A weighted cyclic quadratic ratio sum is at most three quarters

Proved

WorkbookSource.base_26445

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

lean-workbooksource-checked

Let $a, b, c$ be positive real numbers. Prove that: $\frac{a^2}{2a^2+ab+c^2}+\frac{b^2}{2b^2+bc+a^2}+\frac{c^2}{2c^2+ca+b^2} \le \frac{3}{4}$

Source: InternLM Lean-Workbook, record lean_workbook_26445 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_26445 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 / (2 * a^2 + b * a + c^2) + b^2 / (2 * b^2 + c * b + a^2) + c^2 / (2 * c^2 + a * c + b^2)) ≤ 3 / 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_26445; Apache-2.0

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