A symmetric quadratic ratio with a cyclic quartic correction

Proved

WorkbookSource.base_26251

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers . Prove that $\frac{3(a^2+b^2+c^2)}{ab+bc+ca}+\frac{ab^3+bc^3+ca^3}{a^3b+b^3c+c^3a}\geq 4.$

Source: InternLM Lean-Workbook, record lean_workbook_26251 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_26251 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (3 * (a ^ 2 + b ^ 2 + c ^ 2) / (a * b + b * c + c * a) + (a * b ^ 3 + b * c ^ 3 + c * a ^ 3) / (a ^ 3 * b + b ^ 3 * c + c ^ 3 * a)) ≥ 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_26251; Apache-2.0

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