A quadratic sum times three shifted reciprocals bounds twice a triple sum

Proved

WorkbookSource.base_4672

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

lean-workbooksource-checked

Let $a,b,c,d$ be positive real numbers . Prove that $(a^2+b^2+c^2 +d^2 )\left(\frac{1}{a+d}+\frac{1}{b+d}+\frac{1}{c+d} \right) \geq 2(a+b+c).$

Source: InternLM Lean-Workbook, record lean_workbook_4672 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_4672 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :  (a^2 + b^2 + c^2 + d^2) * (1 / (a + d) + 1 / (b + d) + 1 / (c + d)) ≥ 2 * (a + b + c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_4672; Apache-2.0

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