A comparison of cyclic and weighted pairwise ratio sums

Proved

WorkbookSource.base_25958

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

lean-workbooksource-checked

Given $a,b,c>0$ . Prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{a}{b+2c}+\frac{b}{c+2a}+\frac{c}{a+2b}+2$

Source: InternLM Lean-Workbook, record lean_workbook_25958 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_25958 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / b + b / c + c / a) ≥ (a / (b + 2 * c) + b / (c + 2 * a) + c / (a + 2 * b)) + 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_25958; Apache-2.0

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