A weighted quadratic reciprocal sum is at most one fifth at fixed sum three

Proved

WorkbookSource.plus_17559

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

lean-workbooksource-checked

Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 3.$ Prove that $\frac {1}{a^2 + 7b^2 + 7c^2} + \frac {1}{b^2 + 7c^2 + 7a^2} + \frac {1}{c^2 + 7a^2 + 7b^2} \le \frac {1}{5}.$

Source: InternLM Lean-Workbook, record lean_workbook_plus_17559 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_17559 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : 1 / (a ^ 2 + 7 * b ^ 2 + 7 * c ^ 2) + 1 / (b ^ 2 + 7 * c ^ 2 + 7 * a ^ 2) + 1 / (c ^ 2 + 7 * a ^ 2 + 7 * b ^ 2) ≤ 1 / 5   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_17559; Apache-2.0

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