A quadratic reciprocal sum with a product correction at fixed sum three

Proved

WorkbookSource.plus_78775

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

lean-workbooksource-checked

Let a,b,c>0 such that $a + b + c = 3$ . Prove that $\frac{{{a^2}}}{b} + \frac{{{b^2}}}{c} + \frac{{{c^2}}}{a} + \frac{{9abc}}{{ab + bc + ca}} \ge 6$

Source: InternLM Lean-Workbook, record lean_workbook_plus_78775 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_78775 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (habc : a + b + c = 3) : a^2 / b + b^2 / c + c^2 / a + 9 * a * b * c / (a * b + b * c + c * a) ≥ 6   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_78775; Apache-2.0

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