A pairwise ratio sum bounds a normalized symmetric cubic expression

Proved

WorkbookSource.base_33090

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

lean-workbooksource-checked

If $a,b,c>0$ prove that: $\sum_{cyc}\frac{a}{b+c}\ge\frac{27[a^2(b+c)+b^2(c+a)+c^2(a+b)]}{4(a+b+c)^3}$

Source: InternLM Lean-Workbook, record lean_workbook_33090 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_33090 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (b + c) + b / (c + a) + c / (a + b)) ≥ 27 * (a^2 * (b + c) + b^2 * (c + a) + c^2 * (a + b)) / (4 * (a + b + c)^3)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_33090; Apache-2.0

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