A shifted three-variable quadratic ratio sum is at least three

Proved

WorkbookSource.base_44304

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: $\frac{d+a^2}{d+bc}+\frac{d+b^2}{d+ca}+\frac{d+c^2}{d+ab} \ge 3.$

Source: InternLM Lean-Workbook, record lean_workbook_44304 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_44304 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (d + a ^ 2) / (d + b * c) + (d + b ^ 2) / (d + c * a) + (d + c ^ 2) / (d + a * b) ≥ 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_44304; Apache-2.0

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