Found problems: 3597
Consider a polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+d$ with integer coefficients.
Prove that if $f(x)$ has exactly one real root, then it can be factored into nonconstant polynomials with rational coefficients
Given $n\ge 2$ a natural number, $(K,+,\cdot )$ a body with commutative property that $\underbrace{1+...+}_{m}1\ne 0,m=2,...,n,f\in K[X]$ a polynomial of degree $n$ and $G$ a subgroup of the additive group $(K,+,\cdot )$, $G\ne K.$Show that there is $a\in K$ so$f(a)\notin G$.
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.
\[ \begin{array}{c} {x^{2} \plus{} y^{2} \plus{} z^{2} \equal{} 21} \\
{x \plus{} y \plus{} z \plus{} xyz \equal{} \minus{} 3} \\
{x^{2} yz \plus{} y^{2} xz \plus{} z^{2} xy \equal{} \minus{} 40} \end{array}
\]
The number of real triples $ \left(x,y,z\right)$ satisfying above system is
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None}$
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$?
[i]Proposed by Andre Arslan[/i]
Given 2 positive reals $a$ and $b$. There exists 2 polynomials $F(x)=x^2+ax+b$ and $G(x)=x^2+bx+a$ such that all roots of polynomials $F(G(x))$ and $G(F(x))$ are real. Show that $a$ and $b$ are more than $6$.
[i]Raja Oktovin, Pekanbaru[/i]
Find all polynomials $P(x)$ with real coefficients that satisfy the relation
$$1+P(x)=\frac{P(x-1)+P(x+1)}2.$$
Fix a prime number $ p > 5$. Let $ a,b,c$ be integers no two of which have their difference divisible by $ p$. Let $ i,j,k$ be nonnegative integers such that $ i \plus{} j \plus{} k$ is divisible by $ p \minus{} 1$. Suppose that for all integers $ x$, the quantity
\[ (x \minus{} a)(x \minus{} b)(x \minus{} c)[(x \minus{} a)^i(x \minus{} b)^j(x \minus{} c)^k \minus{} 1]\]
is divisible by $ p$. Prove that each of $ i,j,k$ must be divisible by $ p \minus{} 1$.
[i]Kiran Kedlaya and Peter Shor.[/i]
The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations
\[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
Show that $\cos \frac{\pi}{7}$ is irrational.
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial
\[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n.
\]
For the sequence of real numbers $a_1,a_2,\dots ,a_k$ we say it is [i]invested[/i] on the interval $[b,c]$ if there exists numbers $x_0,x_1,\dots ,x_k$ in the interval $[b,c]$ such that $|x_i-x_{i-1}|=a_i$ for $i=1,2,3,\dots k$ .
A sequence is [i]normed[/i] if all its members are not greater than $1$ . For a given natural $n$ , prove :
a)Every [i]normed[/i] sequence of length $2n+1$ is [i]invested[/i] in the interval $\left[ 0, 2-\frac{1}{2^n} \right ]$.
b) there exists [i]normed[/i] sequence of length $4n+3$ wich is not [i]invested[/i] on $\left[ 0, 2-\frac{1}{2^n} \right ]$.
Square trinomial $f(x)$ is such that the polynomial $(f(x)) ^3- f(x)$ has exactly three real roots. Find the ordinate of the vertex of the graph of this trinomial.
Let $P(X) = a_n X^n + a_{n-1} X^{n-1} + \cdots + a_1 X + a_0$ be a polynomial with real coefficients such that $0 \leqslant a_i \leqslant a_0$ for $i = 1, 2, \ldots, n$. Prove that, if $P(X)^2 = b_{2n} X^{2n} + b_{2n-1} X^{2n-1} + \cdots + b_{n+1} X^{n+1} + \cdots + b_1 X + b_0$, then $4 b_{n+1} \leqslant P(1)^2$.
In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.
Let $d$ and $m$ be two fixed positive integers. Pinocchio and Geppetto know the values of $d$ and $m$ and play the following game: In the beginning, Pinocchio chooses a polynomial $P$ of degree at most $d$ with integer coefficients. Then Geppetto asks him questions of the following form "What is the value of $P(n)$?'' for $n \in \mathbb{Z}$. Pinocchio usually says the truth, but he can lie up to $m$ times. What is, as a function of $d$ and $m$, the minimal number of questions that Geppetto needs to ask to be sure to determine $P$, no matter how Pinocchio chooses to reply?
Given the polynomial $ P(x) = \frac{1}{2} - \frac{1}{3}x + \frac{1}{6}x^2 $. Let $ Q(x) = \sum_{k=0}^{m} b_k x^k $ be a polynomial given by $$ Q(x) = P(x) \cdot P(x^3) \cdot P(x^9) \cdot P(x^{27}) \cdot P(x^{81}).
$$
Calculate $ \sum_{k=0}^m |b_k| $.
A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$.
[i]Proposed by Aaron Lin[/i]
Let $b$, $c$ be integer numbers, and define $f(x)=(x+b)^2-c$.
i) If $p$ is a prime number such that $c$ is divisible by $p$ but not by $p^{2}$, show that for every integer $n$, $f(n)$ is not divisible by $p^{2}$.
ii) Let $q \neq 2$ be a prime divisor of $c$. If $q$ divides $f(n)$ for some integer $n$, show that for every integer $r$ there exists an integer $n'$ such that $f(n')$ is divisible by $qr$.
Let $ \varphi(x)$ be a cubic polynomial with integer coefficients. Given that $ \varphi(x)$ has have 3 distinct real roots $u,v,w $ and $u,v,w $ are not rational number. there are integers $ a, b,c$ such that $u=av^2+bv+c$. Prove that $b^2 -2b -4ac - 7$ is a square number .
Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.
For a complex number $z=1+2\sqrt{6}i$ and natural number $n=1,\ 2,\ 3,\ \cdots$, express the complex number $z^n$ in using real numbers $a_n,\ b_n$ as $z^n=a_n+b_ni$.
Answer the following questions.
(1) Show that $a_n^2+b_n^2=5^{2n}\ (n=1,\ 2,\ 3,\ \cdots).$
(2) Find the constants $p,\ q$ such that $a_{n+2}=pa_{n+1}+qa_n$ holds for all $n$.
(3) Show that $a_n$ is not a multiple of $5$ for any $n$.
(4) Show that $z^n\ (n=1,\ 2,\ 3,\ \cdots)$ is not a real number.
Find all numbers $\alpha$ for which the equation
\[x^2 - 2x[x] + x -\alpha = 0\]
has two nonnegative roots. ($[x]$ denotes the largest integer less than or equal to x.)
An ordered quadruple $(y_1,y_2,y_3,y_4)$ is $\textbf{quadratic}$ if there exist real numbers $a$, $b$, and $c$ such that \[y_n=an^2+bn+c\] for $n=1,2,3,4$.
Prove that if $16$ numbers are placed in a $4\times 4$ grid such that all four rows are quadratic and the first three columns are also quadratic then the fourth column must also be quadratic.
[i](We say that a row is quadratic if its entries, in order, are quadratic. We say the same for a column.)[/i]
[asy]
size(100);
defaultpen(linewidth(0.8));
for(int i=0;i<=4;i=i+1)
draw((i,0)--(i,4));
for(int i=0;i<=4;i=i+1)
draw((0,i)--(4,i));
[/asy]