A comparison of mixed quadratic ratio sums

Proved

WorkbookSource.plus_5224

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

lean-workbooksource-checked

Prove that for any $a,b,c>0$ we have

$3\sum_{cyc}\frac{a(b+c)}{b^{2}+bc+c^{2}}\le5+\sum_{cyc}\frac{a^{2}}{b^{2}+bc+c^{2}}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_5224 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_5224 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 3 * (a * (b + c) / (b ^ 2 + b * c + c ^ 2) + b * (c + a) / (c ^ 2 + c * a + a ^ 2) + c * (a + b) / (a ^ 2 + a * b + b ^ 2)) ≤ 5 + a ^ 2 / (b ^ 2 + b * c + c ^ 2) + b ^ 2 / (c ^ 2 + c * a + a ^ 2) + c ^ 2 / (a ^ 2 + a * b + b ^ 2)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_5224; Apache-2.0

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