# 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 $P(1983) = F_{1983} - 1.$

Problem 12.  Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that

$\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1.$

Problem 13. Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$

Problem 14. Given is a natural number $n \geq 3$. What is the smallest possible value of $k$ if the following statements are true.

For every $n$ points $A_i = (x_i, y_i)$ on a plane, where no three points are collinear, and for any real numbers $c_i (1 \le i \le n)$ there exists such polynomial $P(x, y)$, the degree of which is no more than $k$, where $P(x_i, y_i) = c_i$ for every $i = 1, . . . , n$. (The degree of a nonzero monomial $a_{i,j} x^{i}y^{j}$ is $i+j$, while the degree of polynomial $P(x, y)$ is the greatest degree of the degrees of its monomials.)

Problem 15. Find all $P(x)\in\mathbb{R}[x]$ such that $\forall r\in\mathbb{Q}\, \exists x\in\mathbb{Q}:\, P(x)=r$.

Problem 16. Is it true that for any two increasing sequences $a_1 and $b_1 we can find a strictly increasing polynomial $P[X] \in \mathbb{R}[X]$ s.t. $P(a_i)=b_i$ for $i=1,2,\cdots,n$ ?