A weighted cyclic quadratic reciprocal upper bound

Proved

WorkbookSource.plus_7119

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

lean-workbooksource-checked

Let a,b,c be positives real number prove that : $\frac{2a}{3a^{2}+b^{2}+2ca}+\frac{2b}{3b^{2}+c^{2}+2ab}+\frac{2c}{3c^{2}+a^{2}+2bc} \leq \frac{3}{a+b+c}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_7119 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_7119 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (2 * a / (3 * a ^ 2 + b ^ 2 + 2 * c * a) + 2 * b / (3 * b ^ 2 + c ^ 2 + 2 * a * b) + 2 * c / (3 * c ^ 2 + a ^ 2 + 2 * b * c)) ≤ 3 / (a + b + c)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_7119; Apache-2.0

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