Prove2Me

A shifted squared pairwise reciprocal lower bound at fixed sum three

Proved
WorkbookSource.base_37690

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

lean-workbooksource-checked

If x,y,z>0 and x+y+z=3 prove that 14+(x+y)2+14+(y+z)2+14+(z+x)2≥38\frac{1}{4+(x+y)^2}+\frac{1}{4+(y+z)^2}+\frac{1}{4+(z+x)^2}\geq \frac{3}{8}

Source: InternLM Lean-Workbook, record lean_workbook_37690 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_37690 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (hx1 : x + y + z = 3) : 1 / (4 + (x + y) ^ 2) + 1 / (4 + (y + z) ^ 2) + 1 / (4 + (z + x) ^ 2) ≥ 3 / 8  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_37690; Apache-2.0

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