Found problems: 4275
Let be two polynoms $ P,Q\in\mathbb{C} [X] $ with degree at least $ 1, $ and such that $ P $ has only simple roots. Prove that the following affirmations are equivalent:
$ \text{(i)} P\circ Q $ is divisible by $ P. $
$ \text{(ii)} $ The evaluation of $ Q $ at any root of $ P $ is a root of $ P. $
[i]Marcel Țena[/i]
Find all integer $n$ such that $n-5$ divide $n^2+n-27$.
Let $P$ be a polynomial with integer coefficients. Assume that there exists a positive integer $n$ with $P(n^2)=2022$. Prove that there cannot be a positive rational number $r$ with $P(r^2)=2024$.
Let $p > 3$ be a prime number, and let $F_p$ denote the (fi nite) set of residue classes modulo $p$.
Let $S_d$ denote the set of $2$-variable polynomials $P(x, y)$ with coefficients in $F_p$, total degree $\le d$, and satisfying $P(x, y) = P(y,- x -y)$. Show that $$|S_d| = p^{\lceil (d+1)(d+2)/6 \rceil}$$.
[i]The total degree of a $2$-variable polynomial $P(x, y)$ is the largest value of $i + j$ among monomials $x^iy^j$
[/i] appearing in $P$.
Sofiya and Marquis are playing a game. Sofiya announces to Marquis that she's thinking of a polynomial of the form $f(x)=x^3+px+q$ with three integer roots that are not necessarily distinct. She also explains that all of the integer roots have absolute value less than (and not equal to) $N$, where $N$ is some fixed number which she tells Marquis. As a "move" in this game, Marquis can ask Sofiya about any number $x$ and Sofiya will tell him whether $f(x)$ is positive negative, or zero. Marquis's goal is to figure out Sofiya's polynomial.
If $N=3\cdot 2^k$ for some positive integer $k$, prove that there is a strategy which allows Marquis to identify the polynomial after making at most $2k+1$ "moves".
Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[
\left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}.
\] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$
Let $Q$ be a quadriatic polynomial having two different real zeros. Prove that there is a non-constant monic polynomial $P$ such that all coefficients of the polynomial $Q(P(x))$ except the leading one are (by absolute value) less than $0.001$.
[b]p1.[/b] Find the smallest positive integer which is $1$ more than multiple of $3$, $2$ more than a multiple of $4$, and $4$ more than a multiple of $7$.
[b]p2.[/b] Let $p = 4$, and let $a =\sqrt1$, $b =\sqrt2$, $c =\sqrt3$, $...$. Compute the value of $(p-a)(p-b) ... (p-z)$.
[b]p3.[/b] There are $6$ points on the circumference of a circle. How many convex polygons are there having vertices on these points?
[b]p4.[/b] David and I each have a sheet of computer paper, mine evenly spaced by $19$ parallel lines into $20$ sections, and his evenly spaced by $29$ parallel lines into $30$ sections. If our two sheets are overlayed, how many pairs of lines are perfectly incident?
[b]p5.[/b] A pyramid is created by stacking equilateral triangles of balls, each layer having one fewer ball per side than the triangle immediately beneath it. How many balls are used if the pyramid’s base has $5$ balls to a side?
[b]p6.[/b] Call a positive integer $n$ good if it has $3$ digits which add to $4$ and if it can be written in the form $n = k^2$, where $k$ is also a positive integer. Compute the average of all good numbers.
[b]p7.[/b] John’s birthday cake is a scrumptious cylinder of radius $6$ inches and height $3$ inches. If his friends cut the cake into $8$ equal sectors, what is the total surface area of a piece of birthday cake?
[b]p8.[/b] Evaluate $\sum^{10}_{i=1}\sum^{10}_{j=1} ij$.
[b]p9.[/b] If three numbers $a$, $b$, and $c$ are randomly selected from the interval $[-2, 2]$, what is the probability that $a^2 + b^2 + c^2 \ge 4$?
[b]p10.[/b] Evaluate $\sum^{\infty}_{x=2} \frac{2}{x^2 - 1}.$
[b]p11.[/b] Consider $4x^2 - kx - 1 = 0$. If the roots of this polynomial are $\sin \theta$ and $\cos \theta$, compute $|k|$.
[b]p12.[/b] Given that $65537 = 2^{16} + 1$ is a prime number, compute the number of primes of the form $2^n + 1$ (for $n \ge 0$) between $1$ and $10^6$.
