A cubic squared-reciprocal sum bounds twice the quadratic sum minus three

Proved

WorkbookSource.plus_78043

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

lean-workbooksource-checked

For $a,b,c>0$ such that $a+b+c=3$ . Prove that

\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2}+3\ge 2(a^2+b^2+c^2)

Source: InternLM Lean-Workbook, record lean_workbook_plus_78043 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_78043 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : a ^ 3 / b ^ 2 + b ^ 3 / c ^ 2 + c ^ 3 / a ^ 2 + 3 ≥ 2 * (a ^ 2 + b ^ 2 + c ^ 2)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_78043; Apache-2.0

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