A quadratic reciprocal lower bound involving pairwise products

Proved

WorkbookSource.base_18343

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

lean-workbooksource-checked

If $x, y, z$ are positive real numbers, then prove that $\frac{1}{x^{2}+y^{2}}+\frac{1}{y^{2}+z^{2}}+\frac{1}{z^{2}+x^{2}}\ge \frac{5}{2(xy+yz+zx)}$

Source: InternLM Lean-Workbook, record lean_workbook_18343 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_18343 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : 1 / (x ^ 2 + y ^ 2) + 1 / (y ^ 2 + z ^ 2) + 1 / (z ^ 2 + x ^ 2) ≥ 5 / (2 * (x * y + y * z + z * x))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_18343; Apache-2.0

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