A weighted quadratic reciprocal upper bound

Proved

WorkbookSource.base_8606

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

lean-workbooksource-checked

For $a,b,c$ positive reals, show: $\frac{6a}{9a^2+5(a+b+c)^2} + \frac{6b}{9b^2+5(a+b+c)^2} + \frac{6c}{9c^2+5(a+b+c)^2} \leq \frac{1}{a+b+c}$

Source: InternLM Lean-Workbook, record lean_workbook_8606 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_8606 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (6 * a) / (9 * a ^ 2 + 5 * (a + b + c) ^ 2) + (6 * b) / (9 * b ^ 2 + 5 * (a + b + c) ^ 2) + (6 * c) / (9 * c ^ 2 + 5 * (a + b + c) ^ 2) ≤ 1 / (a + b + c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_8606; Apache-2.0

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