A pairwise reciprocal sum bounds a normalized cubic reciprocal

Proved

WorkbookSource.plus_39846

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

lean-workbooksource-checked

Let $x,y,z$ be positive real numbers. Prove that

\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x} \geq \frac{(x+y+z)^2}{x^3+y^3+z^3+3xyz}

Source: InternLM Lean-Workbook, record lean_workbook_plus_39846 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_39846 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) ≥ (x + y + z) ^ 2 / (x ^ 3 + y ^ 3 + z ^ 3 + 3 * x * y * z)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_39846; Apache-2.0

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