A cyclic quadratic reciprocal lower bound at fixed sum three

Proved

WorkbookSource.base_13448

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

lean-workbooksource-checked

For $a,b,c>0$ such that $a+b+c=3$ . Prove that:

[2] $\dfrac{1}{5a^2+ab+bc}+\dfrac{1}{5b^2+bc+ca}+\dfrac{1}{5c^2+ca+ab} \ge \dfrac{3}{7}$

Source: InternLM Lean-Workbook, record lean_workbook_13448 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_13448 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (1 / (5 * a ^ 2 + a * b + b * c) + 1 / (5 * b ^ 2 + b * c + c * a) + 1 / (5 * c ^ 2 + c * a + a * b)) ≥ 3 / 7  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_13448; Apache-2.0

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