A quadratic bound at real unit sum

Proved

WorkbookSource.base_33591

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

lean-workbooksource-checked

Let $a,b,c$ be real numbers with $a+b+c=1$ . Show that $\frac{a^2+b^2+c^2}{4}+\frac{ab+bc+ca}{5}\ \geq\ \frac{3}{20}$ (NB: $a,b,c$ can be negative/zero as well as positive.)

Source: InternLM Lean-Workbook, record lean_workbook_33591 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_33591 (a b c : ℝ) (h : a + b + c = 1) :
  3 / 20 ≤ a ^ 2 / 4 + b ^ 2 / 4 + c ^ 2 / 4 + a * b / 5 + b * c / 5 + c * a / 5  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_33591; Apache-2.0

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