A cubic refinement of the sum of pairwise ratios

Proved

WorkbookSource.base_204

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

lean-workbooksource-checked

Given that $a,b,c$ are positive reals, prove that:

b) $\frac {a}{b + c} + \frac {b}{c + a} + \frac {c}{a + b} \geq \frac {a^3 + b^3 + c^3 + 3abc}{a^2(b + c) + b^2(c + a) + c^2(a + b)} + \frac {1}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_204 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_204 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (b + c) + b / (c + a) + c / (a + b)) ≥ (a ^ 3 + b ^ 3 + c ^ 3 + 3 * a * b * c) / (a ^ 2 * (b + c) + b ^ 2 * (c + a) + c ^ 2 * (a + b)) + 1 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_204; Apache-2.0

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