A cyclic ratio sum bounds a weighted complementary sum

Proved

WorkbookSource.base_51901

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

lean-workbooksource-checked

Let $a,b,c$ be positive reals, show that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac92 \ge \frac{5(b+c)}{b+c+2a}+\frac{5(c+a)}{c+a+2b}+\frac{5(a+b)}{a+b+2c}$

Source: InternLM Lean-Workbook, record lean_workbook_51901 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_51901 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / b + b / c + c / a + 9 / 2) ≥ 5 * (b + c) / (b + c + 2 * a) + 5 * (c + a) / (c + a + 2 * b) + 5 * (a + b) / (a + b + 2 * c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_51901; Apache-2.0

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