A quadratic lower bound under a weighted linear constraint

Proved

WorkbookSource.base_42167

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

lean-workbooksource-checked

Let $a,b,c$ be real numbers such that $a+2b+3c=10$ , prove that: $a^2+b^2+c^2+ab+bc+ca\ge10$ .

Source: InternLM Lean-Workbook, record lean_workbook_42167 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_42167 (a b c : ℝ) (h : a + 2 * b + 3 * c = 10) : a ^ 2 + b ^ 2 + c ^ 2 + a * b + b * c + c * a ≥ 10  :=  by sorry

Source