A weighted quadratic pair-product ratio sum is at most one half

Proved

WorkbookSource.plus_73163

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

lean-workbooksource-checked

Prove that if $x,y,z>0$ then $\frac{xy}{3x^2+2y^2+z^2}+\frac{yz}{3y^2+2z^2+x^2}+\frac{zx}{3z^2+2x^2+y^2} \leq \frac{1}{2}.$

Source: InternLM Lean-Workbook, record lean_workbook_plus_73163 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_73163 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x * y / (3 * x ^ 2 + 2 * y ^ 2 + z ^ 2) + y * z / (3 * y ^ 2 + 2 * z ^ 2 + x ^ 2) + z * x / (3 * z ^ 2 + 2 * x ^ 2 + y ^ 2)) ≤ 1 / 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_73163; Apache-2.0

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