A mixed pair-product ratio sum is at most three fifths

Proved

WorkbookSource.plus_73744

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

lean-workbooksource-checked

If $a, b, c>0$ prove that $\frac{ab}{(a+b)^2+bc}+\frac{bc}{(b+c)^2+ca}+\frac{ca}{(c+a)^2+ab}\le\frac{3}{5}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_73744 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_73744 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * b / ((a + b) ^ 2 + b * c) + b * c / ((b + c) ^ 2 + c * a) + c * a / ((c + a) ^ 2 + a * b)) ≤ 3 / 5   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_73744; Apache-2.0

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