A cyclic linear-over-quadratic lower bound

Proved

WorkbookSource.base_9638

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

lean-workbooksource-checked

Let a,b,c be positive numbers. Prove that $\frac {a(b + c)}{b^2 + bc + c^2} + \frac {b(a + c)}{a^2 + ac + c^2} + \frac {c(a + b)}{a^2 + ab + b^2}\geq2.$

Source: InternLM Lean-Workbook, record lean_workbook_9638 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_9638 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * (b + c) / (b ^ 2 + b * c + c ^ 2) + b * (a + c) / (a ^ 2 + a * c + c ^ 2) + c * (a + b) / (a ^ 2 + a * b + b ^ 2)) ≥ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_9638; Apache-2.0

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