Define a sequence of integers by $a_0=1$ , and $a_n=\sum_{k=0}^{n-1} \binom{n}{k}a_k$ , $n \geq 1$ . Let $m$ be a positive integer , let $p$ be a prime , and let $q$ and $r$ be non-negative integers . Prove that : $$a_{p^mq+r} \equiv a_{p^{m-1}q+r} \pmod{p^m}$$

Determine all integers $ a$ and $ b$, so that $ (a^3\plus{}b)(a\plus{}b^3)\equal{}(a\plus{}b)^4$

Let $x$ and $y$ be real numbers such that \[ 2 < \frac{x - y}{x + y} < 5. \] If $\frac{x}{y}$ is an integer, what is its value?

Prove that if $ \cos \pi x =\frac{1}{3} $ then $ x $ is an irrational number.

Two circumferences, with the distance $d$ between centres, intersect in points $P$ and $Q$ . Two lines are drawn through the point $A$ on the first circumference ($Q\ne A\ne P$) and points $P$ and $Q$ . They intersect the second circumference in the points $B$ and $C$ . a) Prove that the radius of the circle, circumscribed around the triangle$ABC$ , equals $d$. b) Describe the set of the new circle's centres, if thepoint $A$ moves along all the first circumference.

$\frac{n(n + 1)}{2}$ distinct numbers are arranged at random into $n$ rows. The first row has $1$ number, the second has $2$ numbers, the third has $3$ numbers and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers.

[b]p1.[/b] Calculate $A \cdot B +M \cdot C$, where $A = 1$, $B = 2$, $C = 3$, $M = 13$. [b]p2.[/b] What is the remainder of $\frac{2022\cdot2023}{10}$ ? [b]p3.[/b] Daniel and Bryan are rolling fair $7$-sided dice. If the probability that the sum of the numbers that Daniel and Bryan roll is greater than $11$ can be represented as the fraction $\frac{a}{b}$ where $a$, $b$ are relatively prime positive integers, what is $a + b$? [b]p4.[/b] Billy can swim the breaststroke at $25$ meters per minute, the butterfly at $30$ meters per minute, and the front crawl at $40$ meters per minute. One day, he swam without stopping or slowing down, swimming $1130$ meters. If he swam the butterfly for twice as long as the breaststroke, plus one additional minute, and the front crawl for three times as long as the butterfly, minus eight minutes, for how many minutes did he swim? [b]p5.[/b] Elon Musk is walking around the circumference of Mars trying to find aliens. If the radius of Mars is $3396.2$ km and Elon Musk is $73$ inches tall, the difference in distance traveled between the top of his head and the bottom of his feet in inches can be expressed as $a\pi$ for an integer $a$. Find $a$. ($1$ yard is exactly $0.9144$ meters). [b]p6.[/b] Lukas is picking balls out of his five baskets labeled $1$,$2$,$3$,$4$,$5$. Each basket has $27$ balls, each labeled with the number of its respective basket. What is the least number of times Lukas must take one ball out of a random basket to guarantee that he has chosen at least $5$ balls labeled ”$1$”? If there are no balls in a chosen basket, Lukas will choose another random basket. [b]p7.[/b] Given $35_a = 42_b$, where positive integers $a$, $b$ are bases, find the minimum possible value of the sum $a + b$ in base $10$. [b]p8.[/b] Jason is playing golf. If he misses a shot, he has a $50$ percent chance of slamming his club into the ground. If a club is slammed into the ground, there is an $80$ percent chance that it breaks. Jason has a $40$ percent chance of hitting each shot. Given Jason must successfully hit five shots to win a prize, what is the expected number of clubs Jason will break before he wins a prize? [b]p9.[/b] Circle $O$ with radius $1$ is rolling around the inside of a rectangle with side lengths $5$ and $6$. Given the total area swept out by the circle can be represented as $a + b\pi$ for positive integers $a$, $b$ find $a + b$. [b]p10.[/b] Quadrilateral $ABCD$ has $\angle ABC = 90^o$, $\angle ADC = 120^o$, $AB = 5$, $BC = 18$, and $CD = 3$. Find $AD$. [b]p11.[/b] Raymond is eating huge burgers. He has been trained in the art of burger consumption, so he can eat one every minute. There are $100$ burgers to start with. However, at the end of every $20$ minutes, one of Raymond’s friends comes over and starts making burgers. Raymond starts with $1$ friend. If each of his friends makes $1$ burger every $20$ minutes, after how long in minutes will there be $0$ burgers left for the first time? [b]p12.[/b] Find the number of pairs of positive integers $(a, b)$ and $b\le a \le 2022$ such that $a\cdot lcm(a, b) = b \cdot gcd(a, b)^2$. [b]p13.[/b] Triangle $ABC$ has sides $AB = 6$, $BC = 10$, and $CA = 14$. If a point $D$ is placed on the opposite side of $AC$ from $B$ such that $\vartriangle ADC$ is equilateral, find the length of $BD$. [b]p14.[/b] If the product of all real solutions to the equation $(x-1)(x-2)(x-4)(x-5)(x-7)(x-8) = -x^2+9x-64$ can be written as $\frac{a-b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, b, d) = 1$ and $c$ is squarefree, compute $a + b + c + d$. [b]p15.[/b] Joe has a calculator with the keys $1, 2, 3, 4, 5, 6, 7, 8, 9,+,-$. However, Joe is blind. If he presses $4$ keys at random, and the expected value of the result can be written as $\frac{x}{11^4}$ , compute the last $3$ digits of $x$ when $x$ divided by $1000$. (If there are consecutive signs, they are interpreted as the sign obtained when multiplying the two signs values together, e.g $3$,$+$,$-$,$-$, $2$ would return $3 + (-(-(2))) = 3 + 2 = 5$. Also, if a sign is pressed last, it is ignored.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Points $M$ and $N$ are the midpoints of the sides $BC$ and $AD$, respectively, of a convex quadrilateral $ABCD$. Is it possible that $$ AB+CD>\max(AM+DM,BN+CN)? $$ [i](Folklore)[/i]

