A squared ratio sum bounds a symmetric quadratic ratio

Proved

WorkbookSource.base_15999

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

lean-workbooksource-checked

Prove that $(\frac{a}{b+c})^2+(\frac{b}{c+a})^2+(\frac{c}{a+b})^2+3\geq \frac{5(a+b+c)^2}{4(ab+bc+ca)}$ for $a,b,c>0$ .

Source: InternLM Lean-Workbook, record lean_workbook_15999 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_15999 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (b + c)) ^ 2 + (b / (c + a)) ^ 2 + (c / (a + b)) ^ 2 + 3 ≥ 5 * (a + b + c) ^ 2 / (4 * (a * b + b * c + a * c))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_15999; Apache-2.0

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