Prove2Me

A shifted pairwise quadratic ratio upper bound at fixed sum three

Proved
WorkbookSource.base_37161

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

lean-workbooksource-checked

If a,b,c>0,a+b+c=3a, b, c>0, a+b+c=3 prove that ∑cyca+b1+a2+b2+ab≤92(ab+bc+ca)\sum_{cyc}{\frac{a+b}{1+a^2+b^2+ab}}\le \frac9{2(ab+bc+ca)}

Source: InternLM Lean-Workbook, record lean_workbook_37161 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_37161 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a + b) / (1 + a^2 + b^2 + a * b) + (b + c) / (1 + b^2 + c^2 + b * c) + (c + a) / (1 + c^2 + a^2 + c * a) ≤ 9 / (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_37161; Apache-2.0

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