Found problems: 4275
[b]Problem 6.[/b] Let $p>2$ be prime. Find the number of the subsets $B$ of the set $A=\{1,2,\ldots,p-1\}$ such that, the sum of the elements of $B$ is divisible by $p.$
[i] Ivan Landgev[/i]
Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial with even degree. Prove that, if for infinitely many integers $x$, the number $P(x)$ is a square of a positive integer, then there exists a polynomial $Q(x)\in\mathbb{Z}[x]$ such that $P(x)=Q(x)^2$.
Suppose that $P(x)$ is a polynomial with the property that there exists another polynomial $Q(x)$ to satisfy $P(x)Q(x)=P(x^2)$. $P(x)$ and $Q(x)$ may have complex coefficients. If $P(x)$ is a quintic with distinct complex roots $r_1,\dots,r_5$, find all possible values of $|r_1|+\dots+|r_5|$.
Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ is:
$ \textbf{(A)}\ x^2-bx-ac=0$
$\qquad\textbf{(B)}\ x^2-bx+ac=0$
$\qquad\textbf{(C)}\ x^2+3bx+ca+2b^2=0$
${\qquad\textbf{(D)}\ x^2+3bx-ca+2b^2=0 }$
${\qquad\textbf{(E)}\ x^2+bx(2-a)+a^2c+b^2(a+1)=0} $
Can the equation $x^3-2x^2-2x+m = 0$ have three different rational roots?
Assume that every root of polynomial $P(x) = x^d - a_1x^{d-1} + ... + (-1)^{d-k}a_d$ is in $[0,1]$. Show that for every $k = 1,2,...,d$ the following inequality holds:
$ a_k - a_{k+1} + ... + (-1)^{d-k}a_d \geq 0 $
Let $n\ge 2$ be an even integer and $a,b$ real numbers such that $b^n=3a+1$. Show that the polynomial $P(X)=(X^2+X+1)^n-X^n-a$ is divisible by $Q(X)=X^3+X^2+X+b$ if and only if $b=1$.
The function $f(x) = x^2+ax+b$ has two distinct zeros. If $f(x^2+2x-1)=0$ has four distinct zeros $x_1<x_2<x_3<x_4$ that form an arithmetic sequence, compute the range of $a-b$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 11)[/i]
Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.
Suppose that $ x_1, x_2, x_3, \ldots$ are positive real numbers for which \[ x^n_n \equal{} \sum^{n\minus{}1}_{j\equal{}0} x^j_n\] for $ n \equal{} 1, 2, 3, \ldots$ Prove that $ \forall n,$ \[ 2 \minus{} \frac{1}{2^{n\minus{}1}} \leq x_n < 2 \minus{} \frac{1}{2^n}.\]
$f(x) = x^3+3x^2 - 1$. Find the number of real solutions of $f(f(x)) = 0$.
Let $P(x)=x^6-x^5-x^3-x^2-x$ and $a,b,c$ and $d$ be the roots of the equation $x^4-x^3-x^2-1=0$, then determine the value of $P(a)+P(b)+P(c)+P(d)$
(A)$5$
(B)$6$
(C)$7$
(D)$8$
If $p(x)$ is a polynomial with integer coeffcients, let $q(x) = \frac{p(x)}{x(1-x)}$ . If $q(x) = q\left(\frac{1}{1-x}\right)$ for every $x \ne 0$, and $p(2) = -7, p(3) = -11$, find $p(10)$.
Let $P$ and $Q$ be polynomials with integer coefficients. Suppose that the integers $a$ and $a+1997$ are roots of $P$, and that $Q(1998)=2000$. Prove that the equation $Q(P(x))=1$ has no integer solutions.
Let $P(x)$ be a nonconstant polynomial of degree $n$ with rational coefficients which can not be presented as a product of two nonconstant polynomials with rational coefficients. Prove that the number of polynomials $Q(x)$ of degree less than $n$ with rational coefficients such that $P(x)$ divides $P(Q(x))$
a) is finite
b) does not exceed $n$.
Find all pairs of polynomials $p(x),q(x)\in R[x]$ satisfying the equality $p(x^2)=p(x)q(1-x)+p(1-x)q(x)$ for all real $x$.
I.Voronovich
The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$. Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$, where $m$, $n$ are relatively prime integers. Find $m + n$.
A liquid is moving in an infinite pipe. For each molecule if it is at point with coordinate $x$ then after $t$ seconds it will be at a point of $p(t,x)$. Prove that if $p(t,x)$ is a polynomial of $t,x$ then speed of all molecules are equal and constant.
A polynomial $P(x)$ with integer coefficients and a positive integer $a>1$, are such that for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$. Find all such pairs of $(P(x),a)$.
[b]p1.[/b] Farmer James goes to Kristy’s Krispy Chicken to order a crispy chicken sandwich. He can choose from $3$ types of buns, $2$ types of sauces, $4$ types of vegetables, and $4$ types of cheese. He can only choose one type of bun and cheese, but can choose any nonzero number of sauces, and the same with vegetables. How many different chicken sandwiches can Farmer James order?
[b]p2.[/b] A line with slope $2$ and a line with slope $3$ intersect at the point $(m, n)$, where $m, n > 0$. These lines intersect the $x$ axis at points $A$ and $B$, and they intersect the y axis at points $C$ and $D$. If $AB = CD$, find $m/n$.
