Petya and Vasya came up with ten polynomials of the fifth degree. Then Vasya sequentially called out natural numbers (starting from some number), and Petya substituted each called number into one of the polynomials of his choice and wrote down the obtained values on the board from left to right. It turned out that the numbers written on the board form an arithmetic progression (in this exact order). What is the maximum number of numbers Vasya could have called?
Evaluation. Since a fifth-degree equation has no more than five roots, it is similarly proven as in problem $\underline{64349}$ that the total number of numbers could not have been more than 50.

Example. Let the polynomials $P_{k}(x)=x+(x-5 k+4)(x-5 k+3) \ldots(x-5 k)$ for $k=1,2, \ldots, 10$ and

Vasya called out the numbers $1,2, \ldots, 50$. Since

$P_{k}(5 k-i)=5 k-i$ for $i=0,1, \ldots, 4$, Petya could have sequentially obtained the numbers $1,2, \ldots, 50$.

## Answer

50 numbers.