A cyclic cubic maximum at fixed sum

Proved

WorkbookCorrected.base_12747

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

corrected-formalizationlean-workbooksource-checked

Prove that the maximum value of $x^2 y + y^2 z + z^2 x$ given $x+y+z = 9$ and $x,y,z$ are non-negative real numbers is 108

Source: InternLM Lean-Workbook, record lean_workbook_12747 (Apache-2.0). Natural-language proposition preserved; the source bound is completed with an exact attainment witness. Proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookCorrected.base_12747 : (∀ (x y z : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) (h : x + y + z = 9), x ^ 2 * y + y ^ 2 * z + z ^ 2 * x ≤ 108) ∧ (∃ x y z : ℝ, (0 ≤ x) ∧ (0 ≤ y) ∧ (0 ≤ z) ∧ (x + y + z = 9) ∧ ( x ^ 2 * y + y ^ 2 * z + z ^ 2 * x  =  108  )) := by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_12747; Apache-2.0

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