Positive numbers $x, y, z$ satisfy the condition $$xy + yz + zx + 2xyz = 1.$$ Prove that $4x + y + z \ge 2.$ [i]A. Khrabrov[/i]

In $\triangle ABC$, three lines are drawn parallel to side $BC$ dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$, what is the area of the whole $\triangle ABC$? [asy] defaultpen(linewidth(0.7)); size(120); pair B = (0,0), C = (1,0), A = (0.7,1); pair[] AB, AC; draw(A--B--C--cycle); for(int i = 1; i < 4; ++i) { AB.push((i*A + (4-i)*B)/4); AC.push((i*A + (4-i)*C)/4); draw(AB[i-1] -- AC[i-1]); } filldraw(AB[1]--AB[0]--AC[0]--AC[1]--cycle, gray(0.7)); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S);[/asy]

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

An urn contains $n$ balls labeled $1, 2, ... , n$. We draw the balls out one by one (without replacing them) until we obtain a ball whose number is divisible by $k$. Find all $k$ such that the expected number of balls removed is $k$.

There are $2^n$ words of length $n$ over the alphabet $\{0, 1\}$. Prove that the following algorithm generates the sequence $w_0, w_1, \ldots, w_{2^n-1}$ of all these words such that any two consecutive words differ in exactly one digit. (1) $w_0 = 00 \ldots 0$ ($n$ zeros). (2) Suppose $w_{m-1} = a_1a_2 \ldots a_n,\quad a_i \in \{0, 1\}$. Let $e(m)$ be the exponent of $2$ in the representation of $n$ as a product of primes, and let $j = 1 + e(m)$. Replace the digit $a_j$ in the word $w_{m-1}$ by $1 - a_j$. The obtained word is $w_m$.

Find all functions $f: R\to R$ with the property that $xf (y) + yf (x) = (x + y) f (xy)$ for all $x, y \in R$.

Let $a_1,a_2,\dots,a_{2018}$ be a strictly increasing sequence of positive integers such that $$a_1+a_2+\cdots+a_{2018}=2018^{2018}.$$ What is the remainder when $a_1^3+a_2^3+\cdots+a_{2018}^3$ is divided by $6$? $\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$

Square $S$ is the unit square with vertices at $(0, 0)$, $(0, 1)$, $(1, 0)$ and $(1, 1)$. We choose a random point $(x, y)$ inside $S$ and construct a rectangle with length $x$ and width $y$. What is the average of $\lfloor p \rfloor$ where $p$ is the perimeter of the rectangle? $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.

In the quadrilateral $ABCD$ the side $AB$ has length $7, BC$ length $14, CD$ length $26$, and $DA$ length $23$. Show that the diagonals are perpendicular. You may assume that the quadrilateral is convex (all internal angles are less than $180^o$).

Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$? $\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty) $

Let $u_1=1,u_2=2,u_3=24,$ and $u_{n+1}=\frac{6u_n^2 u_{n-2}-8u_nu_{n-1}^2}{u_{n-1}u_{n-2}}, n \geq 3.$ Prove that the elements of the sequence are natural numbers and that $n\mid u_n$ for all $n$.

Let $f(x) = |x - p| + |x - 15| + |x - p - 15|$, where $0 < p < 15$. Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \le x \le 15$.

Integers $1, 2, 3, ... ,n$, where $n > 2$, are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed and the average of the remaining numbers is found to be $17$. What is the maximum sum of the two removed numbers?

For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1, a_2, . . .$ and an infinite geometric sequence of integers $g_1, g_2, . . .$ satisfying the following properties? [list] [*] $a_n - g_n$ is divisible by $m$ for all integers $n \ge 1$; [*] $a_2 - a_1$ is not divisible by $m$. [/list] [i]Holden Mui[/i]

Find all polynomials $P\in \mathbb{R}[x]$, for which $P(P(x))=\lfloor P^2 (x)\rfloor$ is true for $\forall x\in \mathbb{Z}$.

Let $F$ be the set of all $n-tuples$ $(A_1,A_2,…,A_n)$ such that each $A_i$ is a subset of ${1,2,…,2019}$. Let $\mid{A}\mid$ denote the number of elements o the set $A$ . Find $\sum_{(A_1,…,A_n)\in{F}}^{}\mid{A_1\cup{A_2}\cup...\cup{A_n}}\mid$

Let $ABCD$ be an inscribed quadrilateral in a circle $c(O,R)$ (of circle $O$ and radius $R$). With centers the vertices $A,B,C,D$, we consider the circles $C_{A},C_{B},C_{C},C_{D}$ respectively, that do not intersect to each other . Circle $C_{A}$ intersects the sides of the quadrilateral at points $A_{1} , A_{2}$ , circle $C_{B}$ intersects the sides of the quadrilateral at points $B_{1} , B_{2}$ , circle $C_{C}$ at points $C_{1} , C_{2}$ and circle $C_{D}$ at points $C_{1} , C_{2}$ . Prove that the quadrilateral defined by lines $A_{1}A_{2} , B_{1}B_{2} , C_{1}C_{2} , D_{1}D_{2}$ is cyclic.