Prove2Me

A shifted quadratic ratio upper bound at fixed sum three

Proved
WorkbookSource.base_22608

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

lean-workbooksource-checked

Let a,b,ca,b,c be positive real numbers such that a+b+c=3a+b+c=3 . Prove that a2+92a2+(b+c)2+b2+92b2+(c+a)2+c2+92c2+(a+b)2≤5.\frac{a^2+9}{2a^2+(b+c)^2}+\frac{b^2+9}{2b^2+(c+a)^2}+\frac{c^2+9}{2c^2+(a+b)^2}\leq 5.

Source: InternLM Lean-Workbook, record lean_workbook_22608 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_22608 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (a^2 + 9) / (2 * a^2 + (b + c)^2) + (b^2 + 9) / (2 * b^2 + (c + a)^2) + (c^2 + 9) / (2 * c^2 + (a + b)^2) ≤ 5  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_22608; Apache-2.0

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