A comparison of cyclic weighted reciprocal sums

Proved

WorkbookSource.base_4630

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers. Prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{5}{3a+b}+\frac{5}{3b+c}+\frac{5}{3c+a}\ge \frac{9}{a+3b}+\frac{9}{b+3c}+\frac{9}{c+3a}.$

Source: InternLM Lean-Workbook, record lean_workbook_4630 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_4630 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 1 / a + 1 / b + 1 / c + 5 / (3 * a + b) + 5 / (3 * b + c) + 5 / (3 * c + a) ≥ 9 / (a + 3 * b) + 9 / (b + 3 * c) + 9 / (c + 3 * a)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_4630; Apache-2.0

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