A cyclic pair-product reciprocal sum bounds the quadratic sum at fixed total four

Proved

WorkbookSource.base_24784

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

lean-workbooksource-checked

Let $a,b,c,d$ be positive numbers sach that $a+b+c+d=4$ . Prove that $\frac 1{ab}+\frac 1{bc}+\frac 1{cd}+\frac 1{da}\ge a^2+b^2+c^2+d^2$ .

Source: InternLM Lean-Workbook, record lean_workbook_24784 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_24784 (a b c d : ℝ) (h : a + b + c + d = 4) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : 1 / (a * b) + 1 / (b * c) + 1 / (c * d) + 1 / (d * a) ≥ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_24784; Apache-2.0

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