We say that a positive integer is a [i]good number[/i] if the digit $2$ appears more often than the digit $3$ and that it is a [i]bad number[/i] if the digit $3$ appears more often than the digit $2$. For example, $2023$ is a good number and $123$ is neither good nor bad. Calculate the difference between the quantity of good numbers and the quantity of bad numbers for integers less than or equal to $2023$.

Set $X$ has $56$ elements. Determine the least positive integer $n$ such that for any 15 subsets of $X$, if the union of any $7$ of the subsets has at least $n$ elements, then 3 of the subsets have nonempty intersection.

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

[b]p19.[/b] $A_1A_2...A_{12}$ is a regular dodecagon with side length $1$ and center at point $O$. What is the area of the region covered by circles $(A_1A_2O)$, $(A_3A_4O)$, $(A_5A_6O)$, $(A_7A_8O)$, $(A_9A_{10}O)$, and $(A_{11}A_{12}O)$? $(ABC)$ denotes the circle passing through points $A,B$, and $C$. [b]p20.[/b] Let $N = 2000... 0x0 ... 00023$ be a $2023$-digit number where the $x$ is the $23$rd digit from the right. If$ N$ is divisible by $13$, compute $x$. [b]p21.[/b] Alice and Bob each visit the dining hall to get a grilled cheese at a uniformly random time between $12$ PM and $1$ PM (their arrival times are independent) and, after arrival, will wait there for a uniformly random amount of time between $0$ and $30$ minutes. What is the probability that they will meet? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n; \] \[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. \] [i]Author: Dusan Dukic, Serbia[/i]

[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].

Find the largest number $n$ that for which there exists $n$ positive integers such that non of them divides another one, but between every three of them, one divides the sum of the other two. [i]Proposed by Morteza Saghafian[/i]

Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces and eight equilateral triangular faces. How many faces are on the polyhedron after the fifth step?

Nine points of the plane, located at the vertices of a regular nonagon, are pairwise connected by segments, each of which is colored either red or blue. It is known that in any triangle with vertices at the vertices of the nonagon at least one side is red. Prove that there are four points, any two of which are connected by red lines.

Polina and Yan have $n$ cards, on the first card on one side $1$ is written, on the other side $n+1$, on the second card on one side $2$ is written, on the other side $n+2$, etc. Polina laid all cards in a circle in some order. Yan wants to turn some cards such that the numbers on the top sides of adjacent cards were not coprime. For every positive integer $n \geq 3$ determine can Yan accomplish that regardless of the actions of Polina. [i]M. Shutro[/i]

Three persons $A,B,C$, are playing the following game: A $k$-element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$. For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)

Numbers $1,2,\ldots,1997^2$ are written in the cells of a $1997\times1997$ table. It is allowed to apply the following transformations: exchange places of any two rows or any two columns, or reverse a row or column. (When a row or column is reversed, the first and last entry exchange their positions, so do the second and second last, etc.) Is it possible that, after finitely many such transformations, arbitrary two numbers exchange their positions and no other number changes its position?

