A cyclic quartic ratio lower bound at fixed sum three

Proved

WorkbookSource.base_50971

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

lean-workbooksource-checked

If $a,b,c>0$ and $a+b+c=3$ prove that:

$\frac{{a^3 \left( {a + b} \right)}}{{a^2 + ab + b^2 }} + \frac{{b^3 \left( {b + c} \right)}}{{b^2 + bc + c^2 }} + \frac{{c^3 \left( {c + a} \right)}}{{c^2 + ac + a^2 }} \ge 2$

Source: InternLM Lean-Workbook, record lean_workbook_50971 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_50971 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a^3 * (a + b) / (a^2 + a * b + b^2) + b^3 * (b + c) / (b^2 + b * c + c^2) + c^3 * (c + a) / (c^2 + c * a + a^2)) ≥ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_50971; Apache-2.0

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