In a group of $6$ people playing the card game Tractor, all $54$ cards from $3$ decks are dealt evenly to all the players at random. Each deck is dealt individually. Let the probability that no one has at least two of the same card be $X$. Find the largest integer $n$ such that the $n$th root of $X$ is rational. [i]Proposed by Sammy Charney[/i] [b]Due to the problem having infinitely many solutions, all teams who inputted answers received points.[/b]

Let $k$ and $n$ be integers such that $k\ge 2$ and $k \le n \le 2k-1$. Place rectangular tiles, each of size $1 \times k$, or $k \times 1$ on a $n \times n$ chessboard so that each tile covers exactly $k$ cells and no two tiles overlap. Do this until no further tile can be placed in this way. For each such $k$ and $n$, determine the minimum number of tiles that such an arrangement may contain.

[b]p1.[/b] The least prime factor of $a$ is $3$, the least prime factor of $b$ is $7$. Find the least prime factor of $a + b$. [b]p2.[/b] In a Cartesian coordinate system, the two tangent lines from $P = (39, 52)$ meet the circle defined by $x^2 + y^2 = 625$ at points $Q$ and $R$. Find the length $QR$. [b]p3.[/b] For a positive integer $n$, there is a sequence $(a_0, a_1, a_2,..., a_n)$ of real values such that $a_0 = 11$ and $(a_k + a_{k+1}) (a_k - a_{k+1}) = 5$ for every $k$ with $0 \le k \le n-1$. Find the maximum possible value of $n$. (Be careful that your answer isn’t off by one!) [b]p4.[/b] Persons $A$ and $B$ stand at point $P$ on line $\ell$. Point $Q$ lies at a distance of $10$ from point $P$ in the direction perpendicular to $\ell$. Both persons intially face towards $Q$. Person $A$ walks forward and to the left at an angle of $25^o$ with $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the right, and continues walking. Person $B$ walks forward and to the right at an angle of $55^o$ with line $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the left, and continues walking. Their paths cross at point $R$. Find the distance $PR$. [b]p5.[/b] Compute $$\frac{lcm (1,2, 3,..., 200)}{lcm (102, 103, 104, ..., 200)}.$$ [b]p6.[/b] There is a unique real value $A$ such that for all $x$ with $1 < x < 3$ and $x \ne 2$, $$\left| \frac{A}{x^2-x - 2} +\frac{1}{x^2 - 6x + 8} \right|< 1999.$$ Compute $A$. [b]p7.[/b] Nine poles of height $1, 2,..., 9$ are placed in a line in random order. A pole is called [i]dominant [/i] if it is taller than the pole immediately to the left of it, or if it is the pole farthest to the left. Count the number of possible orderings in which there are exactly $2$ dominant poles. [b]p8.[/b] $\tan (11x) = \tan (34^o)$ and $\tan (19x) = \tan (21^o)$. Compute $\tan (5x)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A $4 \times 4$ board has all its squares painted white. An allowed operation is to choose a rectangle that contains $3$ squares and paint each of the boxes as follows: a) If the box is white then it is painted black. b) If the box is black then it is painted white. Prove that by applying the allowed operation several times, it is impossible get the entire board painted black.

