A shifted cyclic quadratic ratio lower bound at fixed sum three

Proved

WorkbookSource.base_25410

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

lean-workbooksource-checked

Let $a,b,c$ be positive reals number and $a+b+c=3.$ ，Prove that

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

Source: InternLM Lean-Workbook, record lean_workbook_25410 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_25410 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (a^2 + b)/(a^2 + 1) + (b^2 + c)/(b^2 + 1) + (c^2 + a)/(c^2 + 1) ≥ 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_25410; Apache-2.0

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