A shifted pair-product ratio upper bound at fixed sum three

Proved

WorkbookSource.plus_46949

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

lean-workbooksource-checked

Let $a, b, c>0, a+b+c=3$ . Prove that $\frac{ab}{2c^2+3(a+b)}+\frac{bc}{2a^2+3(b+c)}+\frac{ca}{2b^2+3(c+a)}\le\frac{ab+bc+ca}{8}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_46949 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_46949 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a * b / (2 * c ^ 2 + 3 * (a + b)) + b * c / (2 * a ^ 2 + 3 * (b + c)) + c * a / (2 * b ^ 2 + 3 * (c + a))) ≤ (a * b + b * c + c * a) / 8   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_46949; Apache-2.0

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