lean_workbook_plus_48286

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Find $k > 0$ largest such that $\left(\frac ab+\frac bc+\frac ca\right)^2 \geqslant k\left [ (a+b+c)\left(\frac 1a+\frac 1b+\frac 1c\right)-9 \right ]+9,$ for all $a,\,b,\,c$ are positive real numbers.\n\nFind $k > 0$ largest such that $\left(\frac ab+\frac bc+\frac ca\right)^2 \geqslant k\left [ (a+b+c)\left(\frac 1a+\frac 1b+\frac 1c\right)-9 \right ]+9,$ for all $a,\,b,\,c$ are positive real numbers.\n\n\n $k\le 3$\n\nId est, it remains to prove the last inequality for $y=x$ and $z=\frac{1}{x^2}$ , which gives $(x-1)^2(13x^4-28x^3+3x^2+20x+10)\geq0$ , which is true.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_48286 :
  IsGreatest {k : ℝ | 0 < k ∧ ∀ a b c : ℝ, (1 / a + 1 / b + 1 / c) * (a + b + c) - 9 ≥ k * ((1 / a + 1 / b + 1 / c) ^ 2 - 9) } 3   :=  by sorry

The formal statement changes or incorrectly supplies part of the source problem; the accepted note identifies a concrete witness or definitional mechanism.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Find the largest k>0 such that for all positive real a,b,c, (a/b+b/c+c/a)²≥k[(a+b+c)(1/a+1/b+1/c)−9]+9. Restore this exact cyclic-ratio inequality before proving the optimum.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.