A squared total times reciprocal pairwise squares is at least ten

Proved

WorkbookSource.base_1996

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

lean-workbooksource-checked

Prove or find the minimum of LHS: $(x+y+z)^2(\frac 1{x^2+y^2}+\frac 1{y^2+z^2}+\frac 1{z^2+x^2}) \geq 10$ where $x, y, z > 0$ .

Source: InternLM Lean-Workbook, record lean_workbook_1996 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1996 (x y z : ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (x + y + z) ^ 2 * (1 / (x ^ 2 + y ^ 2) + 1 / (y ^ 2 + z ^ 2) + 1 / (z ^ 2 + x ^ 2)) ≥ 10  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1996; Apache-2.0

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