A cyclic quadratic ratio lower bound at fixed sum three

Proved

WorkbookSource.base_12018

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

lean-workbooksource-checked

Let $a,b,c> 0$ and $a+b+c=3$ . Prove that: $\frac{a^2}{a+ 2b^2}+\frac{b^2}{b+2c^2}+\frac{c^2}{c+2a^2}\geq 1$

Source: InternLM Lean-Workbook, record lean_workbook_12018 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_12018 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : a^2 / (a + 2 * b^2) + b^2 / (b + 2 * c^2) + c^2 / (c + 2 * a^2) ≥ 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_12018; Apache-2.0

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