A cyclic rational sum bounds the normalized quadratic sum

Proved

WorkbookSource.base_12954

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

lean-workbooksource-checked

Let $a,b,c$ be postive real numbers. Prove that $\frac {a^2}{b} + \frac {b^2}{c} + \frac {c^2}{a} + 4\left(\frac {bc}{a} + \frac {ca}{b} + \frac {ab}{c}\right) + a + b + c \ge \frac {18(a^2 + b^2 + c^2)}{a + b + c}.$

Source: InternLM Lean-Workbook, record lean_workbook_12954 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_12954 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : a + b + c + 4 * (b * c / a + c * a / b + a * b / c) + a^2 / b + b^2 / c + c^2 / a ≥ 18 * (a^2 + b^2 + c^2) / (a + b + c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_12954; Apache-2.0

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