A quadratic rational sum bounded by the squared total

Proved

WorkbookSource.base_218

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

lean-workbooksource-checked

If $a,$ $b,$ $c$ are positive real numbers, then $\frac{a(b+c)}{a^2+2bc}+\frac{b(c+a)}{b^2+2ca}+\frac{c(a+b)}{c^2+2ab}+1 \le \frac{(a+b+c)^2}{ab+bc+ca}.$

Source: InternLM Lean-Workbook, record lean_workbook_218 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_218 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * (b + c) / (a ^ 2 + 2 * b * c) + b * (c + a) / (b ^ 2 + 2 * c * a) + c * (a + b) / (c ^ 2 + 2 * a * b) + 1) ≤ (a + b + c) ^ 2 / (a * b + b * c + a * c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_218; Apache-2.0

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