A shifted mixed cyclic ratio sum is at least three

Proved

WorkbookSource.base_50181

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

lean-workbooksource-checked

If $a$ , $b$ , $c$ are positive reals, prove that

\frac{a+bc}{a+a^2}+\frac{b+ca}{b+b^2}+\frac{c+ab}{c+c^2} \geq 3

Source: InternLM Lean-Workbook, record lean_workbook_50181 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_50181 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b * c) / (a + a^2) + (b + c * a) / (b + b^2) + (c + a * b) / (c + c^2) ≥ 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_50181; Apache-2.0

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