A cyclic square ratio sum has a quartic upper bound

Proved

WorkbookSource.plus_20512

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

lean-workbooksource-checked

Let $a, b, c>0$ . Prove that $\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\le\frac{(a^2+b^2+c^2)^2}{6abc}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_20512 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_20512 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 / (b + c) + b^2 / (c + a) + c^2 / (a + b)) ≤ (a^2 + b^2 + c^2)^2 / (6 * a * b * c)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_20512; Apache-2.0

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