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.

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Lagrange’s interpolation polynomial (3)

Part 2’s link

Problem 9. Let t and n be fixed integers each at least 2. Find the largest positive integer m for which there exists a polynomial P, of degree n and with rational coefficients, such that the following property holds: exactly one of  \displaystyle\frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}}  is an integer for each k = 0,1, ..., m.

Problem 10. Let f\left ( x \right )=x^{n}+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+...+a_{1}x+a_{0} be a polynomial. Prove that we have an \displaystyle i\in \left \{ 1,2,...,n \right \}\mid  \left | f\left ( i \right ) \right |\geq \frac{n!}{\binom{n}{i}}.

Problem 11.  Let (F_n)_{n\geq 1} be the Fibonacci sequence F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1), and P(x) the polynomial of degree 990 satisfying

P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982. Prove that P(1983) = F_{1983} - 1.

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Lagrange’s interpolation polynomial (2)

Part 1’s link

Problem 1. Let P be a polynomial of degree at most n satisfying \displaystyle P(k)=\frac{1}{C^k_{n+1}}\,\,\forall k=\overline{0,n}. Determine P(n+1).

Problem 2. A polynomial P(x) has degree at most 2k, where k = 0, 1,2,\cdots. Given that for an integer i, the inequality -k \le i \le k implies |P(i)| \le 1, prove that for all real numbers x, with -k \le x \le k, the following inequality holds |P(x)| \leq 2^{2k}.

Problems 3. Prove that at least one of the numbers |f(1)|,|f(2)|,\cdots, |f(n+1)| is greater than or equal to \dfrac{n!}{2n}. Where

f(x) = x^n + a_1x^{n-1} + \cdots+ a_n\,\,  (\quad a_i \in  \mathbb R, \quad  i = 1, \ldots , n,n\in\mathbb{N}.)

Problem 4. Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree n with real coefficients is the average of two monic polynomials of degree n with n real roots.

Problem 5. Let p be a prime number and f an integer polynomial of degree d such that f(0) = 0,f(1) = 1 and f(n) is congruent to 0 or 1 modulo p for every integer n. Prove that d\geq p - 1.

Problem 6. Let P be a polynomial of degree n\in\mathbb{N} satisfying P(k)=2^k\,\,\forall k=\overline{0,n}. Prove that P(n+1)=2^{n+1}-1.

Problem 7.  P(x) is a polynomial of degree 3n\,\, (n\in\mathbb{N}) such that

P(0) = P(3) = \cdots = P(3n) = 2,\,\,\, P(1) = P(4) = \cdots = P(3n-2) = 1,

P(2) = P(5) = \cdots = P(3n-1) = 0, \quad\text{and}\quad P(3n+1) = 730.

Determine n.

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Lagrange’s interpolation polynomial

In this article, I will use Lagrange polynomial to solve some polynomial problems from Mathematical Olympiads.

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