[b]p13.[/b] Compute $\sin^{-1}(36/85) + \cos^{-1}(4/5) + \cos^{-1}(15/17).$
[b]p14.[/b] Find the number of integers $n$, $1\le n \le 2003$, such that $n^{2003} - 1$ is a multiple of $10$.
[b]p15.[/b] Find the number of integers $n,$ $1 \le n \le 120$, such that $n^2$ leaves remainder $1$ when divided by $120$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Determine all real polynomials $p$ such that $p(x+p(x))=x^2p(x)$ for all $x$.
We say that an integer $m$ is a perfect power if there are $a\in\mathbf{Z}$, $b\in\mathbf{N}$ with $b > 1$ such that $m = a^b$.
Find all polynomials $P\in\mathbf{Z}[x]$ such that $P(n)$ is a perfect power for every $n\in\mathbf{N}$.
[b]p1.[/b] Find the sum of the seventeenth powers of the seventeen roots of the seventeeth degree polynomial equation $x^{17} - 17x + 17 = 0$.
[b]p2.[/b] Four identical spherical cows, each of radius $17$ meters, are arranged in a tetrahedral pyramid (their centers are the vertices of a regular tetrahedron, and each one is tangent to the other three). The pyramid of cows is put on the ground, with three of them laying on it. What is the distance between the ground and the top of the topmost cow?
[b]p3.[/b] If $a_n$ is the last digit of $\sum^{n}_{i=1} i$, what would the value of $\sum^{1000}_{i=1}a_i$ be?
[b]p4.[/b] If there are $15$ teams to play in a tournament, $2$ teams per game, in how many ways can the tournament be organized if each team is to participate in exactly $5$ games against dierent opponents?
[b]p5.[/b] For $n = 20$ and $k = 6$, calculate $$2^k {n \choose 0}{n \choose k}- 2^{k-1}{n \choose 1}{{n - 1} \choose {k - 1}} + 2^{k-2}{n \choose 2}{{n - 2} \choose {k - 2}} +...+ (-1)^k {n \choose k}{{n - k} \choose 0}$$ where ${n \choose k}$ is the number of ways to choose $k$ things from a set of $n$.
[b]p6.[/b] Given a function $f(x) = ax^2 + b$, with a$, b$ real numbers such that $$f(f(f(x))) = -128x^8 + \frac{128}{3}x^6 - \frac{16}{22}x^2 +\frac{23}{102}$$ , find $b^a$.
[b]p7.[/b] Simplify the following fraction $$\frac{(2^3-1)(3^3-1)...(100^3-1)}{(2^3+1)(3^3+1)...(100^3+1)}$$
[b]p8.[/b] Simplify the following expression
$$\frac{\sqrt{3 + \sqrt5} + \sqrt{3 - \sqrt5}}{\sqrt{3 - \sqrt8}} -\frac{4}{ \sqrt{8 - 2\sqrt{15}}}$$
[b]p9.[/b] Suppose that $p(x)$ is a polynomial of degree $100$ such that $p(k) = k2^{k-1}$ , $k =1, 2, 3 ,... , 100$. What is the value of $p(101)$ ?
[b]p10. [/b] Find all $17$ real solutions $(w, x, y, z)$ to the following system of equalities:
$$ 2w + w^2x = x$$
$$ 2x + x^2y=y $$
$$ 2y + y^2z=z $$
$$ -2z+z^2w=w $$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial
\[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n.
\]
Find all pairs $(a, b)$ of real numbers such that the roots of polynomials $6x^2 -24x -4a$ and $x^3 + ax^2 + bx - 8$ are all non-negative real numbers.
For polynomials $P(x) = a_nx^n + \cdots + a_0$, let $f(P) = a_n\cdots a_0$ be the product of the coefficients of $P$. The polynomials $P_1,P_2,P_3,Q$ satisfy $P_1(x) = (x-a)(x-b)$, $P_2(x) = (x-a)(x-c)$, $P_3(x) = (x-b)(x-c)$, $Q(x) = (x-a)(x-b)(x-c)$ for some complex numbers $a,b,c$. Given $f(Q) = 8$, $f(P_1) + f(P_2) + f(P_3) = 10$, and $abc>0$, find the value of $f(P_1)f(P_2)f(P_3)$.
