A weighted quadratic reciprocal comparison

Proved

WorkbookSource.plus_44568

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

lean-workbooksource-checked

With positive $a,b,c,$ prove:

\frac{a}{8a^2+5b^2+3c^2} +\frac{b}{8b^2+5c^2+3a^2} +\frac{c}{8c^2+5a^2+3b^2}\le\frac{1}{16}\left( \frac{1}{a} +\frac{1}{b} +\frac{1}{c} \right)

Source: InternLM Lean-Workbook, record lean_workbook_plus_44568 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_44568 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (8 * a ^ 2 + 5 * b ^ 2 + 3 * c ^ 2) + b / (8 * b ^ 2 + 5 * c ^ 2 + 3 * a ^ 2) + c / (8 * c ^ 2 + 5 * a ^ 2 + 3 * b ^ 2)) ≤ (1 / 16) * (1 / a + 1 / b + 1 / c)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_44568; Apache-2.0

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