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

Prove that

**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 ?

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