Found problems: 4275
Suppose that $P$ is a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$ and $P(3) = 2016$. If $N$ is the smallest possible positive value of $P(2016)$, find the remainder when $N$ is divided by $2016$.
Find all the polynomials $P(x)$ of a degree $\leq n$ with real non-negative coefficients such that $P(x) \cdot P(\frac{1}{x}) \leq [P(1)]^2$ , $ \forall x>0$.
Let $f$ be an irreducible monic polynomial with integer coefficients such that $f(0)$ is
not equal to $1$. Let $z$ be a complex number that is a root of $f$. Show that if $w$ is another complex
root of $f$, then $\frac{z}{w}$ cannot be a positive integer greater than $1$.
[b]p1.[/b] Given a group of $n$ people. An $A$-list celebrity is one that is known by everybody else (that is, $n - 1$ of them) but does not know anybody. A $B$-list celebrity is one that is known by exactly $n - 2$ people but knows at most one person.
(a) What is the maximum number of $A$-list celebrities? You must prove that this number is attainable.
(b) What is the maximum number of $B$-list celebrities? You must prove that this number is attainable.
[b]p2.[/b] A polynomial $p(x)$ has a remainder of $2$, $-13$ and $5$ respectively when divided by $x+1$, $x-4$ and $x-2$. What is the remainder when $p(x)$ is divided by $(x + 1)(x - 4)(x - 2)$?
[b]p3.[/b] (a) Let $x$ and y be positive integers satisfying $x^2 + y = 4p$ and $y^2 + x = 2p$, where $p$ is an odd prime number. Prove: $x + y = p + 1$.
(b) Find all values of $x, y$ and $p$ that satisfy the conditions of part (a). You will need to prove that you have found all such solutions.
[b]p4.[/b] Let function $f(x, y, z)$ be defined as following:
$$f(x, y, z) = \cos^2(x - y) + \cos^2(y - z) + \cos^2(z - x), x, y, z \in R.$$
Find the minimum value and prove the result.
[b]p5.[/b] In the diagram below, $ABC$ is a triangle with side lengths $a = 5$, $b = 12$,$ c = 13$. Let $P$ and $Q$ be points on $AB$ and $AC$, respectively, chosen so that the segment $PQ$ bisects the area of $\vartriangle ABC$. Find the minimum possible value for the length $PQ$.
[img]https://cdn.artofproblemsolving.com/attachments/b/2/91a09dd3d831b299b844b07cd695ddf51cb12b.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]. Thanks to gauss202 for sending the problems.
Let $P(x)$ be a polynomial with integer coefficients. We denote $\deg(P)$ its degree which is $\geq 1.$ Let $n(P)$ be the number of all the integers $k$ for which we have $(P(k))^{2}=1.$ Prove that $n(P)- \deg(P) \leq 2.$
Determine all polynomials $p(x)$ with non-negative integer coefficients such that $p (1) = 7$ and $p (10) = 2014$.
Let $d(n)$ be the number of divisors of a nonnegative integer $n$ (we set $d(0)=0$). Find all positive integers $d$ such that there exists a two-variable polynomial $P(x,y)$ of degree $d$ with integer coefficients such that:
[list]
[*] for any positive integer $y$, there are infinitely many positive integers $x$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid x$, and
[*] for any positive integer $x$, there are infinitely many positive integers $y$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid y$.
[/list]
[i]Allen Wang[/i]
Let $f \in Z[X]$, $f = X^2 + aX + b$, be a quadratic polynomial. Prove that $f$ has integer zeros if and only if for each positive integer $n$ there is an integer $u_n$ such that $n | f(u_n)$.
Prove that the equation $x^6 - 100x+1 = 0$ has two roots, and both of these roots are positive.
a) Find the first non-zero digit in the decimal notation of the lesser root of this equation.
b) Find the first two non-zero digits in the decimal notation of the lesser root of this equation.
A polynomial $P$ is of the form $\pm x^6 \pm x^5 \pm x^4 \pm x^3 \pm x^2 \pm x \pm 1$. Given that $P(2) = 27$, what is $P(3)$?
Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication.
[i]By Mohsen Jamali[/i]
The function $f(n)$ satisfies $f(0)=0$, $f(n)=n-f \left( f(n-1) \right)$, $n=1,2,3 \cdots$. Find all polynomials $g(x)$ with real coefficient such that
\[ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots \]
Where $[ g(n) ]$ denote the greatest integer that does not exceed $g(n)$.
