A shifted cyclic quadratic ratio sum is at least three

Proved

WorkbookSource.plus_16317

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

lean-workbooksource-checked

Let $a, b, c$ be positive real numbers satisfying $a+b+c=3$ . Prove that: $\frac{{{a^2} + b}}{{{b^2} + a}} + \frac{{{b^2} + c}}{{{c^2} + b}} + \frac{{{c^2} + a}}{{{a^2} + c}} \ge 3.$

Source: InternLM Lean-Workbook, record lean_workbook_plus_16317 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_16317 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a^2 + b)/(b^2 + a) + (b^2 + c)/(c^2 + b) + (c^2 + a)/(a^2 + c) ≥ 3   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_16317; Apache-2.0

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