A cyclic squared-product ratio upper bound

Proved

WorkbookSource.base_21878

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

lean-workbooksource-checked

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

Source: InternLM Lean-Workbook, record lean_workbook_21878 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_21878 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 * (b + c)^2 / (a^2 + b * c) + b^2 * (c + a)^2 / (b^2 + c * a) + c^2 * (a + b)^2 / (c^2 + a * b)) ≤ a^2 + b^2 + c^2 + a * b + b * c + c * a  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_21878; Apache-2.0

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