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OlympiadBench / 1810 / An integer n≥3 is given. We call an n-tuple of real numbers…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

An integer n≥3n \geqslant 3 is given. We call an nn-tuple of real numbers (x1,x2,…,xn)\left(x_{1}, x_{2}, \ldots, x_{n}\right) Shiny if for each permutation y1,y2,…,yny_{1}, y_{2}, \ldots, y_{n} of these numbers we have ∑i=1n−1yiyi+1=y1y2+y2y3+y3y4+⋯+yn−1yn≥−1 \sum_{i=1}^{n-1} y_{i} y_{i+1}=y_{1} y_{2}+y_{2} y_{3}+y_{3} y_{4}+\cdots+y_{n-1} y_{n} \geqslant-1 Find the largest constant K=K(n)K=K(n) such that ∑1≤i<j≤nxixj≥K \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j} \geqslant K holds for every Shiny nn-tuple (x1,x2,…,xn)\left(x_{1}, x_{2}, \ldots, x_{n}\right).
Plain-text mathematical notation (without MathML)
An integer n≥3 is given. We call an n-tuple of real numbers (x₁,x₂,…,x_(n)) Shiny if for each permutation y₁,y₂,…,y_(n) of these numbers we have

∑_(i=1)^(n−1)y_(i)y_(i+1)=y₁y₂+y₂y₃+y₃y₄+⋯+y_(n−1)y_(n)≥−1

Find the largest constant K=K(n) such that

∑_(1≤i<j≤n)x_(i)x_(j)≥K

holds for every Shiny n-tuple (x₁,x₂,…,x_(n)).
Original LaTeX notation
An integer $n \geqslant 3$ is given. We call an $n$-tuple of real numbers $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ Shiny if for each permutation $y_{1}, y_{2}, \ldots, y_{n}$ of these numbers we have

$$
\sum_{i=1}^{n-1} y_{i} y_{i+1}=y_{1} y_{2}+y_{2} y_{3}+y_{3} y_{4}+\cdots+y_{n-1} y_{n} \geqslant-1
$$

Find the largest constant $K=K(n)$ such that

$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j} \geqslant K
$$

holds for every Shiny $n$-tuple $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$.

answer type

Expression

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID
1810
task_id
adacba10-89d7-5c30-9798-ace5d9175922
task_key
OE~5fTO~5fmaths~5fen~5fCOMP--train--1810
task_revision_id
3
config / split
OE_TO_maths_en_COMP / train

Official source

initial import