A shifted pairwise cubic ratio lower bound at fixed sum three

Proved

WorkbookSource.base_40646

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

lean-workbooksource-checked

Prove that $\frac{a^3+b^3}{ab+1}+\frac{b^3+c^3}{bc+1}+\frac{c^3+a^3}{ca+1}\geq3$ given $a,b,c>0$ and $a+b+c=3$ .

Source: InternLM Lean-Workbook, record lean_workbook_40646 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_40646 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a^3 + b^3) / (a * b + 1) + (b^3 + c^3) / (b * c + 1) + (c^3 + a^3) / (c * a + 1) ≥ 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_40646; Apache-2.0

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