Question 14
A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 \lt \dots \lt x_{2n}$, 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.
Determine all $n$ such that the magician can find a strategy to perform such a trick.
Determine all $n$ such that the magician can find a strategy to perform such a trick.