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

Proved

WorkbookSource.base_33663

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

lean-workbooksource-checked

For positives $a$ , $b$ and $c$ such that $a+b+c=3$ prove that: $\frac{a}{4a+5bc}+\frac{b}{4b+5ca}+\frac{c}{4c+5ab}\le\frac{a^2+b^2+c^2}{9}$

Source: InternLM Lean-Workbook, record lean_workbook_33663 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_33663 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : a / (4 * a + 5 * b * c) + b / (4 * b + 5 * c * a) + c / (4 * c + 5 * a * b) ≤ (a ^ 2 + b ^ 2 + c ^ 2) / 9  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_33663; Apache-2.0

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