A quadratic sum times pairwise reciprocals bounds the total

Proved

WorkbookSource.base_48569

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

lean-workbooksource-checked

Prove that for $x, y, z > 0$ , $(x^2 + y^2 + z^2)\cdot \sum \frac 1{y + z} \ge \frac 32 \cdot (x + y + z)$

Source: InternLM Lean-Workbook, record lean_workbook_48569 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_48569 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x^2 + y^2 + z^2) * (1/(y + z) + 1/(x + z) + 1/(x + y)) ≥ 3/2 * (x + y + z)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_48569; Apache-2.0

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