$P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{eqnarray*} Determine $n$.
Today I just stumbled upon this problem while I was reading a book. I spent about an hour on solving it but to no avail. On searching the internet I found that the problem is from USAMO 1984 test. I have tried using all the fundamental polynomial theory, use of binomial coefficients and some basic algebra. The internet did give me a solution but it used something named Lagrange interpolation formula or something which I don't have any idea of. Can someone please help me to work out this problem.
$\mathbf{Edit: }$ Some of you have posted very useful links. I liked the link
https://mks.mff.cuni.cz/kalva/usa/usoln/usol845.html.
But I could not understand what the answer means to say. Can someone please explain whatever the link tries to answer in a better way