Let $P(x)$ and $Q(x)$ be polynomials of degree $p$ and $q$ respectively such that every coefficient is $1$ or $2023$. If $P(x)$ divides $Q(x)$, prove that $p+1$ divides $q+1$.
For integers $a$ and $b$ with $0 \leq a,b < {2010}^{18}$ let $S$ be the set of all polynomials in the form of $P(x)=ax^2+bx.$ For a polynomial $P$ in $S,$ if for all integers n with $0 \leq n <{2010}^{18}$ there exists a polynomial $Q$ in $S$ satisfying $Q(P(n)) \equiv n \pmod {2010^{18}},$ then we call $P$ as a [i]good polynomial.[/i]
Find the number of [i]good polynomials.[/i]
Consider the sequence of polynomials $P_0(x) = 2$, $P_1(x) = x$ and $P_n(x) = xP_{n-1}(x) - P_{n-2}(x)$ for $n \geq 2$. Let $x_n$ be the greatest zero of $P_n$ in the the interval $|x| \leq 2$. Show that $$\lim \limits_{n \to \infty}n^2\left(4-2\pi +n^2\int \limits_{x_n}^2P_n(x)dx\right)=2\pi - 4-\frac{\pi^3}{12}$$
Find the least positive integer \(M\) for which there exist a positive integer \(n\) and polynomials \(P_1(x)\), \(P_2(x)\), \(\ldots\), \(P_n(x)\) with integer coefficients satisfying \[Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.\]
[i]Proposed by Karthik Vedula[/i]
Let $P(x)$ be a polynomial with integer coefficients. It is known that the number $\sqrt2+\sqrt3$ is its root. Prove that the number $\sqrt2-\sqrt3$ is also its root.
A polynomial $P(x,y)$ with integer coefficients satisfies two following conditions:
1. for every integer $a$ there exists exactly one integer $y$, such that $P(a,y)=0$
2. for every integer $b$ there exists exactly one integer $x$, such that $P(x,b)=0$
a) Prove that if the degree of $P$ is $2$, then it is divisible by either $x-y+C$ for some integer $C$, or $x+y+C$ for some integer $C$
b) Is there a polynomial $P$ that isn't divisible by any of $x-y+C$ or $x+y+C$ for integers $C$?
Let $P(x)$ be a polynomial with positive integer coefficients such that $deg(P)=699$. Prove that if $P(1) \le 2022$, then there exist some consecutive coefficients such that their sum is $22$, $55$, or $77$.
Given a positive integer $ m$ and $ 0 < \delta <\pi$, construct a trigonometric polynomial $ f(x)\equal{}a_0\plus{} \sum_{n\equal{}1}^m (a_n \cos nx\plus{}b_n \sin nx)$ of degree $ m$ such that $ f(0)\equal{}1, \int_{ \delta \leq |x| \leq \pi} |f(x)|dx \leq c/m,$ and $ \max_{\minus{}\pi \leq x \leq \pi}|f'(x)| \leq c/{\delta}$, for some universal constant $ c$.
[i]G. Halasz[/i]
Determine all non-constant monic polynomials $f(x)$ with integer coefficients for which there exists a natural number $M$ such that for all $n \geq M$, $f(n)$ divides $f(2^n) - 2^{f(n)}$
[i] Proposed by Anant Mudgal [/i]
Polynomials $(P_n(X))_{n\in\mathbb{N}}$ are defined as: $$P_0(X)=0, \quad P_1(X)=X+2,$$ $$P_n(X)=P_{n-1}(X)+3P_{n-1}(X)\cdot P_{n-2}(X)+P_{n-2}(X), \quad (\forall) n\geq2.$$ Show that if $ k $ divides $m$ then $P_k(X)$ divides $P_m(X).$
Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$. For a given positive integer $n$, $n\geq 2$ consider the system of equations: \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivalent if there exists a real number $\lambda \neq 0$, $x_1=\lambda y_1$, $\ldots$, $x_n= \lambda y_n$. How many nonequivalent solutions does the system have?
[i]Mircea Becheanu[/i]
Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers
$c_nr, c_nr^2+c_{n-1}r, c_nr^3+c_{n-1}r^2+c_{n-1}r, \dots, c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$
are all integers.
Problem:
Find all polynomials satisfying the equation
$ f(x^2) = (f(x))^2 $
for all real numbers x.
I'm not exactly sure where to start though it doesn't look too difficult. Thanks!