Found problems: 4275
Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a, b,$ and $c,$ and that the roots of $x^3+rx^2+sx+t=0$ are $a+b, b+c,$ and $c+a.$ Find $t.$
Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial \[a_{99}x^{99} + \cdots + a_1x + a_0\]with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve?
Calculate $f(\sqrt[3]{2}-1) $, where
$$f(x) = x^{1999} + 3x^{1998} + 4x^{1997} + 2x^{1996} + 4x^{1995} + 2x^{1994} + ...$$
$$... + 4x^3 + 2x^2 + 3x+ 1.$$
The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as
\[P_0(x)=x,\quad P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1),\quad n\ge 1.\]
Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$.
Let $a, b_1, b_2, \dots, b_n, c_1, c_2, \dots, c_n$ be real numbers such that \[x^{2n} + ax^{2n - 1} + ax^{2n - 2} + \dots + ax + 1 = \prod_{i = 1}^{n}{(x^2 + b_ix + c_i)}\]
Prove that $c_1 = c_2 = \dots = c_n = 1$.
As a consequence, all complex zeroes of this polynomial must lie on the unit circle.
Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form \[ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0 \] where each of $c_0,c_1,\ldots,c_{n-1}$ is equal to $1$ or $-1$.
Let $g(x)$ be a polynomial with leading coefficient $1,$ whose three roots are the reciprocals of the three roots of $f(x)=x^3+ax^2+bx+c,$ where $1<a<b<c.$ What is $g(1)$ in terms of $a,b,$ and $c?$
$\textbf{(A) }\frac{1+a+b+c}c \qquad \textbf{(B) }1+a+b+c \qquad \textbf{(C) }\frac{1+a+b+c}{c^2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}$
Let $ R$ be a finite commutative ring. Prove that $ R$ has a multiplicative identity element $ (1)$ if and only if the annihilator of $ R$ is $ 0$ (that is, $ aR\equal{}0, \;a\in R $ imply $ a\equal{}0$).
Let $a, b, c$ be integers and let $p$ be an odd prime with \[p \not\vert a \;\; \text{and}\;\; p \not\vert b^{2}-4ac.\] Show that \[\sum_{k=1}^{p}\left( \frac{ak^{2}+bk+c}{p}\right) =-\left( \frac{a}{p}\right).\]
The equation \[2^{333x-2}+2^{111x+2}=2^{222x+1}+1\] has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
Let $\mathbb{R}[x]$ be the set of all polynomials with real coefficients. Find all functions $f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ satisfying the following conditions:
[list]
[*] $f$ maps the zero polynomial to itself,
[*] for any non-zero polynomial $P \in \mathbb{R}[x]$, $\text{deg} \, f(P) \le 1+ \text{deg} \, P$, and
[*] for any two polynomials $P, Q \in \mathbb{R}[x]$, the polynomials $P-f(Q)$ and $Q-f(P)$ have the same set of real roots.
[/list]
[i]Proposed by Anant Mudgal, Sutanay Bhattacharya, Pulkit Sinha[/i]
Determine the lowest positive integer n such that following statement is true:
If polynomial with integer coefficients gets value 2 for n different integers,
then it can't take value 4 for any integer.
Let $\alpha, \beta, \gamma \in C$ be the roots of the polynomial $x^3 - 3x2 + 3x + 7$. For any complex number $z$, let $f(z)$ be defined as follows:
$$f(z) = |z -\alpha | + |z - \beta|+ |z-\gamma | - 2 \underbrace{\max}_{w \in \{\alpha, \beta, \gamma}\} |z - w|.$$
Let $A$ be the area of the region bounded by the locus of all $z \in C$ at which $f(z)$ attains its global minimum. Find $\lfloor A \rfloor$.
find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \]
for every natural numbers $ a $ and $ b $
Let $A(x,y)$ be the number of points $(m,n)$ in the plane with integer coordinates $m$ and $n$ satisfying $m^2+n^2\le x^2+y^2$. Let $g=\sum_{k=1}^\infty e^{-k^2}$. Express
$$\int^\infty_{-\infty}\int^\infty_{-\infty}A(x,y)e^{-x^2-y^2}dxdy$$
as a polynomial in $g$.
A polynomial $p(x)$ of degree $1000$ is such that $p(n) = (n+1)2^n$ for all nonnegative integers $n$ such that $n\leq 1000$. Given that \[p(1001) = a\cdot 2^b - c,\] where $a$ is an odd integer, and $0 < c < 2007$, find $c-(a+b)$.
The sequence $f_0, f_1, \dots$ of polynomials in $\mathbb{F}_{11}[x]$ is defined by $f_0(x) = x$ and $f_{n+1}(x) = f_n(x)^{11} - f_n(x)$ for all $n \ge 0$. Compute the remainder when the number of nonconstant monic irreducible divisors of $f_{1000}(x)$ is divided by $1000$.
[i]Proposed by Ankan Bhattacharya[/i]
Non-zero polynomials $P(x)$, $Q(x)$, and $R(x)$ with real coefficients satisfy the identities
$$ P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0. $$
Prove that the degrees of the three polynomials are all even.
A real number $ to $ is randomly and uniformly chosen from the $ [- 3,4] $ interval. What is the probability that all roots of the polynomial $ x ^ 3 + ax ^ 2 + ax + 1 $ are real?
Let $ P(x)$ be a polynomial with real coefficients such that $ P(x) > 0$ for all $ x \geq 0.$ Prove that there exists a positive integer n such that $ (1 \plus{} x)^n \cdot P(x)$ is a polynomial with nonnegative coefficients.
Find all non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients such that $P(Q(x)^2)=P(x)\cdot Q(x)^2$.
[i](I. Voronovich)[/i]
Pete chooses $ 1004$ monic quadratic polynomial $ f_{1},\cdots,f_{1004}$, such that each integer from $ 0$ to $ 2007$ is a root of at least one of them. Vasya considers all equations of the form $ f_{i}\equal{}f_{j}(i\not \equal{}j)$ and computes their roots; for each such root , Pete has to pay to Vasya $ 1$ ruble . Find the least possible value of Vasya's income.
Let \( f \) and \( g \) be two nonzero polynomials with integer coefficients such that \( \deg(f) > \deg(g) \). Suppose that for infinitely many prime numbers \( p \), the polynomial \( pf + g \) has a rational root. Prove that \( f \) has a rational root.
Clarification: A rational root of a polynomial \( f \) is a number \( q \in \mathbb{Q} \) such that \( f(q) = 0 \).
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
Square trinomial $f(x)$ is such that the polynomial (f(x))^5 - f(x) has exactly three real roots. Find the ordinate of the vertex of the graph of this trinomial.