A squared pair-product ratio upper bound at fixed sum three

Proved

WorkbookSource.plus_44524

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

lean-workbooksource-checked

Let $a,cb,c>0,a+b+c=3$ ,prove that: $\frac{b^2c^2}{a+3bc}+\frac{c^2a^2}{b+3ac}+\frac{a^2b^2}{c+3ab}\le \frac{3}{4}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_44524 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_44524 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (b^2 * c^2 / (a + 3 * b * c) + c^2 * a^2 / (b + 3 * c * a) + a^2 * b^2 / (c + 3 * a * b)) ≤ 3 / 4   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_44524; Apache-2.0

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