A weighted cyclic ratio sum bounded by symmetric quadratic forms

Proved

WorkbookSource.base_27353

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers. Prove that $\frac {a^2 + b^2 + c^2}{ab + bc + ca} \ge \frac {a}{b + 2c} + \frac {b}{c + 2a} + \frac {c}{a + 2b}.$

Source: InternLM Lean-Workbook, record lean_workbook_27353 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_27353 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : (a^2 + b^2 + c^2) / (a * b + b * c + 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_27353; Apache-2.0

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