A product of shifted ratios with a cubic reciprocal correction

Proved

WorkbookSource.base_37591

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

lean-workbooksource-checked

Let $a, b, c>0$ . Prove that $\left( \frac{a}{b+c}+\frac{1}{2} \right)\left( \frac{b}{c+a}+\frac{1}{2} \right)\left( \frac{c}{a+b}+\frac{1}{2} \right)+\frac{abc}{6(a^3+b^3+c^3)} \geq\frac{19}{18}$

Source: InternLM Lean-Workbook, record lean_workbook_37591 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_37591 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (b + c) + 1 / 2) * (b / (c + a) + 1 / 2) * (c / (a + b) + 1 / 2) + a * b * c / (6 * (a ^ 3 + b ^ 3 + c ^ 3)) ≥ 19 / 18  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_37591; Apache-2.0

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