A pair-product rational upper bound at fixed sum three

Proved

WorkbookSource.base_23786

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers such that $a+b+c=3$ . Prove that: $\frac{27}{2}\ge 4(ab+bc+ca)+\frac{a^{2}b^{2}}{a+b}+\frac{b^{2}c^{2}}{b+c}+\frac{c^{2}a^{2}}{a+c}$

Source: InternLM Lean-Workbook, record lean_workbook_23786 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_23786 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : 27 / 2 ≥ 4 * (a * b + b * c + c * a) + a ^ 2 * b ^ 2 / (a + b) + b ^ 2 * c ^ 2 / (b + c) + c ^ 2 * a ^ 2 / (a + c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_23786; Apache-2.0

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