A mixed quadratic ratio lower bound at fixed sum three

Proved

WorkbookSource.base_50327

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: $\frac{a^{2}+bc}{a+bc}+\frac{b^{2}+ac}{b+ac}+\frac{c^{2}+ab}{c+ab}\geq 3$

Source: InternLM Lean-Workbook, record lean_workbook_50327 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_50327 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a^2 + b * c) / (a + b * c) + (b^2 + c * a) / (b + c * a) + (c^2 + a * b) / (c + a * b) ≥ 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_50327; Apache-2.0

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