A shifted squared-product reciprocal bound at fixed sum three

Proved

WorkbookSource.base_7161

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

lean-workbooksource-checked

Let $a,b,c$ be positive numbers such that $a+b+c=3$ . Prove that

\frac{(1+a)^2(1+b)^2}{2+c^2} +\frac{(1+b)^2(1+c)^2}{2+a^2} +\frac{(1+c)^2(1+a)^2}{2+b^2} \geq 16

Source: InternLM Lean-Workbook, record lean_workbook_7161 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_7161 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (1 + a) ^ 2 * (1 + b) ^ 2 / (2 + c ^ 2) + (1 + b) ^ 2 * (1 + c) ^ 2 / (2 + a ^ 2) + (1 + c) ^ 2 * (1 + a) ^ 2 / (2 + b ^ 2) ≥ 16  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_7161; Apache-2.0

View graph