Lagrange’s interpolation polynomial (3)


Part 2’s link https://nttuan.org/2016/11/13/topic-832/


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.

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