Found problems: 4275
is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?
Prove that if $m$ is a natural number and $P,Q,R$ polynomials of degrees less than $m$ satisfying
$$x^{2m}P(x,y)+y^{2m}Q(x,y) = (x+y)^{2m}R(x,y),$$
then each of the polynomials is zero.
Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$.
[i]Proposed by Evan Chen[/i]
For a prime $p$, a subset $S$ of residues modulo $p$ is called a [i]sum-free multiplicative subgroup[/i] of $\mathbb F_p$ if
$\bullet$ there is a nonzero residue $\alpha$ modulo $p$ such that $S = \left\{ 1, \alpha^1, \alpha^2, \dots \right\}$ (all considered mod $p$), and
$\bullet$ there are no $a,b,c \in S$ (not necessarily distinct) such that $a+b \equiv c \pmod p$.
Prove that for every integer $N$, there is a prime $p$ and a sum-free multiplicative subgroup $S$ of $\mathbb F_p$ such that $\left\lvert S \right\rvert \ge N$.
[i]Proposed by Noga Alon and Jean Bourgain[/i]
A real polynomial of odd degree has all positive coefficients. Prove that there is a (possibly trivial) permutation of the coefficients such that the resulting polynomial has exactly one real zero.
If $a,b,c$ are non-zero real numbers such that \[\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a},\] and \[x=\frac{(a+b)(b+c)(c+a)}{abc},\] and $x<0$, then $x$ equals
$\textbf{(A) }-1\qquad\textbf{(B) }-2\qquad\textbf{(C) }-4\qquad\textbf{(D) }-6\qquad \textbf{(E) }-8$
Determine all the complex numbers $w = a + bi$ with $a, b \in \mathbb{R}$, such that there exists a polinomial $p(z)$ whose coefficients are real and positive such that $p(w) = 0.$
For integers $0\le a\le n$, let $f(n,a)$ denote the number of coefficients in the expansion of $(x+1)^a(x+2)^{n-a}$ that is divisible by $3.$ For example, $(x+1)^3(x+2)^1=x^4+5x^3+9x^2+7x+2$, so $f(4,3)=1$. For each positive integer $n$, let $F(n)$ be the minimum of $f(n,0),f(n,1),\ldots ,f(n,n)$.
(1) Prove that there exist infinitely many positive integer $n$ such that $F(n)\ge \frac{n-1}{3}$.
(2) Prove that for any positive integer $n$, $F(n)\le \frac{n-1}{3}$.
For $-1\leq x\leq 1$ and $n\in\mathbb N$ define $T_{n}(x)=\frac{1}{2^{n}}[(x+\sqrt{1-x^{2}})^{n}+(x-\sqrt{1-x^{2}})^{n}]$.
a)Prove that $T_{n}$ is a monic polynomial of degree $n$ in $x$ and that the maximum value of $|T_{n}(x)|$ is $\frac{1}{2^{n-1}}$.
b)Suppose that $p(x)=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\in\mathbb{R}[x]$ is a monic polynomial of degree $n$ such that $p(x)>-\frac{1}{2^{n-1}}$ forall $x$, $-1\leq x\leq 1$. Prove that there exists $x_{0}$, $-1\leq x_{0}\leq 1$ such that $p(x_{0})\geq\frac{1}{2^{n-1}}$.
Let $p(x)$ be a polynomial of degree $4$ with leading coefficient $1$. Suppose $p(1)=1$, $p(2)=2$, $p(3)=3$ and $p(4)=4$. Then $p(5)=$?
$\textbf{(A)}~5$
$\textbf{(B)}~\frac{25}6$
$\textbf{(C)}~29$
$\textbf{(D)}~35$
We consider a matrix $A\in M_n(\textbf{C})$ with rank $r$, where $n\ge 2$ and $1\le r\le n-1$.
a) Show that there exist $B\in M_{n,r}(\textbf{C}), C\in M_{r,n}(\textbf{C})$, with $%Error. "rank" is a bad command.
B=%Error. "rank" is a bad command.
C = r$, such that $A=BC$.
b) Show that the matrix $A$ verifies a polynomial equation of degree $r+1$, with complex coefficients.
[b]p1.[/b] What is $20\times 20 - 19\times 19$?
[b]p2.[/b] Andover has a total of $1440$ students and teachers as well as a $1 : 5$ teacher-to-student ratio (for every teacher, there are exactly $5$ students). In addition, every student is either a boarding student or a day student, and $70\%$ of the students are boarding students. How many day students does Andover have?
