A cyclic ratio upper bound involving pairwise products

Proved

WorkbookSource.base_27556

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers such that $a+b+c=3$ . Prove that $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\leq\frac{9}{3+bc+ca+ab}$

Source: InternLM Lean-Workbook, record lean_workbook_27556 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_27556 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : a / (a + b) + b / (b + c) + c / (c + a) ≤ 9 / (3 + b * c + c * a + a * b)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_27556; Apache-2.0

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