A shifted quadratic ratio sum is at least nine at fixed sum three

Proved

WorkbookSource.plus_62905

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

lean-workbooksource-checked

Let $a,b,c$ be poistive reals such that $a + b + c = 3$ . Prove that

$\frac {2a^2 + 3a + 1}{b^2 + 1} + \frac {2b^2 + 3b + 1}{c^2 + 1} + \frac {2c^2 + 3c + 1}{a^2 + 1}\geq9$

Source: InternLM Lean-Workbook, record lean_workbook_plus_62905 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_62905 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (2 * a ^ 2 + 3 * a + 1) / (b ^ 2 + 1) + (2 * b ^ 2 + 3 * b + 1) / (c ^ 2 + 1) + (2 * c ^ 2 + 3 * c + 1) / (a ^ 2 + 1) ≥ 9   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_62905; Apache-2.0

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