A shifted cubic ratio sum is at least three halves

Proved

WorkbookSource.plus_2308

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

lean-workbooksource-checked

Let $a$ , $b$ and $c$ be positive real numbers such that $a+b+c=3$ . Prove the inequality

\frac{a^3}{a^2+1}+\frac{b^3}{b^2+1}+\frac{c^3}{c^2+1} \geq \frac{3}{2}.

Proposed by Anastasija Trajanova

Source: InternLM Lean-Workbook, record lean_workbook_plus_2308 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_2308 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : a^3 / (a^2 + 1) + b^3 / (b^2 + 1) + c^3 / (c^2 + 1) ≥ 3 / 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_2308; Apache-2.0

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