A symmetric quadratic ratio bounds a weighted cyclic sum

Proved

WorkbookSource.base_50909

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

lean-workbooksource-checked

The following inequality is also true. For positives $a$ , $b$ and $c$ prove that: $\frac{a^2+b^2+c^2}{3(ab+ac+bc)}\geq\frac{a}{2a+7b}+\frac{b}{2b+7c}+\frac{c}{2c+7a}$

Source: InternLM Lean-Workbook, record lean_workbook_50909 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_50909 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b^2 + c^2) / (3 * (a * b + a * c + b * c)) ≥ a / (2 * a + 7 * b) + b / (2 * b + 7 * c) + c / (2 * c + 7 * a)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_50909; Apache-2.0

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