# Lagrange’s interpolation polynomial (4)

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

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

$P(x^{2}-2)=P^{2}(x)-2.$

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?