A mixed sixth-degree lower bound at fixed sum one

Proved

WorkbookSource.base_47198

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

lean-workbooksource-checked

Let $a,b,c\ge 0$ such that $a+b+c=1$ . Show that: $243a^2b^2c^2+45(a^2b^2+b^2c^2+c^2a^2)+6(a^2+b^2+c^2)\ge 4$

Source: InternLM Lean-Workbook, record lean_workbook_47198 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_47198 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hab : a + b + c = 1) : 243 * a ^ 2 * b ^ 2 * c ^ 2 + 45 * (a ^ 2 * b ^ 2 + b ^ 2 * c ^ 2 + c ^ 2 * a ^ 2) + 6 * (a ^ 2 + b ^ 2 + c ^ 2) ≥ 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_47198; Apache-2.0

View graph