A cyclic shifted-product ratio bound at fixed sum three

Proved

WorkbookSource.base_5863

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

lean-workbooksource-checked

For $a, b, c>0, a+b+c=3$ prove that $\frac{a(b+2)}{a(c+2)+b+2}+\frac{b(c+2)}{b(a+2)+c+2}+\frac{c(a+2)}{c(b+2)+a+2}\le\frac{a^2+b^2+c^2}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_5863 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_5863 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a * (b + 2) / (a * (c + 2) + b + 2) + b * (c + 2) / (b * (a + 2) + c + 2) + c * (a + 2) / (c * (b + 2) + a + 2)) ≤ (a ^ 2 + b ^ 2 + c ^ 2) / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_5863; Apache-2.0

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