Lagrange’s interpolation polynomial (2)

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 $n$.