A cyclic product-over-linear upper bound at fixed sum

Proved

WorkbookSource.base_631

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

lean-workbooksource-checked

Let $a,b,c$ are positive numbers satisfies $a+b+c=3.$ Prove that $\frac{ab}{2b+1}+\frac{bc}{2c+1}+\frac{ca}{2a+1} \leq 1.$

Source: InternLM Lean-Workbook, record lean_workbook_631 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_631 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a * b / (2 * b + 1) + b * c / (2 * c + 1) + c * a / (2 * a + 1)) ≤ 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_631; Apache-2.0

View graph