A cyclic quadratic ratio sum bounded by a symmetric ratio

Proved

WorkbookSource.base_20232

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

lean-workbooksource-checked

Given $a,b,c>0$ .Prove that: $\frac{5}{4}. \frac{a^2+b^2+c^2}{ab+bc+ca} \ge \frac{a^2}{a^2+ab+bc}+ \frac{b^2}{b^2+bc+ca}+ \frac{c^2}{c^2+ca+ab}+ \frac{1}{4}$

Source: InternLM Lean-Workbook, record lean_workbook_20232 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_20232 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (5 / 4) * (a ^ 2 + b ^ 2 + c ^ 2) / (a * b + b * c + a * c) ≥ a ^ 2 / (a ^ 2 + a * b + b * c) + b ^ 2 / (b ^ 2 + b * c + a * c) + c ^ 2 / (c ^ 2 + a * c + a * b) + 1 / 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_20232; Apache-2.0

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