[b]p1.[/b] There are $2024$ apples in a very large basket. First, Julie takes away half of the apples in the basket; then, Diane takes away $202$ apples from the remaining bunch. How many apples remain in the basket? [b]p2.[/b] The set of all permutations (different arrangements) of the letters in ”ABMC” are listed in alphabetical order. The first item on the list is numbered $1$, the second item is numbered $2$, and in general, the kth item on the list is numbered $k$. What number is given to ”ABMC”? [b]p3.[/b] Daniel has a water bottle that is three-quarters full. After drinking $3$ ounces of water, the water bottle is three-fifths full. The density of water is $1$ gram per milliliter, and there are around $28$ grams per ounce. How many milliliters of water could the bottle fit at full capacity? [b]p4.[/b] How many ways can four distinct $2$-by-$1$ rectangles fit on a $2$-by-$4$ board such that each rectangle is fully on the board? [b]p5.[/b] Iris and Ivy start reading a $240$ page textbook with $120$ left-hand pages and $120$ right-hand pages. Iris takes $4$ minutes to read each page, while Ivy takes $5$ minutes to read a left-hand page and $3$ minutes to read a right-hand page. Iris and Ivy move onto the next page only when both sisters have completed reading. If a sister finishes reading a page first, the other sister will start reading three times as fast until she completes the page. How many minutes after they start reading will both sisters finish the textbook? [b]p6.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $24$. Then, let $M$ be the midpoint of $BC$. Define $P$ to be the set of all points $P$ such that $2PM = BC$. The minimum value of $AP$ can be expressed as $\sqrt{a}- b$, where $a$ and $b$ are positive integers. Find $a + b$. [b]p7.[/b] Jonathan has $10$ songs in his playlist: $4$ rap songs and $6$ pop songs. He will select three unique songs to listen to while he studies. Let $p$ be the probability that at least two songs are rap, and let $q$ be the probability that none of them are rap. Find $\frac{p}{q}$ . [b]p8.[/b] A number $K$ is called $6,8$-similar if $K$ written in base $6$ and $K$ written in base $8$ have the same number of digits. Find the number of $6,8$-similar values between $1$ and $1000$, inclusive. [b]p9.[/b] Quadrilateral $ABCD$ has $\angle ABC = 90^o$, $\angle ADC = 120^o$, $AB = 5$, $BC = 18$, and $CD = 3$. Find $AD^2$. [b]p10.[/b] Bob, Eric, and Raymond are playing a game. Each player rolls a fair $6$-sided die, and whoever has the highest roll wins. If players are tied for the highest roll, the ones that are tied reroll until one wins. At the start, Bob rolls a $4$. The probability that Eric wins the game can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [b]p11.[/b] Define the following infinite sequence $s$: $$s = \left\{\frac92,\frac{99}{2^2},\frac{999}{2^3} , ... , \overbrace{\frac{999...999}{2^k}}^{k\,\,nines}, ...\right\}$$ The sum of the first $2024$ terms in $s$, denoted $S$, can be expressed as $$S =\frac{5^a - b}{4}+\frac{1}{2^c},$$ where $a, b$, and $c$ are positive integers. Find $a + b + c$. [b]p12.[/b] Andy is adding numbers in base $5$. However, he accidentally forgets to write the units digit of each number. If he writes all the consecutive integers starting at $0$ and ending at $50$ (base $10$) and adds them together, what is the difference between Andy’s sum and the correct sum? (Express your answer in base-$10$.) [b]p13.[/b] Let $n$ be the positive real number such that the system of equations $$y =\frac{1}{\sqrt{2024 - x^2}}$$ $$y =\sqrt{x^2 - n}$$ has exactly two real solutions for $(x, y)$: $(a, b)$ and $(-a, b)$. Then, $|a|$ can be expressed as $j\sqrt{k}$, where $j$ and $k$ are integers such that $k$ is not divisible by any perfect square other than $1$. Find $j · k$. [b]p14.[/b] Nakio is playing a game with three fair $4$-sided dice. But being the cheater he is, he has secretly replaced one of the three die with his own $4$-sided die, such that there is a $1/2$ chance of rolling a $4$, and a $1/6$ chance to roll each number from $1$ to $3$. To play, a random die is chosen with equal probability and rolled. If Nakio guesses the number that is on the die, he wins. Unfortunately for him, Nakio’s friends have an anti-cheating mechanism in place: when the die is picked, they will roll it three times. If each roll lands on the same number, that die is thrown out and one of the two unused dice is chosen instead with equal probability. If Nakio always guesses $4$, the probability that he wins the game can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime. Find $m + n$. [b]p15.[/b] A particle starts in the center of a $2$m-by-$2$m square. It moves in a random direction such that the angle between its direction and a side of the square is a multiple of $30^o$. It travels in that direction at $1$ m/s, bouncing off of the walls of the square. After a minute, the position of the particle is recorded. The expected distance from this point to the start point can be written as $$\frac{1}{a}\left(b - c\sqrt{d}\right),$$ where $a$ and $b$ are relatively prime, and d is not divisible by any perfect square. Find $a + b + c + d$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a small town, there are $n \times n$ houses indexed by $(i, j)$ for $1 \leq i, j \leq n$ with $(1, 1)$ being the house at the top left corner, where $i$ and $j$ are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by $(1, c)$, where $c \leq \frac{n}{2}$. During each subsequent time interval $[t, t+1]$, the fire fighters defend a house which is not yet on fire while the fire spreads to all undefended [i]neighbors[/i] of each house which was on fire at time t. Once a house is defended, it remains so all the time. The process ends when the fire can no longer spread. At most how many houses can be saved by the fire fighters? A house indexed by $(i, j)$ is a [i]neighbor[/i] of a house indexed by $(k, l)$ if $|i - k| + |j - l|=1$.