[i]Proposed by Justin Hsieh[/i]
find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.
Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$.
[i]Proposed by Mohammad Ahmadi[/i]
Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations:
\[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.
[u]Set 7[/u]
[b]p19.[/b] Let circles $\omega_1$ and $\omega_2$, with centers $O_1$ and $O_2$, respectively, intersect at $X$ and $Y$ . A lies on $\omega_1$ and $B$ lies on $\omega_2$ such that $AO_1$ and $BO_2$ are both parallel to $XY$, and $A$ and $B$ lie on the same side of $O_1O_2$. If $XY = 60$, $\angle XAY = 45^o$, and $\angle XBY = 30^o$, then the length of $AB$ can be expressed in the form $\sqrt{a - b\sqrt2 + c\sqrt3}$, where $a, b, c$ are positive integers. Determine $a + b + c$.
[b]p20.[/b] If $x$ is a positive real number such that $x^{x^2}= 2^{80}$, find the largest integer not greater than $x^3$.
[b]p21.[/b] Justin has a bag containing $750$ balls, each colored red or blue. Sneaky Sam takes out a random number of balls and replaces them all with green balls. Sam notices that of the balls left in the bag, there are $15$ more red balls than blue balls. Justin then takes out $500$ of the balls chosen randomly. If $E$ is the expected number of green balls that Justin takes out, determine the greatest integer less than or equal to $E$.
[u]Set 8[/u]
These three problems are interdependent; each problem statement in this set will use the answers to the other two problems in this set. As such, let the positive integers $A, B, C$ be the answers to problems $22$, $23$, and $24$, respectively, for this set.
[b]p22.[/b] Let $WXYZ$ be a rectangle with $WX =\sqrt{5B}$ and $XY =\sqrt{5C}$. Let the midpoint of $XY$ be $M$ and the midpoint of $YZ$ be $N$. If $XN$ and $W Y$ intersect at $P$, determine the area of $MPNY$ .
[b]p23.[/b] Positive integers $x, y, z$ satisfy $$xy \equiv A \,\, (mod 5)$$
$$yz \equiv 2A + C\,\, (mod 7)$$
$$zx \equiv C + 3 \,\, (mod 9).$$ (Here, writing $a \equiv b \,\, (mod m)$ is equivalent to writing $m | a - b$.)
Given that $3 \nmid x$, $3 \nmid z$, and $9 | y$, find the minimum possible value of the product $xyz$.
[b]p24.[/b] Suppose $x$ and $y$ are real numbers such that $$x + y = A$$
$$xy =\frac{1}{36}B^2.$$ Determine $|x - y|$.
[u]Set 9[/u]
[b]p25. [/b]The integer $2017$ is a prime which can be uniquely represented as the sum of the squares of two positive integers: $$9^2 + 44^2 = 2017.$$ If $N = 2017 \cdot 128$ can be uniquely represented as the sum of the squares of two positive integers $a^2 +b^2$, determine $a + b$.
[b]p26.[/b] Chef Celia is planning to unveil her newest creation: a whole-wheat square pyramid filled with maple syrup. She will use a square flatbread with a one meter diagonal and cut out each of the five polygonal faces of the pyramid individually. If each of the triangular faces of the pyramid are to be equilateral triangles, the largest volume of syrup, in cubic meters, that Celia can enclose in her pyramid can be expressed as $\frac{a-\sqrt{b}}{c}$ where $a, b$ and $c$ are the smallest possible possible positive integers. What is $a + b + c$?
[b]p27.[/b] In the Cartesian plane, let $\omega$ be the circle centered at $(24, 7)$ with radius $6$. Points $P, Q$, and $R$ are chosen in the plane such that $P$ lies on $\omega$, $Q$ lies on the line $y = x$, and $R$ lies on the $x$-axis. The minimum possible value of $PQ+QR+RP$ can be expressed in the form $\sqrt{m}$ for some integer $m$. Find m.
[u]Set 10[/u]
[i]Deja vu?[/i]
[b]p28. [/b] Let $ABC$ be a triangle with incircle $\omega$. Let $\omega$ intersect sides $BC$, $CA$, $AB$ at $D, E, F$, respectively. Suppose $AB = 7$, $BC = 12$, and $CA = 13$. If the area of $ABC$ is $K$ and the area of $DEF$ is $\frac{m}{n}\cdot K$, where $m$ and $n$ are relatively prime positive integers, then compute $m + n$.
