[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].

In an ${8}$×${8}$ squares chart , we dig out $n$ squares , then we cannot cut a "T"shaped-5-squares out of the surplus chart . Then find the mininum value of $n$ .

In a country consisting of $2015$ cities, between any two cities there is exactly one direct round flight operated by some air company. Find the minimal possible number of air companies if direct flights between any three cities are operated by three different air companies.

[b]p1.[/b] A tennis net is made of strings tied up together which make a grid consisting of small squares as shown below. [img]https://cdn.artofproblemsolving.com/attachments/9/4/72077777d57408d9fff0ea5e79be5ecb6fe8c3.png[/img] The size of the net is $100\times 10$ small squares. What is the maximal number of sides of small squares which can be cut without breaking the net into two separate pieces? (The side is cut only in the middle, not at the ends). [b]p2.[/b] What number is bigger $2^{300}$ or $3^{200}$ ? [b]p3.[/b] All noble knights participating in a medieval tournament in Camelot used nicknames. In the tournament each knight had combats with all other knights. In each combat one knight won and the second one lost. At the end of tournament the losers reported their real names to the winners and to the winners of their winners. Was there a person who knew the real names of all knights? [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $10$ rocks in the first pile and $12$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] There is an interesting $5$-digit integer. With a $1$ after it, it is three times as large as with a $1$ before it. What is the number? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $100$ members in a ladies' club.Each lady has had tea (in private) with exactly $56$ of her lady friends.The Board,consisting of the $50$ most distinguished ladies,have all had tea with one another.Prove that the entire club may be split into two groups in such a way that,with in each group,any lady has had tea with any other.

Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?

John and James wish to divide $25$ coins, of denominations $1, 2, 3, \ldots , 25$ kopeks. In each move, one of them chooses a coin, and the other player decides who must take this coin. John makes the initial choice of a coin, and in subsequent moves, the choice is made by the player having more kopeks at the time. In the event that there is a tie, the choice is made by the same player in the preceding move. After all the coins have been taken, the player with more kokeps wins. Which player has a winning strategy? [i](6 points)[/i]

Consider a $4\times 4$ array of pairwise distinct positive integers such that on each column, respectively row, one of the numbers is equal to the sum of the other three. Determine the least possible value of the largest number such an array may contain. [i]The Problem Selection Committee[/i]

[i]Superchess[/i] is played on on a $12 \times 12$ board, and it uses [i]superknights[/i], which move between opposite corner cells of any $3\times4$ subboard. Is it possible for a [i]superknight[/i] to visit every other cell of a superchessboard exactly once and return to its starting cell ?

Ten children have several bags of candies. The children begin to divide these candies among them. They take turns picking their shares of candies from each bag, and leave just after that. The size of the share is determined as follows: the current number of candies in the bag is divided by the number of remaining children (including the one taking the turn). If the remainder is nonzero than the quotient is rounded to the lesser integer. Is it possible that all the children receive different numbers of candies if the total number of bags is: a) 8 ; 6) 99 ? Alexey Glebov

The first $360$ natural numbers are separated into $9$ blocks in such a way that the numbers in each block are consecutive. Then, the numbers in each block are added, obtaining $9$ numbers. Is it possible to fill a $3 \times 3$ grid and form a [i]magic square[/i] with these numbers? Note: In a magic square, the sum of the numbers written in any column, diagonal or row of the grid is the same.

For a sequence $\{a_i\}_{i\ge1}$ consisting of only positive integers, prove that if for all different positive integers $i$ and $j$, we have $a_i\nmid a_j$, then $$\{p\mid p\text{ is a prime and }p\mid a_i\text{ for some }i\}$$is a infinite set.

Let be given two positive integers $ m$, $ n$ with $ m < 2001$, $ n < 2002$. Let distinct real numbers be written in the cells of a $ 2001 \times 2002$ board (with $ 2001$ rows and $ 2002$ columns). A cell of the board is called [i]bad[/i] if the corresponding number is smaller than at least $ m$ numbers in the same column and at least $ n$ numbers in the same row. Let $ s$ denote the total number of [i]bad[/i] cells. Find the least possible value of $ s$.

From an urn containing one ball marked with the number 1, two balls marked with the number 2, ..., $ n $ balls marked with the number $ n $ we draw two balls without replacement. We assume that each ball is equally likely to be drawn from the urn. Calculate the probability that both balls drawn have the same number.

