A squared pair-sum ratio bounds a linear pair-sum ratio

Proved

WorkbookSource.plus_81069

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

lean-workbooksource-checked

Let a,b,c>0 Prove that $\sum\frac {(a + b)^2}{c(2c + a + b)} \geq \sum\frac {a+b}{2c}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_81069 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_81069 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b) ^ 2 / (c * (2 * c + a + b)) + (b + c) ^ 2 / (a * (2 * a + b + c)) + (c + a) ^ 2 / (b * (2 * b + c + a)) ≥ (a + b) / (2 * c) + (b + c) / (2 * a) + (c + a) / (2 * b)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_81069; Apache-2.0

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