A weighted cyclic cubic ratio sum bounds one eighth of the total

Proved

WorkbookSource.plus_57739

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $a,b,c >0$ . Prove that $\frac{a^{3}}{5a^{2}+2ab+b^{2}}+\frac{b^{3}}{5b^{2}+2bc+c^{2}}+\frac{c^{3}}{5c^{2}+2ca+a^{2}}\geq \frac{a+b+c}{8}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_57739 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_57739 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^3 / (5 * a^2 + 2 * a * b + b^2) + b^3 / (5 * b^2 + 2 * b * c + c^2) + c^3 / (5 * c^2 + 2 * c * a + a^2)) ≥ (a + b + c) / 8   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_57739; Apache-2.0

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