[b]p3.[/b] A multi-set of $11$ positive integers has a median of $10$, a unique mode of $11$, and a mean of $ 12$. What is the largest possible number that can be in this multi-set? (A multi-set is a set that allows repeated elements.)
[b]p4.[/b] Farmer James is swimming in the Eggs-Eater River, which flows at a constant rate of $5$ miles per hour, and is recording his time. He swims $ 1$ mile upstream, against the current, and then swims $1$ mile back to his starting point, along with the current. The time he recorded was double the time that he would have recorded if he had swum in still water the entire trip. To the nearest integer, how fast can Farmer James swim in still water, in miles per hour?
[b]p5.[/b] $ABCD$ is a square with side length $60$. Point $E$ is on $AD$ and $F$ is on $CD$ such that $\angle BEF = 90^o$. Find the minimum possible length of $CF$.
[b]p6.[/b] Farmer James makes a trianglomino by gluing together $5$ equilateral triangles of side length $ 1$, with adjacent triangles sharing an entire edge. Two trianglominoes are considered the same if they can be matched using only translations and rotations (but not reflections). How many distinct trianglominoes can Farmer James make?
[b]p7.[/b] Two real numbers $x$ and $y$ satisfy $x^2 - y^2 = 2y - 2x$ , and $x + 6 = y^2 + 2y$. What is the sum of all possible values of$ y$?
[b]p8.[/b] Let $N$ be a positive multiple of $840$. When $N$ is written in base $6$, it is of the form $\overline{abcdef}_6$ where $a, b, c, d, e, f$ are distinct base $6$ digits. What is the smallest possible value of $N$, when written in base $6$?
[b]p9.[/b] For $S = \{1, 2,..., 12\}$, find the number of functions $f : S \to S$ that satisfy the following $3$ conditions:
(a) If $n$ is divisible by $3$, $f(n)$ is not divisible by $3$,
(b) If $n$ is not divisible by $3$, $f(n)$ is divisible by $3$, and
(c) $f(f(n)) = n$ holds for exactly $8$ distinct values of $n$ in $S$.
[b]p10.[/b] Regular pentagon $JAMES$ has area $ 1$. Let $O$ lie on line $EM$ and $N$ lie on line $MA$ so that $E, M, O$ and $M, A, N$ lie on their respective lines in that order. Given that $MO = AN$ and $NO = 11 \cdot ME$, find the area of $NOM$.
[b]p11.[/b] Hen Hao is flipping a special coin, which lands on its sunny side and its rainy side each with probability $1/2$. Hen Hao flips her coin ten times. Given that the coin never landed with its rainy side up twice in a row, find the probability that Hen Hao’s last flip had its sunny side up.
[b]p12.[/b] Find the product of all integer values of a such that the polynomial $x^4 + 8x^3 + ax^2 + 2x - 1$ can be factored into two non-constant polynomials with integer coefficients.
[b]p13.[/b] Isosceles trapezoid $ABCD$ has $AB = CD$ and $AD = 6BC$. Point $X$ is the intersection of the diagonals $AC$ and $BD$. There exist a positive real number $k$ and a point $P$ inside $ABCD$ which satisfy
$$[PBC] : [PCD] : [PDA] = 1 : k : 3,$$
where $[XYZ]$ denotes the area of triangle $XYZ$. If $PX \parallel AB$, find the value of $k$.
[b]p14.[/b] How many positive integers $n < 1000$ are there such that in base $10$, every digit in $3n$ (that isn’t a leading zero) is greater than the corresponding place value digit (possibly a leading zero) in $n$? For example, $n = 56$, $3n = 168$ satisfies this property as $1 > 0$, $6 > 5$, and $8 > 6$. On the other hand, $n = 506$, $3n = 1518$ does not work because of the hundreds place.
[b]p15.[/b] Find the greatest integer that is smaller than $$\frac{2018}{37^2}+\frac{2018}{39^2}+ ... +\frac{2018}{
107^2}.$$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Find all polynomials $P(x)$ satisfying $P(x+1)-2P(x)+P(x-1)= x$ for all $x$
Find all values of the positive integer $ m$ such that there exists polynomials $ P(x),Q(x),R(x,y)$ with real coefficient satisfying the condition: For every real numbers $ a,b$ which satisfying $ a^m-b^2=0$, we always have that $ P(R(a,b))=a$ and $ Q(R(a,b))=b$.
Let $b$ be a number with $-2 < b < 0$. Prove that there exists a positive integer $n$ such that all the coefficients of the polynomial $(x + 1)^n(x^2 + bx + 1)$ are positive.
Let $P$ be a polynomial of degree greater than or equal to $4$ with integer coefficients. An integer $x$ is called $P$-[i]representable[/i] if there exists integer numbers $a$ and $b$ such that $x = P(a) - P(b)$. Prove that, if for all $N \geq 0$, more than half of the integers of the set $\{0,1,\dots,N\}$ are $P$-[i]representable[/i], then all the even integers are $P$-[i]representable[/i] or all the odd integers are $P$-[i]representable[/i].
Find out all the integer pairs $(m,n)$ such that there exist two monic polynomials $P(x)$ and $Q(x)$ ,with $\deg{P}=m$ and $\deg{Q}=n$,satisfy that $$P(Q(t))\not=Q(P(t))$$ holds for any real number $t$.