[b]p1.[/b] If $A = 0$, $B = 1$, $C = 2$, $...$, $Z = 25$, then what is the sum of $A + B + M+ C$? [b]p2.[/b] Eric is playing Tetris against Bryan. If Eric wins one-fifth of the games he plays and he plays $15$ games, find the expected number of games Eric will win. [b]p3.[/b] What is the sum of the measures of the exterior angles of a regular $2023$-gon in degrees? [b]p4.[/b] If $N$ is a base $10$ digit of $90N3$, what value of $N$ makes this number divisible by $477$? [b]p5.[/b] What is the rightmost non-zero digit of the decimal expansion of $\frac{1}{2^{2023}}$ ? [b]p6.[/b] if graphs of $y = \frac54 x + m$ and $y = \frac32 x + n$ intersect at $(16, 27)$, what is the value of $m + n$? [b]p7.[/b] Bryan is hitting the alphabet keys on his keyboard at random. If the probability he spells out ABMC at least once after hitting $6$ keys is $\frac{a}{b^c}$ , for positive integers $a$, $b$, $c$ where $b$, $c$ are both as small as possible, find $a+b+c$. Note that the letters ABMC must be adjacent for it to count: AEBMCC should not be considered as correctly spelling out ABMC. [b]p8.[/b] It takes a Daniel twenty minutes to change a light bulb. It takes a Raymond thirty minutes to change a light bulb. It takes a Bryan forty-five minutes to change a light bulb. In the time that it takes two Daniels, three Raymonds, and one and a half Bryans to change $42$ light bulbs, how many light bulbs could half a Raymond change? Assume half a person can work half as productively as a whole person. [b]p9.[/b] Find the value of $5a + 4b + 3c + 2d + e$ given $a, b, c, d, e$ are real numbers satisfying the following equations: $$a^2 = 2e + 23$$ $$b^2 = 10a - 34$$ $$c^2 = 8b - 23$$ $$d^2 = 6c - 14$$ $$e^2 = 4d - 7.$$ [b]p10.[/b] How many integers between $1$ and $1000$ contain exactly two $1$’s when written in base $2$? [b]p11.[/b] Joe has lost his $2$ sets of keys. However, he knows that he placed his keys in one of his $12$ mailboxes, each labeled with a different positive integer from $1$ to $12$. Joe plans on opening the $2$ mailbox labeled $1$ to see if any of his keys are there. However, a strong gust of wind blows by, opening mailboxes $11$ and $12$, revealing that they are empty. If Joe decides to open one of the mailboxes labeled $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ , or $10$, the probability that he finds at least one of his sets of keys can be expressed as $\frac{a}{b}$, where a and b are relatively prime positive integers. Find the sum $a + b$. Note that a single mailbox can contain $0$, $1$, or $2$ sets of keys, and the mailboxes his sets of keys were placed in are determined independently at random. [b]p12.[/b] As we all know, the top scientists have recently proved that the Earth is a flat disc. Bob is standing on Earth. If he takes the shortest path to the edge, he will fall off after walking $1$ meter. If he instead turns $90$ degrees away from the shortest path and walks towards the edge, he will fall off after $3$ meters. Compute the radius of the Earth. [b]p13.[/b] There are $999$ numbers that are repeating decimals of the form $0.abcabcabc...$ . The sum of all of the numbers of this form that do not have a $1$ or $2$ in their decimal representation can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$. Find $a + b$. [b]p14.[/b] An ant is crawling along the edges of a sugar cube. Every second, it travels along an edge to another adjacent vertex randomly, interested in the sugar it notices. Unfortunately, the cube is about to be added to some scalding coffee! In $10$ seconds, it must return to its initial vertex, so it can get off and escape. If the probability the ant will avoid a tragic doom can be expressed as $\frac{a}{3^{10}}$ , where $a$ is a positive integer, find $a$. Clarification: The ant needs to be on its initial vertex in exactly $10$ seconds, no more or less. [b]p15.[/b] Raymond’s new My Little Pony: Friendship is Magic Collector’s book arrived in the mail! The book’s pages measure $4\sqrt3$ inches by $12$ inches, and are bound on the longer side. If Raymond keeps one corner in the same plane as the book, what is the total area one of the corners can travel without ripping the page? If the desired area in square inches is $a\pi+b\sqrt{c}$ where $a$, $b$, and $c$ are integers and $c$ is squarefree, find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Yang has the sequence of integers $1, 2, \dots, 2017$. He makes $2016$ [i]swaps[/i] in order, where a swap changes the positions of two integers in the sequence. His goal is to end with $2, 3, \dots, 2017, 1$. How many different sequences of swaps can Yang do to achieve his goal?

The set of all $10$-digit numbers may be represented as a union of two subsets: the subset $M$ consisting of all $10$-digit numbers, each of which may be represented as a product of two $5$-digit numbers, and the subset $N$ , containing the remaining $10$-digit numbers . Which of the sets $M$ and $N$ contains more elements? (S. Fomin , Leningrad)

