A cyclic quadratic ratio upper bound

Proved

WorkbookSource.base_19167

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

lean-workbooksource-checked

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

Source: InternLM Lean-Workbook, record lean_workbook_19167 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_19167 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 / (a^2 + a * b + b^2) + b^2 / (b^2 + b * c + c^2) + c^2 / (c^2 + c * a + a^2)) ≤ (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_19167; Apache-2.0

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