A cyclic ratio sum with a symmetric quadratic correction

Proved

WorkbookSource.base_18914

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

lean-workbooksource-checked

If $a,b,c>0$ prove or disprove that: $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+2\frac{bc+ca+ab}{a^2+b^2+c^2}\ge 5$

Source: InternLM Lean-Workbook, record lean_workbook_18914 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_18914 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / b + b / c + c / a + (2 * (b * c + c * a + a * b)) / (a ^ 2 + b ^ 2 + c ^ 2)) ≥ 5  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_18914; Apache-2.0

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