p1. Let $AB$ be the diameter of the circle and $P$ is a point outside the circle. The lines $PQ$ and $PR$ are tangent to the circles at points $Q$ and $R$. The lines $PH$ is perpendicular on line $AB$ at $H$ . Line $PH$ intersects $AR$ at $S$. If $\angle QPH =40^o$ and $\angle QSA =30^o$, find $\angle RPS$. p2. There is a meeting consisting of $40$ seats attended by $16$ invited guests. If each invited guest must be limited to at least $ 1$ chair, then determine the number of arrangements. p3. In the crossword puzzle, in the following crossword puzzle, each box can only be filled with numbers from $ 1$ to $9$. [img]https://cdn.artofproblemsolving.com/attachments/2/e/224b79c25305e8ed9a8a4da51059f961b9fbf8.png[/img] Across: 1. Composite factor of $1001$ 3. Non-polyndromic numbers 5. $p\times q^3$, with $p\ne q$ and $p,q$ primes Down: 1. $a-1$ and $b+1$ , $a\ne b$ and $p,q$ primes 2. multiple of $9$ 4. $p^3 \times q$, with $p\ne q$ and $p,q$ primes p4. Given a function $f:R \to R$ and a function $g:R \to R$, so that it fulfills the following figure: [img]https://cdn.artofproblemsolving.com/attachments/b/9/fb8e4e08861a3572412ae958828dce1c1e137a.png[/img] Find the number of values ​​of $x$, such that $(f(x))^2-2g(x)-x \in\{-10,-9,-8,…,9,10\}$. p5. In a garden that is rectangular in shape, there is a watchtower in each corner and in the garden there is a monitoring tower. Small areas will be made in the shape of a triangle so that the corner points are towers (free of monitoring and/or supervisory towers). Let $k(m,n)$ be the number of small areas created if there are $m$ control towers and $n$ monitoring towers. a. Find the values ​​of $k(4,1)$, $k(4,2)$, $k(4,3)$, and $k(4,4)$ b. Find the general formula $k(m,n)$ with $m$ and $n$ natural numbers .

In a small city there are $n$ bus routes, with $n > 1$, and each route has exactly $4$ stops. If any two routes have exactly one common stop, and each pair of stops belongs to exactly one route, find all possible values of $n$.

An [i]operation[/i] on three given non-negative integers consists in increasing two of them by $1$ and decreasing the third by $2$, provided the new numbers are non-negative. The process begins with three non-negative integers that add up to $100$ and are less than $100$. Find the number of distinct triplets that can be obtained by applying the operation. (Triplets that differ only in the order of their members are considered the same).

We have a table in the form of a regular polygon with $1000$ sides, where each side has length $1$. At one of the vertices is a beetle (consider this vertex to be fixed). The $1000$ vertices must be numbered, in some order, using the numbers $1, 2,\ldots ,1000$ such that the beetle is at vertex $1$. The beetle can only move along the edge of the table and always moves clockwise. The beetle moves from vertex $1$ to vertex $2$ and stops there. then it moves from vertex $2$ to vertex $3$, and stops there. So on, until the beetle ends its journey at vertex $1000$. Find the number of ways the numbers can be assigned to the vertices so that the total length of the beetle's journey is $2017$.

Sixty points, of which thirty are coloured red, twenty are coloured blue and ten are coloured green, are marked on a circle. These points divide the circle into sixty arcs. Each of these arcs is assigned a number according to the colours of its endpoints: an arc between a red and a green point is assigned a number $1$, an arc between a red and a blue point is assigned a number $2$, and an arc between a blue and a green point is assigned a number $3$. The arcs between two points of the same colour are assigned a number $0$. What is the greatest possible sum of all the numbers assigned to the arcs?

Three fair six-sided dice are thrown. Determine the probability that the sum of the numbers on the three top faces is $6$.

Given an odd $n\in\mathbb N$. In an $n\times n$ chessboard, you may place many $2\times2$ squares. How many grids, at most, are covered by exactly one square?

There are a lot of different real numbers written on the board. It turned out that for each two numbers written, their product was also written. What is the largest possible number of numbers written on the board?

In a checkered square of size $2021\times 2021$ all cells are initially white. Ivan selects two cells and paints them black. At each step, all the cells that have at least one black neighbor by side are painted black simultaneously. Ivan selects the starting two cells so that the entire square is painted black as fast as possible. How many steps will this take? [i]Ivan Yashchenko[/i]

Form the infinite graph $A$ by taking the set of primes $p$ congruent to $1\pmod{4}$, and connecting $p$ and $q$ if they are quadratic residues modulo each other. Do the same for a graph $B$ with the primes $1\pmod{8}$. Show $A$ and $B$ are isomorphic to each other. [i]Linus Hamilton.[/i]