A comparison of two cyclic linear reciprocal sums

Proved

WorkbookSource.base_15339

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

lean-workbooksource-checked

Given $x,y,z>0$ , prove that $\frac{1}{3x+y}+\frac{1}{3y+z}+\frac{1}{3z+x}\ge\frac{1}{2x+y+z}+\frac{1}{2y+z+x}+\frac{1}{2z+x+y}$ .

Source: InternLM Lean-Workbook, record lean_workbook_15339 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_15339 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (1 / (3 * x + y) + 1 / (3 * y + z) + 1 / (3 * z + x)) ≥ (1 / (2 * x + y + z) + 1 / (2 * y + z + x) + 1 / (2 * z + x + y))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_15339; Apache-2.0

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