A product of cyclic ratio sums is at least twenty-seven

Proved

WorkbookSource.plus_45883

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

lean-workbooksource-checked

Let $a,b,c>0.$ Prove that

\left(\frac{c+a}{b}+\frac{2c}{a+b}\right)\left(\frac{a+b}{c}+\frac{2a}{b+c}\right)\left(\frac{b+c}{a}+\frac{2b}{c+a}\right) \geq27

Source: InternLM Lean-Workbook, record lean_workbook_plus_45883 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_45883 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : ((c + a) / b + 2 * c / (a + b)) * ((a + b) / c + 2 * a / (b + c)) * ((b + c) / a + 2 * b / (c + a)) ≥ 27   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_45883; Apache-2.0

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