[b]p29.[/b] Sebastian is playing the game Split! again, but this time in a three dimensional coordinate system. He begins the game with one token at $(0, 0, 0)$. For each move, he is allowed to select a token on any point $(x, y, z)$ and take it off, replacing it with three tokens, one at $(x + 1, y, z)$, one at $(x, y + 1, z)$, and one at $(x, y, z + 1)$ At the end of the game, for a token on $(a, b, c)$, it is assigned a score $\frac{1}{2^{a+b+c}}$ . These scores are summed for his total score. If the highest total score Sebastian can get in $100$ moves is $m/n$, then determine $m + n$.
[b]p30.[/b] Determine the number of positive $6$ digit integers that satisfy the following properties:
$\bullet$ All six of their digits are $1, 5, 7$, or $8$,
$\bullet$ The sum of all the digits is a multiple of $5$.
[u]Set 11[/u]
[b]p31.[/b] The triangular numbers are defined as $T_n =\frac{n(n+1)}{2}$. We also define $S_n =\frac{n(n+2)}{3}$. If the sum $$\sum_{i=16}^{32} \left(\frac{1}{T_i}+\frac{1}{S_i}\right)= \left(\frac{1}{T_{16}}+\frac{1}{S_{16}}\right)+\left(\frac{1}{T_{17}}+\frac{1}{S_{17}}\right)+...+\left(\frac{1}{T_{32}}+\frac{1}{S_{32}}\right)$$ can be written in the form $a/b$ , where $a$ and $b$ are positive integers with $gcd(a, b) = 1$, then find $a + b$.
[b]p32.[/b] Farmer Will is considering where to build his house in the Cartesian coordinate plane. He wants to build his house on the line $y = x$, but he also has to minimize his travel time for his daily trip to his barnhouse at $(24, 15)$ and back. From his house, he must first travel to the river at $y = 2$ to fetch water for his animals. Then, he heads for his barnhouse, and promptly leaves for the long strip mall at the line $y =\sqrt3 x$ for groceries, before heading home. If he decides to build his house at $(x_0, y_0)$ such that the distance he must travel is minimized, $x_0$ can be written in the form $\frac{a\sqrt{b}-c}{d}$ , where $a, b, c, d$ are positive integers, $b$ is not divisible by the square of a prime, and $gcd(a, c, d) = 1$. Compute $a+b+c+d$.
[b]p33.[/b] Determine the greatest positive integer $n$ such that the following two conditions hold:
$\bullet$ $n^2$ is the difference of consecutive perfect cubes;
$\bullet$ $2n + 287$ is the square of an integer.
[u]Set 12[/u]
The answers to these problems are nonnegative integers that may exceed $1000000$. You will be awarded points as described in the problems.
[b]p34.[/b] The “Collatz sequence” of a positive integer n is the longest sequence of distinct integers $(x_i)_{i\ge 0}$ with $x_0 = n$ and $$x_{n+1} =\begin{cases} \frac{x_n}{2} & if \,\, x_n \,\, is \,\, even \\ 3x_n + 1 & if \,\, x_n \,\, is \,\, odd \end{cases}.$$ It is conjectured that all Collatz sequences have a finite number of elements, terminating at $1$. This has been confirmed via computer program for all numbers up to $2^{64}$. There is a unique positive integer $n < 10^9$ such that its Collatz sequence is longer than the Collatz sequence of any other positive integer less than $10^9$. What is this integer $n$?
An estimate of $e$ gives $\max\{\lfloor 32 - \frac{11}{3}\log_{10}(|n - e| + 1)\rfloor, 0\}$ points.
[b]p35.[/b] We define a graph $G$ as a set $V (G)$ of vertices and a set $E(G)$ of distinct edges connecting those vertices. A graph $H$ is a subgraph of $G$ if the vertex set $V (H)$ is a subset of $V (G)$ and the edge set $E(H)$ is a subset of $E(G)$. Let $ex(k, H)$ denote the maximum number of edges in a graph with $k$ vertices without a subgraph of $H$. If $K_i$ denotes a complete graph on $i$ vertices, that is, a graph with $i$ vertices and all ${i \choose 2}$ edges between them present, determine $$n =\sum_{i=2}^{2018} ex(2018, K_i).$$
An estimate of $e$ gives $\max\{\lfloor 32 - 3\log_{10}(|n - e| + 1)\rfloor, 0\}$ points.
