof.

Acceptance. The site's curator, Thomas Bloom, labels the problem proved and credits Hegyvári's paper, together with Cambie's unpublished solution, for the bound $m^{O(1/\log\log m)}$ and the exact-size pairs. The paper is an arXiv preprint with no journal publication found, so no refereed evidence is listed. This repository has not reviewed the proof.