A cyclic quadratic-product ratio sum is bounded above

Proved

WorkbookSource.base_7010

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

lean-workbooksource-checked

With $x>0, y>0, z>0$ . Prove that: $\frac{xy}{x^2+yz+zx}+\frac{yz}{y^2+zx+xy}+\frac{zx}{z^2+xy+yz} \leq \frac{x^2+y^2+z^2}{xy+yz+zx}$ .

Source: InternLM Lean-Workbook, record lean_workbook_7010 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_7010 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x * y / (x ^ 2 + y * z + z * x) + y * z / (y ^ 2 + z * x + x * y) + z * x / (z ^ 2 + x * y + y * z)) ≤ (x ^ 2 + y ^ 2 + z ^ 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_7010; Apache-2.0

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