A pair-sum product bound at fixed pairwise sum

Proved

WorkbookSource.base_47959

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

lean-workbooksource-checked

For non-negative $a,$ $b$ and $c$ such that $ab + ac + bc = 3$ prove that $8(a + b + c)^2\geq9(a + b)(a + c)(b + c)$

Source: InternLM Lean-Workbook, record lean_workbook_47959 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_47959 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (habc : a * b + a * c + b * c = 3) : 8 * (a + b + c) ^ 2 ≥ 9 * (a + b) * (a + c) * (b + c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_47959; Apache-2.0

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