A sum of quadratic ratios has a constant lower bound

Proved

WorkbookSource.base_1111

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

lean-workbooksource-checked

Let $a,b,c$ positive reals. Prove that

$\frac {4a^2 + 9bc}{(b + c)^2} + \frac {4b^2 + 9ca}{(c + a)^2} + \frac {4c^2 + 9ab}{(a + b)^2}\geq\frac {39}{4}$

Source: InternLM Lean-Workbook, record lean_workbook_1111 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1111 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (4 * a ^ 2 + 9 * b * c) / (b + c) ^ 2 + (4 * b ^ 2 + 9 * c * a) / (c + a) ^ 2 + (4 * c ^ 2 + 9 * a * b) / (a + b) ^ 2 ≥ 39 / 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1111; Apache-2.0

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