A bound with a fixed sum

Proved

WorkbookSource.problem_52646

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $x,y\geq 1$ and $x+y= 3.$ Prove that $(y^2 + y+ 1)(x + 1) + (x^2-x- 1)(y- 1) \ge 9$

Source: InternLM Lean-Workbook, record lean_workbook_52646 (Apache-2.0). The complete source proposition and its explicit variable declarations are preserved.

Preamble

import Mathlib
open Real

Formal statement

theorem WorkbookSource.problem_52646 (x y : ℝ) (hx : x ≥ 1) (hy : y ≥ 1) (hxy : x + y = 3) : (y^2 + y + 1) * (x + 1) + (x^2 - x - 1) * (y - 1) ≥ 9  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_52646; Apache-2.0

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