A quadratic reciprocal product bounds a linear reciprocal product

Proved

WorkbookSource.plus_27662

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

lean-workbooksource-checked

Let $x,y,z$ be positive real numbers. Prove that $(x^2 + y^2 + z^2) \big(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\big)\geq 5(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big)-\frac{73}{2}.$ p/5829942994

Source: InternLM Lean-Workbook, record lean_workbook_plus_27662 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_27662 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x^2 + y^2 + z^2) * (1 / x^2 + 1 / y^2 + 1 / z^2) ≥ 5 * (x + y + z) * (1 / x + 1 / y + 1 / z) - 73 / 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_27662; Apache-2.0

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