A quartic symmetric sum bounds a quadratic sum at total three

Proved

WorkbookSource.base_1812

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

lean-workbooksource-checked

The following inequality is true already. Let $a$ , $b$ and $c$ be real numbers such that $a+b+c=3$ . Prove that:

a^4+b^4+c^4+4(a^2b^2+b^2c^2+c^2a^2) \geqslant 5(a^2+b^2+c^2)

Source: InternLM Lean-Workbook, record lean_workbook_1812 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1812 (a b c : ℝ) (h : a + b + c = 3) : a^4 + b^4 + c^4 + 4 * (a^2 * b^2 + b^2 * c^2 + c^2 * a^2) ≥ 5 * (a^2 + b^2 + c^2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1812; Apache-2.0

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