LeBron James and Carmelo Anthony play a game of one-on-one basketball where the first player to $3$ points or more wins. LeBron James has a $20\%$ chance of making a $3$-point shot; Carmelo has a $10\%$ chance of making a $3$-pointer. LeBron has a $40\%$ chance of making a $2$-point shot from anywhere inside the $3$-point line (excluding dunks, which are also worth $2$ points); Carmelo has a $52\%$ chance of making a $ 2$-point shot from anywhere inside the 3-point line (excluding dunks). LeBron has a $90\%$ chance of dunking on Carmelo; Carmelo has a $95\%$ chance of dunking on LeBron. If each player has $3$ possessions to try to win, LeBron James goes first, and both players follow a rational strategy to try to win, what is the probability that Carmelo Anthony wins the game?

[b]p1.[/b] It takes $12$ months for Santa Claus to pack gifts. It would take $20$ months for his apprentice to do the job. If they work together, how long will it take for them to pack the gifts? [b]p2.[/b] All passengers on a bus sit in pairs. Exactly $2/5$ of all men sit with women, exactly $2/3$ of all women sit with men. What part of passengers are men? [b]p3.[/b] There are $100$ colored balls in a box. Every $10$-tuple of balls contains at least two balls of the same color. Show that there are at least $12$ balls of the same color in the box. [b]p4.[/b] There are $81$ wheels in storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that one can determine which wheel is incorrectly marked with four measurements. [b]p5.[/b] Remove from the figure below the specified number of matches so that there are exactly $5$ squares of equal size left: (a) $8$ matches (b) $4$ matches [img]https://cdn.artofproblemsolving.com/attachments/4/b/0c5a65f2d9b72fbea50df12e328c024a0c7884.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A given rectangle $ R$ is divided into $mn$ small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of $ R$?

Some computers of a computer room have a following network. Each computers are connected by three cable to three computers. Two arbitrary computers can exchange data directly or indirectly (through other computers). Now let's remove $K$ computers so that there are two computers, which can not exchange data, or there is one computer left. Let $k$ be the minimum value of $K$. Let's remove $L$ cable from original network so that there are two computers, which can not exchange data. Let $l$ be the minimum value of $L$. Show that $k=l$.

A binary decision tree is a list of $n$ yes/no questions, together with instructions for the order in which they should be asked (without repetition). For instance, if $n = 3$, there are $12$ possible binary decision trees, one of which asks question $2$ first, then question $3$ (followed by question $ 1$) if the answer was yes or question $1$ (followed by question $3$) if the answer was no. Determine the greatest possible $k$ such that $2^k$ divides the number of binary decision trees on $n = 13$ questions.

Calvin and Hobbes play a game. First, Hobbes picks a family $F$ of subsets of $\{1, 2, . . . , 2020\}$, known to both players. Then, Calvin and Hobbes take turns choosing a number from $\{1, 2, . . . , 2020\}$ which is not already chosen, with Calvin going first, until all numbers are taken (i.e., each player has $1010$ numbers). Calvin wins if he has chosen all the elements of some member of $F$, otherwise Hobbes wins. What is the largest possible size of a family $F$ that Hobbes could pick while still having a winning strategy?

For a special event, the five Vietnamese famous dishes including Phở, (Vietnamese noodle), Nem (spring roll), Bún Chả (grilled pork noodle), Bánh cuốn (stuffed pancake), and Xôi gà (chicken sticky rice) are the options for the main courses for the dinner of Monday, Tuesday, and Wednesday. Every dish must be used exactly one time. How many choices do we have?

We call an integer $ n > 1$ [i]good[/i] if, for any natural numbers $ 1 \le b_1, b_2, \ldots , b_{n\minus{}1} \le n \minus{} 1$ and any $ i \in \{0, 1, \ldots , n \minus{} 1\}$, there is a subset $ I$ of $ \{1, \ldots , n \minus{} 1\}$ such that $ \sum_{k\in I} b_k \equiv i \pmod n$. (The sum over the empty set is zero.) Find all good numbers.

Arne has $2n + 1$ tickets. Each card has one number on it. One card has the number $0$ on it. The natural numbers $1, 2, . . . , n$ occur on exactly two cards each. Prove that Arne can arrange cards in a row so that there are exactly $m$ cards between the two cards with the number $m$, for every $m \in \{1, 2, . . . , n\}$.

