A fixed-sum comparison of cyclic rational sums

Proved

WorkbookSource.base_1491

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

lean-workbooksource-checked

Given $a, b, c$ positive reals such that $a + b + c = 3$ . Prove that:- $\sum_{cyc}{\frac{a}{b^2 + 3a}} \le \sum_{cyc}{\frac{2a^2 + b^2}{3b^2 + 9a}}$

Source: InternLM Lean-Workbook, record lean_workbook_1491 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1491 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : a / (b^2 + 3 * a) + b / (c^2 + 3 * b) + c / (a^2 + 3 * c) ≤ (2 * a^2 + b^2) / (3 * b^2 + 9 * a) + (2 * b^2 + c^2) / (3 * c^2 + 9 * b) + (2 * c^2 + a^2) / (3 * a^2 + 9 * c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1491; Apache-2.0

View graph