A cyclic mixed cubic ratio lower bound at fixed sum three

Proved

WorkbookSource.base_11747

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

lean-workbooksource-checked

Given positive numbers $a,b,c$ satisfying $a+b+c=3.$ Show that $\frac{a^3+b^2+c^2}{a^2+1}+\frac{b^3+c^2+a^2}{b^2+1}+\frac{c^3+a^2+b^2}{c^2+1}\geq\frac{9}{2}.$

Source: InternLM Lean-Workbook, record lean_workbook_11747 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_11747 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a^3 + b^2 + c^2) / (a^2 + 1) + (b^3 + c^2 + a^2) / (b^2 + 1) + (c^3 + a^2 + b^2) / (c^2 + 1) ≥ 9 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_11747; Apache-2.0

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