A cyclic pair-product reciprocal upper bound

Proved

WorkbookSource.base_9359

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

lean-workbooksource-checked

Let $a$ , $b$ , $c$ be positive real numbers. Prove that: $\frac{ab}{a+b+2c}+\frac{bc}{b+c+2a}+\frac{ca}{c+a+2b} \leq \frac{1}{4}\left(a+b+c\right)$

Source: InternLM Lean-Workbook, record lean_workbook_9359 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_9359 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * b) / (a + b + 2 * c) + (b * c) / (b + c + 2 * a) + (c * a) / (c + a + 2 * b) ≤ (1 / 4) * (a + b + c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_9359; Apache-2.0

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