A squared-norm and cubic-product bound at sum nine

Proved

WorkbookSource.base_16964

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

lean-workbooksource-checked

Given $a,b,c>0$ and $a+b+c=9$ shaw that $abc+5(a^2+b^2+c^2)\geqslant{162}$ .

Source: InternLM Lean-Workbook, record lean_workbook_16964 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_16964 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 9) : a * b * c + 5 * (a ^ 2 + b ^ 2 + c ^ 2) ≥ 162  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_16964; Apache-2.0

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