Prove2Me

A sum of squared linear ratios has an upper bound

Proved
WorkbookSource.base_3553

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

lean-workbooksource-checked

Prove that for all real positive numbers x x , y y and z z the following inequality is true: (2x+y+z)22x2+(y+z)2+(2y+z+x)22y2+(z+x)2+(2z+x+y)22z2+(x+y)2≤8 \frac{(2x+y+z)^2}{2x^2+(y+z)^2 }+ \frac{(2y+z+x)^2}{2y^2+(z+x)^2 }+\frac{(2z+x+y)^2}{2z^2+(x+y)^2 }\le 8

Source: InternLM Lean-Workbook, record lean_workbook_3553 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_3553 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (2*x + y + z)^2 / (2 * x^2 + (y + z)^2) + (2*y + z + x)^2 / (2 * y^2 + (z + x)^2) + (2*z + x + y)^2 / (2 * z^2 + (x + y)^2) ≤ 8  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_3553; Apache-2.0

View graph