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

Proved

WorkbookSource.base_51867

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

lean-workbooksource-checked

Prove that $\frac {ab}{7 + 2c^2} + \frac {bc}{7 + 2a^2} + \frac {ca}{7 + 2b^2} \le \frac {1}{3}$ for all $a,b,c > 0: a + b + c = 3$

Source: InternLM Lean-Workbook, record lean_workbook_51867 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_51867 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a * b) / (7 + 2 * c ^ 2) + (b * c) / (7 + 2 * a ^ 2) + (c * a) / (7 + 2 * b ^ 2) ≤ 1 / 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_51867; Apache-2.0

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