A product of mixed ratio sums has a constant lower bound

Proved

WorkbookSource.base_18091

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

lean-workbooksource-checked

Let $a,b,c>0.$ Prove that

\left (\frac{a}{b+c}+\frac{a+b}{c}\right)\left (\frac{b}{c+a}+\frac{b+c}{a}\right)\left (\frac{c}{a+b}+\frac{c+a}{b}\right)\geq \frac{125}{8}

Source: InternLM Lean-Workbook, record lean_workbook_18091 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_18091 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (b + c) + (a + b) / c) * (b / (c + a) + (b + c) / a) * (c / (a + b) + (c + a) / b) ≥ 125 / 8  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_18091; Apache-2.0

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