A pair-product ratio upper bound at fixed sum three

Proved

WorkbookSource.base_11047

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

lean-workbooksource-checked

Let $a,b, c>0, a+b+c=3$ . Prove that $\frac{ab(a+b)}{1+ab}+\frac{bc(b+c)}{1+bc}+\frac{ca(c+a)}{1+ca}\le3$

Source: InternLM Lean-Workbook, record lean_workbook_11047 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_11047 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a * b * (a + b) / (1 + a * b) + b * c * (b + c) / (1 + b * c) + c * a * (c + a) / (1 + c * a)) ≤ 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_11047; Apache-2.0

View graph