A shifted cyclic ratio sum is bounded by the reciprocal product

Proved

WorkbookSource.base_53071

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers such that $a+b+c=3$ . Prove that $\frac{a}{2b+1}+\frac{b}{2c+1}+\frac{c}{2a+1} \le \frac{1}{abc}$

Source: InternLM Lean-Workbook, record lean_workbook_53071 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_53071 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : a / (2 * b + 1) + b / (2 * c + 1) + c / (2 * a + 1) ≤ 1 / (a * b * c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_53071; Apache-2.0

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