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Claim. Let $f(N)$ be the extremal function of Problem 302. The set $A_0=\{a\le N/4:a\text{ odd}\}\cup\{a:N/2\le a\le N\}$ has $5N/8+O(1)$ elements and contains no distinct $a,b,c$ with $1/a=1/b+1/c$, so

$$f(N)\ge\Bigl(\frac58+o(1)\Bigr)N ,$$

and the particular question of the problem, whether $f(N)=(1/2+o(1))N$, is answered in the negative. The verification is elementary: a solution with $a<b,c$ satisfies $(b-a)(c-a)=a^2$; for $a\ge N/2$ the left side is at most $(N-a)^2\le a^2$ with equality only when $b=c=N$, which distinctness excludes; for odd $a\le N/4$ the square $a^2$ is odd, so $b-a$ and $c-a$ are odd, $b$ and $c$ are even and therefore lie in $[N/2,N]$, whence $b-a,c-a\ge N/4\ge a$ and equality forces $b=c=2a$, again excluded. The argument is written out on the problem page.

Covers. The particular question, answered in the negative, and the lower bound $5/8$ for the asymptotic density. Not covered: the estimate of $f(N)$ beyond that; the recorded bounds leave the constant between $5/8$ and $25/28$.

Standing. Claimed. The site's curator, Thomas Bloom, records the construction in the problem's commentary and credits it to Stijn Cambie, but the site labels the problem OPEN and lists no parts, so the credit is not an acceptance and no reviewed evidence is listed. The observation has no written source of its own, so there is no refereed evidence; no Lean built by this corpus checks it, so there is no formalized evidence, and the formal-conjectures statement file for the problem marks the particular question solved on the strength of this bound with a sorry body (its variant lower_five_eighths is sorry too), which is not a formalization. The site's page shows no last-edited date; the observation is absent from the archived copy of the page of 13 July 2024, which carried an earlier formulation of the problem, and present in the archived copy of 25 March 2025, the date this page carries. The construction was checked on the problem page.