[b]p3.[/b] The time is $2:20$. If the acute angle between the hour hand and the minute hand of the clock measures $x$ degrees, find $x$.
[img]https://cdn.artofproblemsolving.com/attachments/b/a/a18b089ae016b15580ec464c3e813d5cb57569.png[/img]
[b]p4.[/b] Point $P$ is located on segment $AC$ of square $ABCD$ with side length $10$ such that $AP >CP$. If the area of quadrilateral $ABPD$ is $70$, what is the area of $\vartriangle PBD$?
[b]p5.[/b] Andrew always sweetens his tea with sugar, and he likes a $1 : 7$ sugar-to-unsweetened tea ratio. One day, he makes a $100$ ml cup of unsweetened tea but realizes that he has run out of sugar. Andrew decides to borrow his sister's jug of pre-made SUPERSWEET tea, which has a $1 : 2$ sugar-to-unsweetened tea ratio. How much SUPERSWEET tea, in ml,does Andrew need to add to his unsweetened tea so that the resulting tea is his desired sweetness?
[b]p6.[/b] Jeremy the architect has built a railroad track across the equator of his spherical home planet which has a radius of exactly $2020$ meters. He wants to raise the entire track $6$ meters off the ground, everywhere around the planet. In order to do this, he must buymore track, which comes from his supplier in bundles of $2$ meters. What is the minimum number of bundles he must purchase? Assume the railroad track was originally built on the ground.
[b]p7.[/b] Mr. DoBa writes the numbers $1, 2, 3,..., 20$ on the board. Will then walks up to the board, chooses two of the numbers, and erases them from the board. Mr. DoBa remarks that the average of the remaining $18$ numbers is exactly $11$. What is the maximum possible value of the larger of the two numbers that Will erased?
[b]p8.[/b] Nathan is thinking of a number. His number happens to be the smallest positive integer such that if Nathan doubles his number, the result is a perfect square, and if Nathan triples his number, the result is a perfect cube. What is Nathan's number?
[b]p9.[/b] Let $S$ be the set of positive integers whose digits are in strictly increasing order when read from left to right. For example, $1$, $24$, and $369$ are all elements of $S$, while $20$ and $667$ are not. If the elements of $S$ are written in increasing order, what is the $100$th number written?
[b]p10.[/b] Find the largest prime factor of the expression $2^{20} + 2^{16} + 2^{12} + 2^{8} + 2^{4} + 1$.
[b]p11.[/b] Christina writes down all the numbers from $1$ to $2020$, inclusive, on a whiteboard. What is the sum of all the digits that she wrote down?
[b]p12.[/b] Triangle $ABC$ has side lengths $AB = AC = 10$ and $BC = 16$. Let $M$ and $N$ be the midpoints of segments $BC$ and $CA$, respectively. There exists a point $P \ne A$ on segment $AM$ such that $2PN = PC$. What is the area of $\vartriangle PBC$?
[b]p13.[/b] Consider the polynomial $$P(x) = x^4 + 3x^3 + 5x^2 + 7x + 9.$$ Let its four roots be $a, b, c, d$. Evaluate the expression $$(a + b + c)(a + b + d)(a + c + d)(b + c + d).$$
[b]p14.[/b] Consider the system of equations $$|y - 1| = 4 -|x - 1|$$
$$|y| =\sqrt{|k - x|}.$$ Find the largest $k$ for which this system has a solution for real values $x$ and $y$.
[b]p16.[/b] Let $T_n = 1 + 2 + ... + n$ denote the $n$th triangular number. Find the number of positive integers $n$ less than $100$ such that $n$ and $T_n$ have the same number of positive integer factors.
[b]p17.[/b] Let $ABCD$ be a square, and let $P$ be a point inside it such that $PA = 4$, $PB = 2$, and $PC = 2\sqrt2$. What is the area of $ABCD$?
[b]p18.[/b] The Fibonacci sequence $\{F_n\}$ is defined as $F_0 = 0$, $F_1 = 1$, and $F_{n+2}= F_{n+1} + F_n$ for all integers $n \ge 0$. Let $$ S =\dfrac{1}{F_6 + \frac{1}{F_6}}+\dfrac{1}{F_8 + \frac{1}{F_8}}+\dfrac{1}{F_{10} +\frac{1}{F_{10}}}+\dfrac{1}{F_{12} + \frac{1}{F_{12}}}+ ... $$ Compute $420S$.
