Found problems: 4275
Let $K$ be a finite field with $q$ elements, $q \ge 3.$ We denote by $M$ the set of polynomials in $K[X]$ of degree $q-2$ whose coefficients are nonzero and pairwise distinct. Find the number of polynomials in $M$ that have $q-2$ distinct roots in $K.$
[i]Marian Andronache[/i]
Let $n$ and $k$ be two natural numbers such that $k$ is even and for each prime $p$ if $p|n$ then $p-1|k$. let $\{a_1,....,a_{\phi(n)}\}$ be all the numbers coprime to $n$. What's the remainder of the number $a_1^k+.....+a_{\phi(n)}^k$ when it's divided by $n$?
[i]proposed by Yahya Motevassel[/i]
Find all polynomials $P(x), Q(x)$ with real coefficients such that for all real numbers $a$, $P(a)$ is a root of the equation $x^{2023}+Q(a)x^2+(a^{2024}+a)x+a^3+2025a=0.$
Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer.
[i]Note: A monic polynomial has a leading coefficient equal to 1.[/i]
[i](Greece - Panagiotis Lolas and Silouanos Brazitikos)[/i]
Find all polynomials $p_n(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ ($n\geq 2$) with real and non-zero coeficients s.t. $p_n(x)-p_1(x)p_2(x)...p_{n-1}(x)$ be a constant polynomial. ;)
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]
Let $F(n)$ be the set of polynomials $P(x) = a_0+a_1x+\cdots+a_nx^n$, with $a_0, a_1, . . . , a_n \in \mathbb R$ and $0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}.$ Prove that if $f \in F(m)$ and $g \in F(n)$, then $fg \in F(m + n).$
Let $f(x)$ be a monic cubic polynomial with $f(0)=-64$, and all roots of $f(x)$ are non-negative real numbers. What is the largest possible value of $f(-1)$? (A polynomial is monic if its leading coefficient is 1.)
Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.
Prove that it is possible to choose different real numbers $a_1, a_2, . . . , a_{10}$ that the equation $$(x - a_1)(x -a_2).... (x -a_{10}) = (x + a_1)(x + a_2) ...(x + a_{10})$$ will have exactly $5$ different real roots.
Let $ p$ be a prime number. Let $ h(x)$ be a polynomial with integer coefficients such that $ h(0),h(1),\dots, h(p^2\minus{}1)$ are distinct modulo $ p^2.$ Show that $ h(0),h(1),\dots, h(p^3\minus{}1)$ are distinct modulo $ p^3.$
There are 6 distinct values of $x$ strictly between $0$ and $\frac{\pi}{2}$ that satisfy the equation
\[
\tan(15 x) = 15 \tan(x) .
\]
Call these 6 values $r_1$, $r_2$, $r_3$, $r_4$, $r_5$, and $r_6$. What is the value of the sum
\[
\frac{1}{\tan^2 r_1} +
\frac{1}{\tan^2 r_2} +
\frac{1}{\tan^2 r_3} +
\frac{1}{\tan^2 r_4} +
\frac{1}{\tan^2 r_5} +
\frac{1}{\tan^2 r_6} \, ?
\]
A polynomial $p(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$ with integer coefficients is said to be divisible by an integer $m$ if $p(x)$ is divisible by m for all integers $x$. Prove that if $p(x)$ is divisible by $m$, then $k!a_0$ is also divisible by $m$. Also prove that if $a_0, k,m$ are non-negative integers for which $k!a_0$ is divisible by $m$, there exists a polynomial $p(x) = a_0x^k+\cdots+ a_k$ divisible by $m.$
All the roots of polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers. What is the value of $B$?
$\textbf{(A)}\ -88 \qquad\textbf{(B)}\ -80 \qquad\textbf{(C)}\ -64\qquad\textbf{(D)}\ -41 \qquad\textbf{(E)}\ -40$
Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.
Determine all polynomials $P$ with integer coefficients satisfying
\[P(P(P(P(P(x)))))=x^{28}\cdot P(P(x))\qquad \forall x\in\mathbb{R}\]
Let $f:\mathbb{Q}\rightarrow \mathbb{Q}$ a monotonic bijective function.
a)Prove that there exist a unique continuous function $F:\mathbb{R}\rightarrow \mathbb{R}$ such that $F(x)=f(x),\ (\forall)x\in \mathbb{Q}$.
b)Give an example of a non-injective polynomial function $G:\mathbb{R}\rightarrow \mathbb{R}$ such that $G(\mathbb{Q})\subset \mathbb{Q}$ and it's restriction defined on $\mathbb{Q}$ is injective.
Let $n$ be the answer to this problem. The polynomial $x^n+ax^2+bx+c$ has real coefficients and exactly $k$ real roots. Find the sum of the possible values of $k$.
Let $ d$ and $ e$ denote the solutions of $ 2x^2\plus{}3x\minus{}5\equal{}0$. What is the value of $ (d\minus{}1)(e\minus{}1)$?
$ \textbf{(A)}\ \minus{}\frac{5}{2} \qquad
\textbf{(B)}\ 0 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ 6$
Triangle $ABC$ has side lengths $AB=12$, $BC=25$, and $CA=17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ=w$, the area of $PQRS$ can be expressed as the quadratic polynomial
\[\text{Area}(PQRS)=\alpha w-\beta\cdot w^2\]
Then the coefficient $\beta=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Does there exist a quadratic trinomial $f(x)$ such that $f(1/2017)=1/2018$, $f(1/2018)=1/2017$, and two of its coefficients are integers?
(A. Khrabrov)
Let $a$ be some integer. Prove that the polynomial $x^4(x-a)^4+1$ can not be a product of two non-constant polynomials with integer coefficients
For every natural number $n\geq{2}$ consider the following affirmation $P_n$:
"Consider a polynomial $P(X)$ (of degree $n$) with real coefficients. If its derivative $P'(X)$ has $n-1$ distinct real roots, then there is a real number $C$ such that the equation $P(x)=C$ has $n$ real,distinct roots."
Are $P_4$ and $P_5$ both true? Justify your answer.
Let $p$ be a prime number, $n$ a natural number which is not divisible by $p$, and $\mathbb{K}$ is a finite field, with $char(K) = p, |K| = p^n, 1_{\mathbb{K}}$ unity element and $\widehat{0} = 0_{\mathbb{K}}.$ For every $m \in \mathbb{N}^{*}$ we note
$ \widehat{m} = \underbrace{1_{\mathbb{K}} + 1_{\mathbb{K}} + \ldots + 1_{\mathbb{K}}}_{m \text{ times}} $ and define the polynomial
\[
f_m = \sum_{k = 0}^{m} (-1)^{m - k} \widehat{\binom{m}{k}} X^{p^k} \in \mathbb{K}[X].
\]
a) Show that roots of $f_1$ are $ \left\{ \widehat{k} | k \in \{0,1,2, \ldots , p - 1 \} \right\}$.
b) Let $m \in \mathbb{N}^{*}.$ Determine the set of roots from $\mathbb{K}$ of polynomial $f_{m}.$
Let $P$ and $Q$ be two polynomials with real coeficients such that $P$ has degree greater than $1$ and \[P(Q(x)) = P(P(x)) + P(x).\]Show that $P(-x) = P(x) + x$.