A cyclic cubic-over-quadratic lower bound at fixed sum three

Proved

WorkbookSource.base_2864

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

lean-workbooksource-checked

If $a, b, c>0, a+b+c=3$ prove that $\frac{a^3}{a^2+bc+ca}+\frac{b^3}{b^2+ca+ab}+\frac{c^3}{c^2+ab+bc}\ge1$

Source: InternLM Lean-Workbook, record lean_workbook_2864 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_2864 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : a^3 / (a^2 + b * c + c * a) + b^3 / (b^2 + c * a + a * b) + c^3 / (c^2 + a * b + b * c) ≥ 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_2864; Apache-2.0

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