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MATH / Find the number of positive integers n≥3 that have the following property: If x₁, x₂, …, x_(n) are real numbers such that x₁+x₂+…+x_(n)=0, then x₁x₂+x₂x₃+…+x_(n−1)x_(n)+x_(n)x₁≤0.

Problem

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problem

Find the number of positive integers n≥3n \ge 3 that have the following property: If x1,x_1, x2,x_2, …,\dots, xnx_n are real numbers such that x1+x2+…+xn=0,x_1 + x_2 + \dots + x_n = 0, then x1x2+x2x3+…+xn−1xn+xnx1≤0.x_1 x_2 + x_2 x_3 + \dots + x_{n - 1} x_n + x_n x_1 \le 0.
Plain-text mathematical notation (without MathML)
Find the number of positive integers n≥3 that have the following property: If x₁, x₂, …, x_(n) are real numbers such that x₁+x₂+…+x_(n)=0, then
x₁x₂+x₂x₃+…+x_(n−1)x_(n)+x_(n)x₁≤0.
Original LaTeX notation
Find the number of positive integers $n \ge 3$ that have the following property: If $x_1,$ $x_2,$ $\dots,$ $x_n$ are real numbers such that $x_1 + x_2 + \dots + x_n = 0,$ then
\[x_1 x_2 + x_2 x_3 + \dots + x_{n - 1} x_n + x_n x_1 \le 0.\]

level

Level 4

type

Intermediate Algebra

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Source and history

task_id
7cbab5c3-958c-53eb-8b1b-ef7c15a34913
task_key
intermediate~5falgebra--test--7cbab5c3-958c-53eb-8b1b-ef7c15a34913
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import