A shifted mixed quadratic ratio upper bound at fixed sum three

Proved

WorkbookSource.base_34270

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

lean-workbooksource-checked

Let $a,b,c$ be positive reals such that $a+b+c=3.$ Prove that $\frac{a}{a^2+bc+1}+\frac{b}{b^2+ca+1}+\frac{c}{c^2+ab+1} \le 1$

Source: InternLM Lean-Workbook, record lean_workbook_34270 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_34270 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : a / (a ^ 2 + b * c + 1) + b / (b ^ 2 + c * a + 1) + c / (c ^ 2 + a * b + 1) ≤ 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_34270; Apache-2.0

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