A cyclic linear ratio sum has a symmetric upper bound

Proved

WorkbookSource.base_52437

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

lean-workbooksource-checked

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

Source: InternLM Lean-Workbook, record lean_workbook_52437 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_52437 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b) / (a + 2 * b + c) + (b + c) / (b + 2 * c + a) + (c + a) / (c + 2 * a + b) ≤ (a + b + c) ^ 2 / (2 * (a * b + b * c + a * c))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_52437; Apache-2.0

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