Let $p$ be a prime and $k$ a positive integer such that $k \le p$. We know that $f(x)$ is a polynomial in $\mathbb Z[x]$ such that for all $x \in \mathbb{Z}$ we have $p^k | f(x)$. [b](a)[/b] Prove that there exist polynomials $A_0(x),\ldots,A_k(x)$ all in $\mathbb Z[x]$ such that \[ f(x)=\sum_{i=0}^{k} (x^p-x)^ip^{k-i}A_i(x),\] [b](b)[/b] Find a counter example for each prime $p$ and each $k > p$.

Let $x_0=2024^{2024}$ and $x_{n+1}=|x_n-\pi|$ for $n \ge 0$. Show that there exists a value of $n$ such that $x_{n+2}=x_n$.

Let $a_0,a_1,\dots,a_{2024}$ be real numbers such that $\left|a_{i+1}-a_i\right| \le 1$ for $i=0,1,\dots,2023$. a) Find the minimum possible value of $$a_0a_1+a_1a_2+\dots+a_{2023}a_{2024}$$ b) Does there exist a real number $C$ such that $$a_0a_1-a_1a_2+a_2a_3-a_3a_4+\dots+a_{2022}a_{2023}-a_{2023}a_{2024} \ge C$$ for all real numbers $a_0,a_1,\dots,a_2024$ such that $\left|a_{i+1}-a_i\right| \le 1$ for $i=0,1,\dots,2023$.

Given a polynomial of degree $2$, $p(x) = ax^2 +bx+c$ define the function $$S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2.$$ Determine the real number$ r$such that, for any polynomial $p(x)$ of degree $2$ with real roots, holds $S(p) \ge ra^2$

The quadratic expression $ax^2+bx+c$ is the $4$-th power (of an integer) for any integer $x$. Prove that $a = b = 0$.

Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality: $$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$ [i]Proposed by Mohammad Jafari[/i]

Find all non-negative real numbers $a$ such that the equation \[ \sqrt[3]{1+x}+\sqrt[3]{1-x}=a \] has at least one real solution $x$ with $0 \leq x \leq 1$. For all such $a$, what is $x$?

Prove that for $ n > 0$ and $ a\neq0$ the polynomial $ p(z) \equal{} az^{2n \plus{} 1} \plus{} bz^{2n} \plus{} \bar bz \plus{} \bar a$ has a root on unit circle

[b]p5.[/b] Find the number of three-digit integers that contain at least one $0$ or $5$. The leading digit of the three-digit integer cannot be zero. [b]p6.[/b] What is the sum of the solutions to $\frac{x+8}{5x+7} =\frac{x+8}{7x+5}$ [b]p7.[/b] Let $BC$ be a diameter of a circle with center $O$ and radius $4$. Point $A$ is on the circle such that $\angle AOB = 45^o$. Point $D$ is on the circle such that line segment$ OD$ intersects line segment $AC$ at $E$ and $OD$ bisects $\angle AOC$. Compute the area of $ADE$, which is enclosed by line segments $AE$ and $ED$ and minor arc $AD$. [b]p8. [/b] William is a bacteria farmer. He would like to give his fiance$ 2021$ bacteria as a wedding gift. Since he is an intelligent and frugal bacteria farmer, he would like to add the least amount of bacteria on his favorite infinite plane petri dish to produce those $2021$ bacteria. The infinite plane petri dish starts off empty and William can add as many bacteria as he wants each day. Each night, all the bacteria reproduce through binary fission, splitting into two. If he has infinite amount of time before his wedding day, how many bacteria should he add to the dish in total to use the least number of bacteria to accomplish his nuptial goals? PS. You should use hide for answers Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$(a)$ Count the number of roots of $\omega$ of the equation $z^{2019} - 1 = 0 $ over complex numbers that satisfy \begin{align*} \vert \omega + 1 \vert \geq \sqrt{2 + \sqrt{2}} \end{align*} $(b)$ Find all real numbers $x$ that satisfy following equation $:$ \begin{align*} \frac{ 8^x + 27^x }{ 12^x + 18^x } = \frac{7}{6} \end{align*}

Determine all positive integers $n$ for which there exists a polynomial $p(x)$ of degree $n$ with integer coefficients such that it takes the value $n$ in $n$ distinct integer points and takes the value $0$ at point $0$.

