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.

Problem 8.  Let S=\{s_1, s_2, s_3,\ldots,s_n\} be a set of n distinct complex numbers n \geq 9, exactly n-3 of which  are real.

Prove that there are at most two quadratic polynomials f(z) with complex

coefficients such that f(S) = S (that is, f permutes the elements of S).


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