A weighted cyclic quadratic ratio upper bound

Proved

WorkbookSource.base_43269

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

lean-workbooksource-checked

For $a, b, c>0$ prove that $\frac{a^2+bc}{3a^2+3b^2+2c^2}+\frac{b^2+ca}{3b^2+3c^2+2a^2}+\frac{c^2+ab}{3c^2+3a^2+2b^2}\le\frac{3}{4}$

Source: InternLM Lean-Workbook, record lean_workbook_43269 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_43269 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b * c) / (3 * a^2 + 3 * b^2 + 2 * c^2) + (b^2 + c * a) / (3 * b^2 + 3 * c^2 + 2 * a^2) + (c^2 + a * b) / (3 * c^2 + 3 * a^2 + 2 * b^2) ≤ 3 / 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_43269; Apache-2.0

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