A product of reverse cyclic quadratic ratio sums is at least nine

Proved

WorkbookSource.plus_13160

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

lean-workbooksource-checked

prove $(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a})(\frac{a^2}{c}+\frac{b^2}{a}+\frac{c^2}{b}) \geq 9$ given $a,b,c >0$ and $a+b+c=3$

Source: InternLM Lean-Workbook, record lean_workbook_plus_13160 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_13160 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a^2 / b + b^2 / c + c^2 / a) * (a^2 / c + b^2 / a + c^2 / b) ≥ 9   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_13160; Apache-2.0

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