A quadratic ratio sum with a triple-product correction

Proved

WorkbookSource.base_13061

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers such that $a+b+c=3$ . Prove that:

\frac{a^2+3bc}{b+c}+\frac{b^2+3ca}{c+a}+\frac{c^2+3ab}{a+b} \geq \frac{21+3abc}{4}

Source: InternLM Lean-Workbook, record lean_workbook_13061 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_13061 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (a^2 + 3 * b * c) / (b + c) + (b^2 + 3 * c * a) / (c + a) + (c^2 + 3 * a * b) / (a + b) ≥ (21 + 3 * a * b * c) / 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_13061; Apache-2.0

View graph