For all positive numbers $x,y,z$ the product $(x+y+z)^{-1}(x^{-1}+y^{-1}+z^{-1})(xy+yz+xz)^{-1}[(xy)^{-1}+(yz)^{-1}+(xz)^{-1}]$ equals $\text{(A)}\ x^{-2}y^{-2}z^{-2} \qquad \text{(B)}\ x^{-2}+y^{-2}+z^{-2} \qquad \text{(C)}\ (x+y+z)^{-1}$ $\text{(D)}\ \frac{1}{xyz} \qquad \text{(E)}\ \frac{1}{xy+yz+xz}$

[u]Round 5[/u] [b]p13.[/b] For a square pyramid whose base has side length $9$, a square is formed by connecting the centroids of the four triangular faces. What is the area of the square formed by the centroids? [b]p14.[/b] Farley picks a real number p uniformly at random in the range $\left( \frac13, \frac23 \right)$. She then creates a special coin that lands on heads with probability $p$ and tails with probability $1 - p$. She flips this coin, and it lands on heads. What is the probability that $p > \frac12$? [b]p15.[/b] Let $ABCD$ be a quadrilateral with $\angle A = \angle C = 90^o$. Extend $AB$ and $CD$ to meet at point $P$. Given that $P B = 3$, $BA = 21$, and $P C = 1$, find $BD^2$ [u]Round 6[/u] [b]p16.[/b] Three congruent, mutually tangent semicircles are inscribed in a larger semicircle, as shown in the diagram below. If the larger semicircle has a radius of $30$ units, what is the radius of one of the smaller semicircles? [img]https://cdn.artofproblemsolving.com/attachments/5/e/1b73791e95dc4ed6342f0151f3f63e1b31ae3c.png[/img] [b]p17.[/b] In isosceles trapezoid $ABCD$ with $BC \parallel AD$, the distances from $A$ and $B$ to line $CD$ are $3$ and $9$, respectively. If the distance between the two bases of trapezoid $ABCD$ is $5$, find the area of quadrilateral $ABCD$. [b]p18.[/b] How many ways are there to tile the “$E$” shape below with dominos? A domino covers two adjacent squares. [img]https://cdn.artofproblemsolving.com/attachments/b/b/82bdb8d8df8bc3d00b9aef9eb39e55358c4bc6.png[/img] [u]Round 7[/u] [b]p19.[/b] In isoceles triangle $ABC$, $AC = BC$ and $\angle ACB = 20^o$. Let $\Omega$ be the circumcircle of triangle $ABC$ with center $O$, and let $M$ be the midpoint of segment $BC$. Ray $\overrightarrow{OM}$ intersects $\Omega$ at $D$. Let $\omega$ be the circle with diameter $OD$. $AD$ intersects $\omega$ again at a point $X$ not equal to $D$. Given $OD = 2$, find the area of triangle $OXD$. [b]p20.[/b] Find the smallest odd prime factor of $2023^{2029} + 2026^{2029} - 1$. [b]p21.[/b] Achyuta, Alan, Andrew, Anish, and Ava are playing in the EMCC games. Each person starts with a paper with their name taped on their back. A person is eliminated from the game when anybody rips their paper off of their back. The game ends when one person remains. The remaining person then rips their paper off of their own back. At the end of the game, each person collects the papers that they ripped off. How many distinct ways can the papers be distributed at the end of the game? [u]Round 8[/u] [b]p22.[/b] Anthony has three random number generators, labelled $A$, $B$ and $C$. $\bullet$ Generator$ A$ returns a random number from the set $\{12, 24, 36, 48, 60\}$. $\bullet$ Generator $B$ returns a random number from the set $ \{15, 30, 45, 60\}$. $\bullet$ Generator $C$ returns a random number from the set $\{20, 40, 60\}$. He uses generator $A$, $B$, and then $C$ in succession, and then repeats this process indefinitely. Anthony keeps a running total of the sum of all previously generated numbers, writing down the new total every time he uses a generator. After he uses each machine $10 $ times, what is the average number of multiples of $60$ that Anthony will have written down? [b]p23.[/b] A laser is shot from one of the corners of a perfectly reflective room shaped like an equilateral triangle. The laser is reflected 2497 times without shining into a corner of the room, but after the 2497th reflection, it shines directly into the corner it started from. How many different angles could the laser have been initially pointed? [b]p24.[/b] We call a k-digit number blissful if the number of positive integers $n$ such that $n^n$ ends in that $k$-digit number happens to be nonzero and finite. What is the smallest value of $k$ such that there exists a blissful $k$-digit number? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3131523p28369592]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If the arithmetic mean of $a$ and $b$ is double their geometric mean, with $a>b>0$, then a possible value for the ratio $\frac{a}{b}$, to the nearest integer, is $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ \text{none of these} $

