A cyclic quadratic difference ratio sum is at least three halves

Proved

WorkbookSource.base_32836

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

lean-workbooksource-checked

Prove the inequality

$\frac{b^2+c^2-a^2}{a\left(b+c\right)}+\frac{c^2+a^2-b^2}{b\left(c+a\right)}+\frac{a^2+b^2-c^2}{c\left(a+b\right)}\geq\frac32$

for any three positive reals $a$ , $b$ , $c$ .

Comment. This was an attempt of creating a contrast to the (rather hard) inequality at the QEDMO before. However, it turned out to be more difficult than I expected (a wrong solution was presented during the competition).

Source: InternLM Lean-Workbook, record lean_workbook_32836 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_32836 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (b^2 + c^2 - a^2) / (a * (b + c)) + (c^2 + a^2 - b^2) / (b * (c + a)) + (a^2 + b^2 - c^2) / (c * (a + b)) ≥ 3 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_32836; Apache-2.0

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