Prove2Me

A mixed squared ratio sum bounds a normalized quadratic expression

Proved
WorkbookSource.plus_58995

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

lean-workbooksource-checked

For a,b,ca,b,c positive reals prove that

4∑a2+bc(b+c)2≥3+(a+b+c)2ab+bc+ca\displaystyle 4\sum\frac{a^2+bc}{(b+c)^2} \ge 3+\frac{(a+b+c)^2}{ab+bc+ca}

Source: InternLM Lean-Workbook, record lean_workbook_plus_58995 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.plus_58995 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : 4 * ((a^2 + b * c) / (b + c) ^ 2 + (b^2 + c * a) / (c + a) ^ 2 + (c^2 + a * b) / (a + b) ^ 2) ≥ 3 + (a + b + c) ^ 2 / (a * b + b * c + c * a)   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_58995; Apache-2.0

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