Prove2Me

A quadratic reciprocal sum bounds a symmetric rational expression

Proved
WorkbookSource.base_41556

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

lean-workbooksource-checked

x,y,z>0x,y,z>0 ,prove xy2+z2+yx2+z2+zx2+y2≥3 x+y+zx2+y2+z2+xy+yz+zx{\frac {x}{{y}^{2}+{z}^{2}}}+{\frac {y}{{x}^{2}+{z}^{2}}}+{\frac {z}{{x}^{2}+{y}^{2}}}\geq 3\,{\frac {x+y+z}{{x}^{2}+{y}^{2}+{z}^{2}+xy+yz+zx}}

Source: InternLM Lean-Workbook, record lean_workbook_41556 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_41556 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : x / (y ^ 2 + z ^ 2) + y / (x ^ 2 + z ^ 2) + z / (x ^ 2 + y ^ 2) ≥ 3 * (x + 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_41556; Apache-2.0

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