A shifted cyclic quadratic ratio product is at least one

Proved

WorkbookSource.base_56185

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

lean-workbooksource-checked

Let $a,b,c>0$ and $a+b+c=3 .$ Prove that ${\left( \frac{a^2}{b+c}+\frac{1}{2} \right)\left( \frac{b^2}{c+a}+\frac{1}{2} \right)\left( \frac{c^2}{a+b}+\frac{1}{2} \right)}\geq 1$

Source: InternLM Lean-Workbook, record lean_workbook_56185 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_56185 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (a^2 / (b + c) + 1 / 2) * (b^2 / (c + a) + 1 / 2) * (c^2 / (a + b) + 1 / 2) ≥ 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_56185; Apache-2.0

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