Part 3’s link https://nttuan.org/2016/11/14/topic-833/

**Problem 17.** Prove that, for infinitely many positive integers , there exists a polynomial of degree with real coefficients such that are different whole powers of .

**Problem 18.** Suppose is an infinite sequence of integers satisfying the following two conditions:

(i) divides for

(ii) there is a polynomial such that for all

Prove that there is a polynomial such that for all .

**Problem 19.** Let such that for infty of integer number : Equation has more than one integer root. Prove that , where and is a polynomial.

**Problem 20.** Find all the polynomials with odd degree such that

**Problem 21.** Suppose is a polynomial with integer coefficients assumes at distinct integral values of that are different form and in absolute value less than Prove that is irreducible.

Prove that the bound may be replaced by where is minimum distance between any two of the integral values of where assumes the integral values considered.

**Problem 22.** Six members of the team of Fatalia for the IMO are selected from candidates. At the TST the candidates got points with if .

The team leader has already candidates and now wants to see them and nobody other in the team. With that end in view he constructs a polynomial and finds the creative potential of each candidate by the formula .

For what minimum can he always find a polynomial of degree not exceeding such that the creative potential of all candidates is strictly more than that of the others?