Found problems: 4275
Let $P(x)=x^n+a_1x^{n-1}+\ldots+a_n$ ($n\ge2$) be a polynomial with integer coefficients having $n$ real roots $b_1,\ldots,b_n$. Prove that for $x_0\ge\max\{b_1,\ldots,b_n\}$,
$$P(x_0+1)\left(\frac1{x_0-b_1}+\ldots+\frac1{x_0-b_n}\right)\ge2n^2.$$
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that the limit
$$g(x)=\lim_{h\rightarrow 0}{\frac{f(x+h)-2f(x)+f(x-h)}{h^2}}$$
exists for all real $x$. Prove that $g(x)$ is constant if and only if $f(x)$ is a polynomial function whose degree is at most $2$.
Find the smallest positive integer $ n$ with the property that the polynomial $ x^4 \minus{} nx \plus{} 63$ can be written as a product of two nonconstant polynomials with integer coefficients.
Suppose $a_1,a_2,a_3,a_4$ are distinct integers and $P(x)$ is a polynomial with integer coefficients satisfying $P(a_1) = P(a_2) = P(a_3) = P(a_4) = 1$.
(a) Prove that there is no integer $n$ such that $P(n) = 12$.
(b) Do there exist such a polynomial and $a_n$ integer $n$ such that $P(n) = 1998$?
Suppose $a_1,a_2,a_3,b_1,b_2,b_3$ are distinct positive integers such that
\[(n \plus{} 1)a_1^n \plus{} na_2^n \plus{} (n \minus{} 1)a_3^n|(n \plus{} 1)b_1^n \plus{} nb_2^n \plus{} (n \minus{} 1)b_3^n\]
holds for all positive integers $n$. Prove that there exists $k\in N$ such that $ b_i \equal{} ka_i$ for $ i \equal{} 1,2,3$.
For positive integers $n$, let $C(n)$ be the number of representation of $n$ as a sum of nonincreasing powers of $2$, where no power can be used more than three times. For example, $C(8)=5$ since the representations of $8$ are:
$$8,4+4,4+2+2,4+2+1+1,\text{ and }2+2+2+1+1.$$Prove or disprove that there is a polynomial $P(x)$ such that $C(n)=\lfloor P(n)\rfloor$ for all positive integers $n$.
Let $x_0$ - is positive root of $x^{2017}-x-1=0$ and $y_0$ - is positive root of $y^{4034}-y=3x_0$
a) Compare $x_0$ and $y_0$
b) Find tenth digit after decimal mark in decimal representation of $|x_0-y_0|$
The polynomial $x^{2n}+1+(x+1)^{2n}$ is not divisible by $x^2+x+1$ if $n$ equals
$\text{(A)} \ 17 \qquad \text{(B)} \ 20 \qquad \text{(C)} \ 21 \qquad \text{(D)} \ 64 \qquad \text{(E)} \ 65$
Let $ (b_0,b_1,b_2,b_3)$ be a permutation of the set $ \{54,72,36,108\}$. Prove that $ x^5\plus{}b_3x^3\plus{}b_2x^2\plus{}b_1x\plus{}b_0$ is irreducible in $ \mathbb Z[x]$.
prove for all $k> 1$ equation $(x+1)(x+2)...(x+k)=y^{2}$ has finite solutions.
Find the remainder when \[\prod_{i=0}^{100}(1-i^2+i^4)\] is divided by $101$.
[i]Victor Wang[/i]
Prove that there exists a positive integer $N$ such that for every polynomial $P(x)$ of degree $2019$, there exist $N$ linear polynomials $p_1,p_2, \ldots p_N$ such that $P(x)=p_1(x)^{2019}+p_2(x)^{2019}+ \ldots + p_N(x)^{2019}$. (Assume all polynomials in this problem have real coefficients, and leading coefficients cannot be zero.)
Let $P$ be a polynomial of degree $6$ and let $a,b$ be real numbers such that $0<a<b$. Suppose that $P(a)=P(-a),P(b)=P(-b),P'(0)=0$. Prove that $P(x)=P(-x)$ for all real $x$.