For each integer $k\geq 2$, determine all infinite sequences of positive integers $a_1$, $a_2$, $\ldots$ for which there exists a polynomial $P$ of the form \[ P(x)=x^k+c_{k-1}x^{k-1}+\dots + c_1 x+c_0, \] where $c_0$, $c_1$, \dots, $c_{k-1}$ are non-negative integers, such that \[ P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k} \] for every integer $n\geq 1$.

Let $p(x)$ be a polynomial with integer coefficients, and let $a, b$ and $c$ be three distinct integers. Show that it is not possible to have $p(a) = b$, $p(b) = c$, and $p(c) = a$.

An arrangement of 12 real numbers in a row is called [i]good[/i] if for any four consecutive numbers the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. How many good arrangements are there in which the first and last numbers are 1, and the second number is the same as the third?

Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.

The positive numbers $a, b, c$ and $d$ satisfy $a\le b\le c\le d$ and $a + b + c + d \le 1$ . Prove that $a^2 + 3b^2 + 5c^2 + 7 d^2 \ge 1$ .

We say that a set $S$ of positive integers is special when there exists a function $f : \mathbb{N}\to \mathbb{N}$ satisfying that: $\bullet$ $f(k)\in S, \forall k\in\mathbb{N}$ $\bullet$ No integer $k$ with $2\le k \le 2021$ can be written as $\frac{af(a)}{bf(b)}$ with $a,b\in \mathbb{N}$ Find the smallest positive integer $n$ such that the set $S = \{ 1, 2021, 2021^2, \ldots , 2021^n \}$ is special or prove that no such integer exists. Note: $\mathbb{N}$ represents the set $\{ 1, 2, 3, \ldots \}$

a) $P(x),R(x)$ are polynomials with rational coefficients and $P(x)$ is not the zero polynomial. Prove that there exist a non-zero polynomial $Q(x)\in\mathbb Q[x]$ that \[P(x)\mid Q(R(x)).\] b) $P,R$ are polynomial with integer coefficients and $P$ is monic. Prove that there exist a monic polynomial $Q(x)\in\mathbb Z[x]$ that \[P(x)\mid Q(R(x)).\]

The roots of the equation $x^2+4x-5 = 0$ are also the roots of the equation $2x^3+9x^2-6x-5 = 0$. What is the third root of the second equation?

Prove that for all positive real numbers $a,b,c$ the following inequality holds \[ \left( \frac ab + \frac bc + \frac ca \right)^2 \geq \frac 32 \cdot \left ( \frac{a+b}c + \frac{b+c}a + \frac{c+a} b \right) . \]

Determine all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the set \[\left \{ \frac{f(x)}{x}: x \neq 0 \textnormal{ and } x \in \mathbb{R}\right \}\] is finite, and for all $x \in \mathbb{R}$ \[f(x-1-f(x)) = f(x) - x - 1\]

A beetle crawls along each of two intersecting straight lines at constant speeds, without changing direction. It is known that projections of the beetles on the $OX$ axis never coincide (neither in the past nor in the future). Prove that the projections of the beetles on the $OY$ axis will necessarily coincide or have coincided before. [hide=oroginal wording] По каждой из двух пересекающихся прямых с постоянными скоростями, не меняя направления, ползет по жуку. Известно, что проекции жуков на ось OX никогда не совпадают (ни в прошлом, ни в будущем). Докажите, что проекции жуков на ось OY обязательно совпадут или совпадали раньше.[/hide]