A pairwise ratio sum bounds two symmetric quadratic expressions

Proved

WorkbookSource.base_39011

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

lean-workbooksource-checked

If $a,b,c$ are positive real numbers, then $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\ge{\frac{18}{5}+\frac{18}{5}\frac{a^2+b^2+c^2}{(a+b+c)^2}+\frac{2}{5}\frac{(a+b+c)^2}{ab+bc+ca}}$

Source: InternLM Lean-Workbook, record lean_workbook_39011 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_39011 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b) / c + (b + c) / a + (c + a) / b ≥ 18 / 5 + 18 / 5 * (a ^ 2 + b ^ 2 + c ^ 2) / (a + b + c) ^ 2 + 2 / 5 * (a + b + c) ^ 2 / (a * b + b * c + c * a)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_39011; Apache-2.0

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