Lagrange’s interpolation polynomial (3)

Part 2’s link

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<a_2<\cdots<a_n and b_1<b_2<\cdots<b_n 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 ?


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