A sum of squared symmetric expressions is at least 147

Proved

WorkbookSource.base_32246

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

lean-workbooksource-checked

Let $a,b,c \ge 0$ and $a+b+c=3$ . Prove that: $2(bc+3a)^2+2(ca+3b)^2+2(ab+3c)^2+\left(2ab+2bc+2ca+3\right)^2 \ge 147$ .

Source: InternLM Lean-Workbook, record lean_workbook_32246 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_32246 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hab : a + b + c = 3) : 2 * (b * c + 3 * a) ^ 2 + 2 * (c * a + 3 * b) ^ 2 + 2 * (a * b + 3 * c) ^ 2 + (2 * a * b + 2 * b * c + 2 * c * a + 3) ^ 2 ≥ 147  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_32246; Apache-2.0

View graph