Part 2’s link https://nttuan.org/2016/11/13/topic-832/
Problem 9. Let and be fixed integers each at least . Find the largest positive integer for which there exists a polynomial , of degree and with rational coefficients, such that the following property holds: exactly one of is an integer for each .
Problem 10. Let be a polynomial. Prove that we have an .
Problem 11. Let be the Fibonacci sequence and the polynomial of degree satisfying
Problem 12. Let be a polynomial of degree with real coefficients and let . Prove that
Problem 13. Let be a polynomial of degree with integral coefficients such that for every there is an integer with . Furthermore, it is given that . Prove that for every integer there is an integer such that
Problem 14. Given is a natural number . What is the smallest possible value of if the following statements are true.
For every points on a plane, where no three points are collinear, and for any real numbers there exists such polynomial , the degree of which is no more than , where for every . (The degree of a nonzero monomial is , while the degree of polynomial is the greatest degree of the degrees of its monomials.)
Problem 15. Find all such that .
Problem 16. Is it true that for any two increasing sequences and we can find a strictly increasing polynomial s.t. for ?