A pair-product ratio upper bound at fixed sum three

Proved

WorkbookSource.base_55136

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 $\frac{ab}{a^2+b^2+3(a+b)}+\frac{bc}{b^2+c^2+3(b+c)}+\frac{ca}{c^2+a^2+3(c+a)}\le\frac{a^2+b^2+c^2}{8}$

Source: InternLM Lean-Workbook, record lean_workbook_55136 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_55136 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a * b / (a ^ 2 + b ^ 2 + 3 * (a + b)) + b * c / (b ^ 2 + c ^ 2 + 3 * (b + c)) + c * a / (c ^ 2 + a ^ 2 + 3 * (c + a))) ≤ (a ^ 2 + b ^ 2 + c ^ 2) / 8  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_55136; Apache-2.0

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