Let $G$ be a planar graph all of whose vertices are of degree $4$. Vasya and Petya walk along its edges. The first time each of them goes as he pleases, and then each of them goes straight (from the three roads they have to choose the middle one). As the result, each vertex was visited by exactly one of them and exactly once. Prove that this graph has an even number of vertices.

To date, in each Mathematics Olympiad Final, no participant has been able to solve all the problems, but every problem has been solved by at least one participant. Prove that in each Final, there was a participant $A$ who solved a problem $P_A$ and another participant $B$ who solved a problem $P_B$ such that $A$ did not solve $P_B$ and $B$ did not solve $P_A$.

A tournament of order $n$, $n \in \mathbb{N}$, consists of $2^n$ players, which are numbered with $1, 2, \ldots, 2^n$, and has $n$ rounds. In each round, the remaining players paired with each other to play a match and the winner from each match advances to the next round. The winner of the $n$-th round is considered the winner of the tournament. Two tournaments are considered different if there is a match that took place in the $k$-th round of one tournament, but not in the $k$-th round of the other, or if the tournaments have different winners. Determine how many different tournaments of order $n$ there are with the property that in each round, the sum of the numbers of the players in each match is the same (but not necessarily the same for all rounds).

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [b]R1.[/b] What is $11^2 - 9^2$? [b]R2.[/b] Write $\frac{9}{15}$ as a decimal. [b]R3.[/b] A $90^o$ sector of a circle is shaded, as shown below. What percent of the circle is shaded? [b]R4.[/b] A fair coin is flipped twice. What is the probability that the results of the two flips are different? [b]R5.[/b] Wayne Dodson has $55$ pounds of tungsten. If each ounce of tungsten is worth $75$ cents, and there are $16$ ounces in a pound, how much money, in dollars, is Wayne Dodson’s tungsten worth? [b]R6.[/b] Tenley Towne has a collection of $28$ sticks. With these $28$ sticks he can build a tower that has $1$ stick in the top row, $2$ in the next row, and so on. Let $n$ be the largest number of rows that Tenley Towne’s tower can have. What is n? [b]R7.[/b] What is the sum of the four smallest primes? [b]R8 / P1.[/b] Let $ABC$ be an isosceles triangle such that $\angle B = 42^o$. What is the sum of all possible degree measures of angle $A$? [b]R9.[/b] Consider a line passing through $(0, 0)$ and $(4, 8)$. This line passes through the point $(2, a)$. What is the value of $a$? [b]R10 / P2.[/b] Brian and Stan are playing a game. In this game, Brian rolls a fair six-sided die, while Stan rolls a fair four-sided die. Neither person shows the other what number they rolled. Brian tells Stan, “The number I rolled is guaranteed to be higher than the number you rolled.” Stan now has to guess Brian’s number. If Stan plays optimally, what is the probability that Stan correctly guesses the number that Brian rolled? [b]R11.[/b] Guang chooses $4$ distinct integers between $0$ and $9$, inclusive. How many ways can he choose the integers such that every pair of chosen integers sums up to an even number? [b]R12 / P4.[/b] David is trying to write a problem for MBMT. He assigns degree measures to every interior angle in a convex $n$-gon, and it so happens that every angle he assigned is less than $144$ degrees. He tells Pratik the value of $n$ and the degree measures in the $n$-gon, and to David’s dismay, Pratik claims that such an $n$-gon does not exist. What is the smallest value of $n \ge 3$ such that Pratik’s claim is necessarily true? [b]R13 / P3.[/b] Consider a triangle $ABC$ with side lengths of $5$, $5$, and $2\sqrt5$. There exists a triangle with side lengths of $5, 5$, and $x$ ($x \ne 2\sqrt5$) which has the same area as $ABC$. What is the value of $x$? [b]R14 / P5.[/b] A mother has $11$ identical apples and $9$ identical bananas to distribute among her $3$ kids. In how many ways can the fruits be allocated so that each child gets at least one apple and one banana? [b]R15 / P7.[/b] Find the sum of the five smallest positive integers that cannot be represented as the sum of two not necessarily distinct primes. [b]P6.[/b] Srinivasa Ramanujan has the polynomial $P(x) = x^5 - 3x^4 - 5x^3 + 15x^2 + 4x - 12$. His friend Hardy tells him that $3$ is one of the roots of $P(x)$. What is the sum of the other roots of $P(x)$? [b]P8.[/b] $ABC$ is an equilateral triangle with side length $10$. Let $P$ be a point which lies on ray $\overrightarrow{BC}$ such that $PB = 20$. Compute the ratio $\frac{PA}{PC}$. [b]P9.[/b] Let $ABC$ be a triangle such that $AB = 10$, $BC = 14$, and $AC = 6$. The median $CD$ and angle bisector $CE$ are both drawn to side $AB$. What is the ratio of the area of triangle $CDE$ to the area of triangle $ABC$? [b]P10.[/b] Find all integer values of $x$ between $0$ and $2017$ inclusive, which satisfy $$2016x^{2017} + 990x^{2016} + 2x + 17 \equiv 0 \,\,\, (mod \,\,\, 2017).$$ [b]P11.[/b] Let $x^2 + ax + b$ be a quadratic polynomial with positive integer roots such that $a^2 - 2b = 97$. Compute $a + b$. [b]P12.[/b] Let $S$ be the set $\{2, 3, ... , 14\}$. We assign a distinct number from $S$ to each side of a six-sided die. We say a numbering is predictable if prime numbers are always opposite prime numbers and composite numbers are always opposite composite numbers. How many predictable numberings are there? (Rotations of a die are not distinct) [b]P13.[/b] In triangle $ABC$, $AB = 10$, $BC = 21$, and $AC = 17$. $D$ is the foot of the altitude from $A$ to $BC$, $E$ is the foot of the altitude from $D$ to $AB$, and $F$ is the foot of the altitude from $D$ to $AC$. Find the area of the smallest circle that contains the quadrilateral $AEDF$. [b]P14.[/b] What is the greatest distance between any two points on the graph of $3x^2 + 4y^2 + z^2 - 12x + 8y + 6z = -11$? [b]P15.[/b] For a positive integer $n$, $\tau (n)$ is defined to be the number of positive divisors of $n$. Given this information, find the largest positive integer $n$ less than $1000$ such that $$\sum_{d|n} \tau (d) = 108.$$ In other words, we take the sum of $\tau (d)$ for every positive divisor $d$ of $n$, which has to be $108$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the number of permutations $( p_1, p_2, p_3 , p_4 , p_5 , p_6)$ of $1, 2 ,3,4,5,6$ such that for any $k, 1 \leq k \leq 5$, $(p_1, \ldots, p_k)$ does not form a permutation of $1 , 2, \ldots, k$.

