A cyclic linear-over-quadratic upper bound

Proved

WorkbookSource.base_9688

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers, prove that: $\frac{a+b+c}{ab+bc+ac}\geq \frac{a}{a^2+bc+b^2}+\frac{b}{b^2+ca+c^2}+\frac{c}{c^2+ab+a^2}$

Source: InternLM Lean-Workbook, record lean_workbook_9688 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_9688 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : (a + b + c) / (a * b + b * c + a * c) ≥ a / (a ^ 2 + b * c + b ^ 2) + b / (b ^ 2 + c * a + c ^ 2) + c / (c ^ 2 + a * b + a ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_9688; Apache-2.0

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