A comparison of two cyclic quadratic ratio sums

Proved

WorkbookSource.base_1089

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

lean-workbooksource-checked

Let a, b, c be three positive reals. Prove the inequality

$\frac{a^2}{2b+a}+\frac{b^2}{2c+b}+\frac{c^2}{2a+c}\geq\frac{a^2}{2a+b}+\frac{b^2}{2b+c}+\frac{c^2}{2c+a}$ .

Source: InternLM Lean-Workbook, record lean_workbook_1089 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1089 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 / (2 * b + a) + b^2 / (2 * c + b) + c^2 / (2 * a + c)) ≥ (a^2 / (2 * a + b) + b^2 / (2 * b + c) + c^2 / (2 * c + a))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1089; Apache-2.0

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