[b]p19.[/b] Let $ABCD$ be a square with side length $5$. Point $P$ is located inside the square such that the distances from $P$ to $AB$ and $AD$ are $1$ and $2$ respectively. A point $T$ is selected uniformly at random inside $ABCD$. Let $p$ be the probability that quadrilaterals $APCT$ and $BPDT$ are both not self-intersecting and have areas that add to no more than $10$. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$.
Note: A quadrilateral is self-intersecting if any two of its edges cross.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
If a polynomial $P$ with integer coefficients has three distinct integer zeroes , then show that $P(n)\neq 1$ for any integer $n$.
(a) Given a positive integer $k$, prove that there do not exist two distinct integers in the open interval $(k^2, (k + 1)^2)$ whose product is a perfect square.
(b) Given an integer $n > 2$, prove that there exist $n$ distinct integers in the open interval $(k^n, (k + 1)^n)$ whose product is the $n$-th power of an integer, for all but a finite number of positive integers $k$.
[i]AMM Magazine[/i]
Given $a, \theta \in \mathbb R, m \in \mathbb N$, and $P(x) = x^{2m}- 2|a|^mx^m \cos \theta +a^{2m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials.
We say that a polynomial $p$ is respectful if $\forall x, y \in Z$, $y - x$ divides $p(y) - p(x)$, and $\forall x \in Z$, $p(x) \in Z$. We say that a respectful polynomial is disguising if it is nonzero, and all of its non-zero coefficients lie between $0$ and $ 1$, exclusive. Determine $\sum deg(f)\cdot f(2)$, where the sum includes all disguising polynomials $f$ of degree at most $5$.
Let $n$ be a positive integer and $a_1, a_2, \dots, a_n$ non-zero real numbers. What is the least number of non-zero coefficients that the polynomial $P(x) = (x - a_1)(x - a_2)\cdots(x - a_n)$ can have?
If $ x$ is a real number such that $ x^2\minus{}x$ and $ x^n\minus{}x$ are integers for some $ n \ge 3$, prove that $ x$ is an integer.
A set \( S \) of two or more positive integers is called [i]almost closed under addition[/i] if the sum of any two distinct elements of \( S \) also belongs to \( S \). Let \( P(x) \) be a polynomial with integer coefficients for which there exists an almost closed under addition set \( S \), such that for any two distinct \( a \) and \( b \) from \( S \), the numbers \( P(a) \) and \( P(b) \) are coprime. Prove that \( P \) is a constant.
For each real number $k$, denote by $f(k)$ the larger of the two roots of the quadratic equation
$$(k^2+1)x^2+10kx-6(9k^2+1)=0.$$Show that the function $f(k)$ attains a minimum and maximum and evaluate these two values.
Define a sequence of polynomials $F_n(x)$ by $F_0(x)=0, F_1(x)=x-1$, and for $n\geq 1$, $$F_{n+1}(x)=2xF_n(x)-F_{n-1}(x)+2F_1(x).$$ For each $n$, $F_n(x)$ can be written in the form $$F_n(x)=c_nP_1(x)P_2(x)\cdots P_{g(n)}(x)$$ where $c_n$ is a constant and $P_1(x),P_2(x)\cdots, P_{g(n)}(x)$ are non-constant polynomials with integer coefficients and $g(n)$ is as large as possible. For all $2< n< 101$, let $t$ be the minimum possible value of $g(n)$ in the above expression; for how many $k$ in the specified range is $g(k)=t$?
We call polynomials $A(x) = a_n x^n +. . .+a_1 x+a_0$ and $B(x) = b_m x^m +. . .+b_1 x+b_0$
($a_n b_m \neq 0$) similar if the following conditions hold:
$(i)$ $n = m$;
$(ii)$ There is a permutation $\pi$ of the set $\{ 0, 1, . . . , n\} $ such that $b_i = a_{\pi (i)}$ for each $i \in {0, 1, . . . , n}$.
Let $P(x)$ and $Q(x)$ be similar polynomials with integer coefficients. Given that
$P(16) = 3^{2012}$, find the smallest possible value of $|Q(3^{2012})|$.
[i]Proposed by Milos Milosavljevic[/i]
Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that \[ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 . \] Prove that \[ \int_0^1 f^2 (x) \, dx \geq 4 . \]
[i]Ion Rasa[/i]
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$, for all $ n \in N$. Let $ r_n$ be the average of $ g_n(x)$'s roots. If $ r_{19}\equal{}99$, find $ r_{99}$.