A shifted pairwise quadratic ratio sum is at most six

Proved

WorkbookSource.plus_22503

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

lean-workbooksource-checked

For $a,b,c>0: a+b+c=3$ . Prove that: $\frac{a^2+4ab+b^2}{a+b+ab}+\frac{b^2+4bc+c^2}{b+c+cb}+\frac{c^2+4ac+a^2}{a+c+ac}\le 6$

Source: InternLM Lean-Workbook, record lean_workbook_plus_22503 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_22503 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (a^2 + 4 * a * b + b^2) / (a + b + a * b) + (b^2 + 4 * b * c + c^2) / (b + c + b * c) + (c^2 + 4 * c * a + a^2) / (c + a + c * a) ≤ 6   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_22503; Apache-2.0

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