Prove2Me

A cyclic ratio sum bounds a symmetric quadratic expression

Proved
WorkbookSource.base_32381

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

lean-workbooksource-checked

Let a,b,c>0. Prove that ab+bc+ca≥12.a2+b2+c2ab+bc+ca+52\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \ge \frac{1}{2}.\frac{{{a^2} + {b^2} + {c^2}}}{{ab + bc + ca}} + \frac{5}{2} .

Source: InternLM Lean-Workbook, record lean_workbook_32381 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_32381 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / b + b / c + c / a) ≥ 1 / 2 * (a ^ 2 + b ^ 2 + c ^ 2) / (a * b + b * c + a * c) + 5 / 2  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_32381; Apache-2.0

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