A weighted quadratic ratio sum is at most five

Proved

WorkbookSource.plus_74571

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

lean-workbooksource-checked

Given $a, b, c$ are positive numbers, prove that: $\frac{a^2+9bc}{2a^2+(b+c)^2}+\frac{b^2+9ca}{2b^2+(c+a)^2}+\frac{c^2+9ab}{2c^2+(a+b)^2}\leq 5$

Source: InternLM Lean-Workbook, record lean_workbook_plus_74571 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_74571 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + 9 * b * c) / (2 * a^2 + (b + c)^2) + (b^2 + 9 * c * a) / (2 * b^2 + (c + a)^2) + (c^2 + 9 * a * b) / (2 * c^2 + (a + b)^2) ≤ 5   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_74571; Apache-2.0

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