## Lagrange’s interpolation polynomial (3)

Problem 9. Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of  $\displaystyle\frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}}$  is an integer for each $k = 0,1, ..., m$.
Problem 10. Let $f\left ( x \right )=x^{n}+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+...+a_{1}x+a_{0}$ be a polynomial. Prove that we have an $\displaystyle i\in \left \{ 1,2,...,n \right \}\mid \left | f\left ( i \right ) \right |\geq \frac{n!}{\binom{n}{i}}$.
Problem 11.  Let $(F_n)_{n\geq 1}$ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying
$P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.$ Prove that $P(1983) = F_{1983} - 1.$