A pairwise-product maximum at fixed sum

Proved

WorkbookCorrected.base_10057

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

corrected-formalizationlean-workbooksource-checked

If $x,y,z>0$ and $x+y+z=9\;,$ Then maximum value of $xy+yz+zx$

Source: InternLM Lean-Workbook, record lean_workbook_10057 (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_10057 : (∀ (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (h : x + y + z = 9), x * y + y * z + z * x ≤ 27) ∧ (∃ x y z : ℝ, (0 < x) ∧ (0 < y) ∧ (0 < z) ∧ (x + y + z = 9) ∧ ( x * y + y * z + z * x  =  27  )) := by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_10057; Apache-2.0

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