Lagrange’s interpolation polynomial (4)

Part 3’s link

Problem 17. Prove that, for infinitely many positive integers n, there exists a polynomial P of degree n with real coefficients such that P(1),P(2),\cdots, P(n+2) are different whole powers of 2.

Problem 18. Suppose q_{0}, \, q_{1}, \, q_{2}, \ldots \; \, is an infinite sequence of integers satisfying the following two conditions:

(i)  m-n \, divides q_{m}-q_{n} for m > n \geq 0,

(ii) there is a polynomial P such that |q_{n}| < P(n) \, for all n

Prove that there is a polynomial Q such that q_{n}= Q(n) for all n.

Problem 19. Let P\in\mathbb{R}[x] such that for infty of integer number c : Equation P(x)=c has more than one integer root. Prove that P(x)=Q((x-a)^{2}), where a\in\mathbb R and Q is a polynomial.

Problem 20. Find all the polynomials P(x) with odd degree such that


Problem 21. Suppose p(x) is a polynomial  with integer coefficients assumes at n distinct integral values of x that are different form 0 and in absolute value less than  \dfrac{(n-[\frac n2])!}{2^{n-[\frac n2]}} . Prove that p(x) is irreducible.

Prove that the bound may be replaced by (\dfrac d2)^{n-[\frac n2]}(n-[\frac n2])! where d is minimum distance between any two of the n integral values of x where p(x) assumes the integral values considered.

Problem 22. Six members of the team of Fatalia for the IMO are selected from 13 candidates. At the TST the candidates got a_1,a_2, \ldots, a_{13} points with a_i \neq a_j if i \neq j.

The team leader has already 6 candidates and now wants to see them and nobody other in the team. With that end in view he constructs a polynomial P(x) and finds the creative potential of each candidate by the formula c_i = P(a_i).

For what minimum n can he always find a polynomial P(x) of degree not exceeding n such that the creative potential of all 6 candidates is strictly more than that of the 7 others?


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