A weighted quadratic reciprocal sum bounds the reciprocal squared total

Proved

WorkbookSource.plus_61932

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

lean-workbooksource-checked

For $a,b,c$ positive real numbers prove that

$\frac {23}{8b^{2} + 7bc + 8c^{2}} + \frac {23}{8c^{2} + 7ca + 8a^{2}} + \frac {23}{8a^{2} + 7ab + 8b^{2}}\ge\frac {27}{(a + b + c)^2}.$

Source: InternLM Lean-Workbook, record lean_workbook_plus_61932 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_61932 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (23 / (8 * b ^ 2 + 7 * b * c + 8 * c ^ 2) + 23 / (8 * c ^ 2 + 7 * c * a + 8 * a ^ 2) + 23 / (8 * a ^ 2 + 7 * a * b + 8 * b ^ 2)) ≥ 27 / (a + b + c) ^ 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_61932; Apache-2.0

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