The plan of a Martian underground is represented by a closed selfintersecting curve, with at most one self-intersection at each point. Prove that a tunnel for such a plan may be constructed in such a way that the train passes consecutively over and under the intersecting parts of the tunnel.

Let $n$ be a positive integer. In a village, $n$ boys and $n$ girls are living. For the yearly ball, $n$ dancing couples need to be formed, each of which consists of one boy and one girl. Every girl submits a list, which consists of the name of the boy with whom she wants to dance the most, together with zero or more names of other boys with whom she wants to dance. It turns out that $n$ dancing couples can be formed in such a way that every girl is paired with a boy who is on her list. Show that it is possible to form $n$ dancing couples in such a way that every girl is paired with a boy who is on her list, and at least one girl is paired with the boy with whom she wants to dance the most.

An equilateral triangle with side $3$ is divided into $9$ small equal equilateral triangles with sides of length $1$. Each vertex of a triangle small (bold dots) is numbered with a different number than the $1$ to $10$. Inside each small triangle, write the sum of the numbers corresponding to its three vertices. Prove that there are three small triangles for which it is verified that the sum of the numbers written inside is at least $48$. [img]https://cdn.artofproblemsolving.com/attachments/2/1/b2f58b6d59cb26e2fe29d0df59c1a42639a496.png[/img]

There is an equilateral triangle $ABC$ on the plane. Three straight lines pass through $A$, $B$ and $C$, respectively, such that the intersections of these lines form an equilateral triangle inside $ABC$. On each turn, Ming chooses a two-line intersection inside $ABC$, and draws the straight line determined by the intersection and one of $A$, $B$ and $C$ of his choice. Find the maximum possible number of three-line intersections within $ABC$ after 300 turns. [i] Proposed by usjl[/i]

Suppose a grid with $2m$ rows and $2n$ columns is given, where $m$ and $n$ are positive integers. You may place one pawn on any square of this grid, except the bottom left one or the top right one. After placing the pawn, a snail wants to undertake a journey on the grid. Starting from the bottom left square, it wants to visit every square exactly once, except the one with the pawn on it, which the snail wants to avoid. Moreover, it wants to finish in the top right square. It can only move horizontally or vertically on the grid. On which squares can you put the pawn for the snail to be able to finish its journey?