A shifted reciprocal sum bounds a symmetric reciprocal

Proved

WorkbookSource.plus_57819

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

lean-workbooksource-checked

Prove that: $\frac1{x+yz}+\frac1{y+zx}+\frac1{z+xy}\leq\frac{9}{2(xy+xz+yz)}$ where $x, y, z$ are positive real numbers such that $x+y+z=3$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_57819 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_57819 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (hx1 : x + y + z = 3) : 1 / (x + y * z) + 1 / (y + z * x) + 1 / (z + x * y) ≤ 9 / (2 * (x * y + x * z + y * z))   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_57819; Apache-2.0

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