A cyclic pair-product ratio sum is at most one

Proved

WorkbookSource.plus_12966

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

lean-workbooksource-checked

Let $a,b,c > 0.$ Prove that $\frac{ab}{ab+b^2+ c^2}+\frac{bc}{bc+c^2 + a^2}+\frac{ca}{ca+a^2+ b^2}\leq 1$

Source: InternLM Lean-Workbook, record lean_workbook_plus_12966 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_12966 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * b / (a * b + b ^ 2 + c ^ 2) + b * c / (b * c + c ^ 2 + a ^ 2) + c * a / (c * a + a ^ 2 + b ^ 2)) ≤ 1   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_12966; Apache-2.0

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