Found problems: 3597
Prove that there are no integers $a,b,c,d$ such that the polynomial $ax^3+bx^2+cx+d$ equals $1$ at $x=19$, and equals $2$ at $x=62$.
Let the product $(12)(15)(16)$, each factor written in base $b$, equal $3146$ in base $b$. Let $s=12+15+16$, each term expressed in base $b$. Then $s$, in base $b$, is
$\textbf{(A)}\ 43\qquad
\textbf{(B)}\ 44\qquad
\textbf{(C)}\ 45\qquad
\textbf{(D)}\ 46\qquad
\textbf{(E)}\ 47$
Denote by $P^{(n)}$ the set of all polynomials of degree $n$ the coefficients of which is a permutation of the set of numbers $\{2^0, 2^1,..., 2^n\}$. Find all pairs of natural numbers $(k,d)$ for which there exists a $n$ such that for any polynomial $p \in P^{(n)}$, number $P(k)$ is divisible by the number $d$.
(Oleksii Masalitin)
Find the smallest positive integer value of $N$ such that field $K=\mathbb{Q}(\sqrt{N},\ \sqrt{i+1})$, where $i=\sqrt{-1}$, is Galois extension on $\mathbb{Q}$, then find the Galois group $Gal(K/\mathbb{Q}).$
$P(x)$ is polynomial with real coefficients such that $\forall n \in \mathbb{Z}, P(n) \in \mathbb{Z}$. Prove that every coefficients of $P(x)$ is rational numbers.
Let $ f(x)$ be a polynomial with integer coefficients and let $ p$ be a prime. Denote by $ z_1,...,z_{p\minus{}1}$ the $ (p\minus{}1)$th complex roots of unity. Prove that
$ f(z_1)\cdots f(z_{p\minus{}1})\equiv f(1)\cdots f(p\minus{}1) \pmod{p}$.
Given $a$, $b$, and $c$ are complex numbers satisfying
\[ a^2+ab+b^2=1+i \]
\[ b^2+bc+c^2=-2 \]
\[ c^2+ca+a^2=1, \]
compute $(ab+bc+ca)^2$. (Here, $i=\sqrt{-1}$)
Define the sequence $(a_n)_{n\geq 1}$ by $a_1=1$, $a_2=p$ and \[a_{n+1}=pa_n-a_{n-1} \textrm { for all } n>1.\]
Prove that for $n>1$ the polynomial $x^n-a_nx+a_{n-1}$ is divisible by $x^2-px+1$.
Using this result, solve the equation \[x^4-56x+15=0.\]
Let $P(x)$ be a monic cubic polynomial. The lines $y = 0$ and $y = m$ intersect $P(x)$ at points $A$, $C$, $E$ and $B$, $D$, $F$ from left to right for a positive real number $m$.
If $AB = \sqrt{7}$, $CD = \sqrt{15}$, and $EF = \sqrt{10}$, what is the value of $m$?
$\textbf{6.1.}$ A monic polynomial is one that has a main coefficient equal to $1$. For example, the polynomial $P(x) = x^3 + 5x^2 - 3x + 7$ is a monic polynomial
[i]Proposed by Lenin Vasquez, Copan[/i]
i) Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$,
the $p$-th term of the progression is also prime.
ii) Find all polynomials $f(X) \in \mathbb{Z}[X]$, such that there exist $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, $| f(p) |$ is also prime.
[i]Dan Schwarz[/i]
To obtain the value of a polynomial of degree $n$, whose coefficients are $$a_0, a_1, . . . ,a_n$$ (starting with the term of highest degree), when the variable $x$ is given the value $b$, the process indicated in the attached flowchart can be applied, which develops the actions required to apply Ruffini's rule. It is requested to build another flowchart analogous that allows to express the calculation of the value of the derivative of the given polynomial, also for $x = b$.
[img]https://cdn.artofproblemsolving.com/attachments/a/a/27563a0e97e74553a270fcd743f22176aed83b.png[/img]
Let $b$ and $c$ be fixed real numbers and let the ten points $(j,y_j )$ for $j=1,2,\ldots,10$ lie on the parabola $y =x^2 +bx+c.$ For $j=1,2,\ldots, 9$ let $I_j$ be the intersection of the tangents to the given parabola at $(j, y_j )$ and $(j+1, y_{j+1}).$ Determine the poynomial function $y=g(x)$ of least degree whose graph passes through all nine points $I_j .$
What is the sum of real roots of $(2+(2+(2+x)^2)^2)^2=2000$ ?
$ \textbf{(A)}\ -4
\qquad\textbf{(B)}\ -2
\qquad\textbf{(C)}\ 0
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ 4
$
Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$.
[i]Proposed by A. Golovanov, M. Ivanov, K. Kokhas[/i]
Let $f(x)$ be a polynomial of nonzero degree. Can it happen that for any real number $a$, an even number of real numbers satisfy the equation $f(x) = a$?
The Bernoulli sequence $\{B_{n}\}_{n \ge 0}$ is defined by \[B_{0}=1, \; B_{n}=-\frac{1}{n+1}\sum^{n}_{k=0}{{n+1}\choose k}B_{k}\;\; (n \ge 1)\] Show that for all $n \in \mathbb{N}$, \[(-1)^{n}B_{n}-\sum \frac{1}{p},\] is an integer where the summation is done over all primes $p$ such that $p| 2k-1$.
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.
Let be three polynomials of degree two $ p_1,p_2,p_3\in\mathbb{R} [X] $ and the function
$$ f:\mathbb{R}\longrightarrow\mathbb{R} ,\quad f(x)=\max\left( p_1(x),p_2(x),p_3(x)\right) . $$
Then, $ f $ is differentiable if and only if any of these three polynomials dominates the other two.
Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.
Let $p=4k+3$ be a prime. Prove that if
\[\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n}\]
(where the fraction $\dfrac {m} {n}$ is in reduced terms), then $p \mid 2m-n$.
[i]Proposed by A. Golovanov[/i]
Quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$, both of which have real roots, are called friendly if for all $t \in [0,1]$ quadratic polynomial $tP(x)+(1-t)Q(x)$ also has real roots.
a) Provide an example of quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ and which have real roots, that are not friendly.
b) Prove that for any two quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ that have real roots, there is a quadratic polynomial $R(x)$ which has a leading coefficient $1$ and which is friendly to both $P$ and $Q$
For a certain complex number $c$, the polynomial
\[ P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\]
has exactly 4 distinct roots. What is $|c|$?
$\textbf{(A) } 2 \qquad \textbf{(B) } \sqrt{6} \qquad \textbf{(C) } 2\sqrt{2} \qquad \textbf{(D) } 3 \qquad \textbf{(E) } \sqrt{10}$
Assume that $P(x)$ is a polynomial with integer non negative coefficients, different from constant. Baron Munchausen claims that he can restore $P(x)$ provided he knows the values of $P(2)$ and $P(P(2))$ only. Is the baron's claim valid?
Given a real number $t\ge3$, suppose a polynomial $f\in\mathbb R[x]$ satisfies
$$\left|f(k)-t^k\right|<1,\enspace\forall k=0,1,\ldots,n.$$Prove that $\deg f\ge n$.
Find all pairs of polynomials $p$, $q$ with complex coefficients so that
\[p(x)\cdot q(x)=p(q(x)).\]