A cyclic pairwise ratio sum is bounded by a reciprocal product

Proved

WorkbookSource.plus_31208

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

lean-workbooksource-checked

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

Source: InternLM Lean-Workbook, record lean_workbook_plus_31208 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_31208 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (a + b) / (b + c) + (b + c) / (c + a) + (c + a) / (a + b) ≤ 3 / (a * b * c)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_31208; Apache-2.0

View graph