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Omni-MATH / A magician intends to perform the following trick. She announces a positive…

Problem

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problem

A magician intends to perform the following trick. She announces a positive integer nn, along with 2n2n real numbers x1<…<x2nx_1 < \dots < x_{2n}, to the audience. A member of the audience then secretly chooses a polynomial P(x)P(x) of degree nn with real coefficients, computes the 2n2n values P(x1),…,P(x2n)P(x_1), \dots , P(x_{2n}), and writes down these 2n2n values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?
Plain-text mathematical notation (without MathML)
A magician intends to perform the following trick. She announces a positive integer n, along with 2n real numbers x₁<…<x_(2n), to the audience. A member of the audience then secretly chooses a polynomial P(x) of degree n with real coefficients, computes the 2n values P(x₁),…,P(x_(2n)), and writes down these 2n values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?
Original LaTeX notation
A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?

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Source and history

task_id
f20619d0-a6fd-52ba-8154-b42b83dd3842
task_key
test--f20619d0-a6fd-52ba-8154-b42b83dd3842
task_revision_id
1
config / split
/ test

Official source

initial import