Two polynomials of the same degree $A(x)=a_nx^n+ \cdots + a_1x+a_0$ and $B(x)=b_nx^n+\cdots+b_1x+b_0$ are called [i]friends[/i] is the coefficients $b_0,b_1, \ldots, b_n$ are a permutation of the coefficients $a_0,a_1, \ldots, a_n$.
$P(x)$ and $Q(x)$ be two friendly polynomials with integer coefficients. If $P(16)=3^{2020}$, the smallest possible value of $|Q(3^{2020})|$.
For each $n \ge 2$ define the polynomial $$f_n(x)=x^n-x^{n-1}-\dots-x-1.$$ Prove that
(a) For each $n \ge 2$, $f_n(x)=0$ has a unique positive real root $\alpha_n$;
(b) $(\alpha_n)_n$ is a strictly increasing sequence;
(c) $\lim_{n \rightarrow \infty} \alpha_n=2.$
Patty is standing on a line of planks playing a game. Define a block to be a sequence of adjacent planks, such that both ends are not adjacent to any planks. Every minute, a plank chosen uniformly at random from the block that Patty is standing on disappears, and if Patty is standing on the plank, the game is over. Otherwise, Patty moves to a plank chosen uniformly at random within the block she is in; note that she could end up at the same plank from which she started. If the line of planks begins with $n$ planks, then for sufficiently large n, the expected number of minutes Patty lasts until the game ends (where the first plank disappears a minute after the game starts) can be written as $P(1/n)f(n) + Q(1/n)$, where $P,Q$ are polynomials and $f(n) =\sum^n_{i=1}\frac{1}{i}$ . Find $P(2023) + Q(2023)$.
Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be a monic polynomial of degree $4$. It is known that all the roots of $P$ are real, distinct and belong to the interval $[-1, 1]$.
(a) Prove that $P(x) > -4$ for all real $x$.
(b) Find the highest value of the real constant $k$ such that $P(x) > k$ for every real $x$ and for every polynomial $P(x)$ satisfying the given conditions.
Say an integer polynomial is $\textit{primitive}$ if the greatest common divisor of its coefficients is $1$. For example, $2x^2+3x+6$ is primitive because $\gcd(2,3,6)=1$. Let $f(x)=a_2x^2+a_1x+a_0$ and $g(x) = b_2x^2+b_1x+b_0$, with $a_i,b_i\in\{1,2,3,4,5\}$ for $i=0,1,2$. If $N$ is the number of pairs of polynomials $(f(x),g(x))$ such that $h(x) = f(x)g(x)$ is primitive, find the last three digits of $N$.
Find all $\mathbb Q$-linear maps $\Phi:\mathbb Q[x]\to\mathbb Q[x]$ such that for any irreducible polynomial $p\in\mathbb Q[x]$ the polynomial $\Phi(p)$ is also irreducible.
Product of square trinomials $ x ^ 2 + a_1x + b_1 $, $ x ^ 2 + a_2x + b_2 $, $ \dots $, $ x ^ 2 + a_n x + b_n $ equals polynomial $ P (x) = x ^ {2n} + c_1x ^ {2n-1} + c_2x ^ {2n-2} + \dots + c_ {2n-1} x + c_ {2n} $, where the coefficients $ c_1 $, $ c_2 $, $ \dots $, $ c_ {2n} $ are positive. Prove that for some $ k $ ($ 1 \leq k \leq n $) the coefficients $ a_k $ and $ b_k $ are positive.
We call polynomial $S(x)\in\mathbb{R}[x]$ sadeh whenever it's divisible by $x$ but not divisible by $x^2$.
For the polynomial $P(x)\in\mathbb{R}[x]$ we know that there exists a sadeh polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists sadeh polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{1401}$.
Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial
$f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients
and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.
Compute the largest root of $x^4-x^3-5x^2+2x+6$.
Prove that for any polynomial $f(x)$ (with real coefficients) there exist polynomials $g(x)$ and $h(x)$ (with real coefficients) such that $f(x) = g(h(x)) - h(g(x))$.
The coefficient of $x^7$ in the polynomial expansion of
\[ (1+2x-x^2)^4 \]
is
$ \textbf{(A)}\ -8 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ -12 \qquad\textbf{(E)}\ \text{none of these} $