[b]p36.[/b] Write down an integer between $1$ and $100$, inclusive. This number will be denoted as $n_i$ , where your Team ID is $i$. Let $S$ be the set of Team ID’s for all teams that submitted an answer to this problem. For every ordered triple of distinct Team ID’s $(a, b, c)$ such that a, b, c ∈ S, if all roots of the polynomial $x^3 + n_ax^2 + n_bx + n_c$ are real, then the teams with ID’s $a, b, c$ will each receive one virtual banana.
If you receive $v_b$ virtual bananas in total and $|S| \ge 3$ teams submit an answer to this problem, you will be awarded $$\left\lfloor \frac{32v_b}{3(|S| - 1)(|S| - 2)}\right\rfloor$$ points for this problem. If $|S| \le 2$, the team(s) that submitted an answer to this problem will receive $32$ points for this problem.
PS. You had better use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c4h2777264p24369138]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
[color=darkred] Let $m$ and $n$ be two nonzero natural numbers. Determine the minimum number of distinct complex roots of the polynomial $\prod_{k=1}^m\, (f+k)$ , when $f$ covers the set of $n^{\text{th}}$ - degree polynomials with complex coefficients.
[/color]
Let $P(x)$ be a real polynomial with $P(x) \ge 0$ for $0 \le x \le 1$. Show that there exist polynomials $P_i (x) (i = 0, 1,2)$ with $P_i (x) \ge 0$ for all real x such that $P (x) = P_0 (x) + xP_1 (x)( 1- x)P_2 (x)$.
Let $p$ be a prime and let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Assume that the numbers $f(1),f(2),\dots,f(p)$ leave exactly $k$ distinct remainders when divided by $p$, and $1<k<p$. Prove that
\[ \frac{p-1}{d}\leq k-1\leq (p-1)\left(1-\frac1d \right) .\]
[i] Dániel Domán, Gauls Károlyi, and Emil Kiss [/i]
Suppose that the real numbers $a_{1},a_{2},...,a_{2002}$ satisfying
$\frac{a_{1}}{2}+\frac{a_{2}}{3}+...+\frac{a_{2002}}{2003}=\frac{4}{3}$
$\frac{a_{1}}{3}+\frac{a_{2}}{4}+...+\frac{a_{2002}}{2004}=\frac{4}{5}$
$...$
$\frac{a_{1}}{2003}+\frac{a_{2}}{2004}+...+\frac{a_{2002}}{4004}=\frac{4}{4005}$
Evaluate the sum $\frac{a_{1}}{3}+\frac{a_{2}}{5}+...+\frac{a_{2002}}{4005}$.
Let the roots $a,b,c$ of
$$f(x)=x^3 +p x^2 + qx+r$$
be real, and let $a\leq b\leq c$. Prove that $f'(x)$ has a root in the interval $\left[\frac{b+c}{2}, \frac{b+2c}{3}\right]$. What will be the form of $f(x)$ if the root in question falls at either end of the interval?
Let $\mathcal{F}$ be the set of polynomials $f(x)$ with integer coefficients for which there exists an integer root of the equation $f(x)=1$. For all $k>1$, let $m_k$ be the smallest integer greater than one for which there exists $f(x)\in \mathcal{F}$ such that $f(x)=m_k$ has exactly $k$ distinct integer roots. If the value of $\sqrt{m_{2021}-m_{2020}}$ can be written as $m\sqrt{n}$ for positive integers $m,n$ where $n$ is squarefree, compute the largest integer value of $k$ such that $2^k$ divides $\frac{m}{n}$.
Let $\displaystyle \mathcal K$ be a finite field such that the polynomial $\displaystyle X^2-5$ is irreducible over $\displaystyle \mathcal K$. Prove that:
(a) $1+1 \neq 0$;
(b) for all $\displaystyle a \in \mathcal K$, the polynomial $\displaystyle X^5+a$ is reducible over $\displaystyle \mathcal K$.
[i]Marian Andronache[/i]
[Edit $1^\circ$] I wanted to post it in "Superior Algebra - Groups, Fields, Rings, Ideals", but I accidentally put it here :blush: Can any mod move it? I'd be very grateful.
[Edit $2^\circ$] OK, thanks.