A symmetric quadratic reciprocal sum bounds a cubic ratio

Proved

WorkbookSource.base_960

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

lean-workbooksource-checked

Let $a,b,c$ are positive numbers, prove inequality $\frac{1}{a^2+ab+b^2}+ \frac{1}{b^2+bc+c^2}+\frac{1}{c^2+ca+a^2} \ge \frac{7}{3} {.} \frac{a+b+c}{a^2(b+c)+b^2(c+a)+c^2(a+b)+abc}$

Source: InternLM Lean-Workbook, record lean_workbook_960 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_960 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (1 / (a ^ 2 + a * b + b ^ 2) + 1 / (b ^ 2 + b * c + c ^ 2) + 1 / (c ^ 2 + c * a + a ^ 2)) ≥ 7 / 3 * (a + b + c) / (a ^ 2 * (b + c) + b ^ 2 * (c + a) + c ^ 2 * (a + b) + a * b * c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_960; Apache-2.0

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