Prove2Me

A weighted quadratic reciprocal upper bound

Proved
WorkbookSource.base_29386

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

lean-workbooksource-checked

Let a,b,ca,b,c be three positive real numbers . Prove that:

a4a2+b2+(b+c)2+b4b2+c2+(c+a)2+c4c2+a2+(a+b)2≤1a+b+c.\frac{a}{4a^2+b^2+(b+c)^2}+\frac{b}{4b^2+c^2+(c+a)^2}+\frac{c}{4c^2+a^2+(a+b)^2} \le \frac{1}{a+b+c}.

Source: InternLM Lean-Workbook, record lean_workbook_29386 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_29386 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : (a / (4 * a ^ 2 + b ^ 2 + (b + c) ^ 2) + b / (4 * b ^ 2 + c ^ 2 + (c + a) ^ 2) + c / (4 * c ^ 2 + a ^ 2 + (a + b) ^ 2)) ≤ 1 / (a + b + c)  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_29386; Apache-2.0

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