A shifted quadratic ratio upper bound at fixed sum three

Proved

WorkbookSource.base_30408

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

lean-workbooksource-checked

For $a, b, c \in \mathbb{R}_{>0}$ such that $a + b + c = 3$ , prove that $\dfrac{a}{a^2+2} + \dfrac{b}{b^2+2} + \dfrac{c}{c^2+2} \le 1$ .

Source: InternLM Lean-Workbook, record lean_workbook_30408 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_30408 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : a / (a^2 + 2) + b / (b^2 + 2) + c / (c^2 + 2) ≤ 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_30408; Apache-2.0

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