A cyclic cubic-over-quadratic sum bounds the total

Proved

WorkbookSource.base_5486

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

lean-workbooksource-checked

Let a, b, c be three positive reals. Prove the inequality

$\frac{a^2\left( b+c\right) }{b^2+bc+c^2}+\frac{b^2\left( c+a\right) }{c^2+ca+a^2}+\frac{c^2\left( a+b\right) }{a^2+ab+b^2}\geq \frac{2}{3}\left( a+b+c\right)$ .

Source: InternLM Lean-Workbook, record lean_workbook_5486 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_5486 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : (a^2 * (b + c) / (b^2 + b * c + c^2) + b^2 * (c + a) / (c^2 + c * a + a^2) + c^2 * (a + b) / (a^2 + a * b + b^2)) ≥ 2 / 3 * (a + b + c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_5486; Apache-2.0

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