A flea jumps in a straight numbered line. It jumps first from point $0$ to point $1$. Afterwards, if its last jump was from $A$ to $B$, then the next jump is from $B$ to one of the points $B + (B - A) - 1$, $B + (B - A)$, $B + (B-A) + 1$. Prove that if the flea arrived twice at the point $n$, $n$ positive integer, then it performed at least $\lceil 2\sqrt n\rceil$ jumps.

Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. At least how many problems are needed?

Several jugs (not necessarily of the same size) with juices are placed along a circle. It is allowed to transfuse any part of juice (maybe nothing or the total content) from any jug to the neighboring one on the right, so that the latter one is not overflowed and the sugariness of its content becomes equal to $10\%$. It is known that at the initial moment such transfusion is possible from each jug. Prove that it is possible to perform several transfusions in some order, at most one transfusion from each jug, such that the sugariness of the content of each non-empty jug will become equal to $10\%$. (Sugariness is the percent of sugar in a jug, by weight. Sugar is always uniformly distributed in a jug.)

Determine the largest integer $N$ for which there exists a table $T$ of integers with $N$ rows and $100$ columns that has the following properties: $\text{(i)}$ Every row contains the numbers $1$, $2$, $\ldots$, $100$ in some order. $\text{(ii)}$ For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r,c) - T(s, c)|\geq 2$. (Here $T(r,c)$ is the entry in row $r$ and column $c$.)

Proctors Andy and Kristin have a PUMaC team of eight students labelled $s_1, s_2, ... , s_8$ (the PUMaC staff being awful with names). The following occurs: $1$. Andy tells the students to arrange themselves in a line in arbitrary order. $2$. Kristin tells each student $s_i$ to move to the current spot of student $s_j$ , where $j \equiv 3i + 1$ (mod $8$). $3$. Andy tells each student $s_i$ to move to the current spot of the student who was in the $i$th position of the line after step $1$. How many possible orders can the students be in now?

The numbers from 1 to $n^2$ are randomly arranged in the cells of a $n \times n$ square ($n \geq 2$). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated. Let us call the [b]characteristic[/b] of the arrangement the smallest of these $n^2\left(n-1\right)$ fractions. What is the highest possible value of the characteristic ?

A real number is written on each square of a $2024 \times 2024$ chessboard. It is given that the sum of all real numbers on the board is $2024$. Then, the board is covered by $1 \times 2$ or $2\times 1$ dominos such that there isn't any square that is covered by two different dominoes. For each domino, Aslı deletes $2$ numbers covered by it and writes $0$ on one of the squares and the sum of the two numbers on the other square. Find the maximum number $k$ such that after Aslı finishes her moves, there exists a column or row where the sum of all the numbers on it is at least $k$ regardless of how dominos were replaced and the real numbers were written initially.

Given an $8 \times 8$ chess board. Each knight is allowed to move between two squares located at opposite vertices of $2 \times 3$ or $3 \times 2$ rectangles. There are four knights that move on the board, evenly start from the same cell $X$ and return to $X$ and then stop. Assume that every square on the chessboard has at least one of these four roosters moving through. Prove that there exists a square $Y$ that is different from $X$ such that it is moved over no less than twice by the same knight or by different knights.

Do there exist positive integers $k$ and $n$ such that for any finite graph $G$ with diameter $k+1$ there exists a set $S$ of at most $n$ vertices such that for any $v\in V(G)\setminus S$, there exists a vertex $u\in S$ of distance at most $k$ from $v$? [i]David Yang.[/i]

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 $1$.

There are $n$ players, $n\ge 2$, which are playing a card game with $np$ cards in $p$ rounds. The cards are coloured in $n$ colours and each colour is labelled with the numbers $1,2,\ldots ,p$. The game submits to the following rules: [list]each player receives $p$ cards. the player who begins the first round throws a card and each player has to discard a card of the same colour, if he has one; otherwise they can give an arbitrary card. the winner of the round is the player who has put the greatest card of the same colour as the first one. the winner of the round starts the next round with a card that he selects and the play continues with the same rules. the played cards are out of the game.[/list] Show that if all cards labelled with number $1$ are winners, then $p\ge 2n$. [i]Barbu Berceanu[/i]