[b]p1.[/b] If $x^2 = 7$, what is $x^4 + x^2 + 1$? [b]p2.[/b] Richard and Alex are competing in a $150$-meter race. If Richard runs at a constant speed of $5$ meters per second and Alex runs at a constant speed of $3$ meters per second, how many more seconds does it take for Alex to finish the race? [b]p3.[/b] David and Emma are playing a game with a chest of $100$ gold coins. They alternate turns, taking one gold coin if the chest has an odd number of gold coins or taking exactly half of the gold coins if the chest has an even number of gold coins. The game ends when there are no more gold coins in the chest. If Emma goes first, how many gold coins does Emma have at the end? [b]p4.[/b] What is the only $3$-digit perfect square whose digits are all different and whose units digit is $5$? [b]p5.[/b] In regular pentagon $ABCDE$, let $F$ be the midpoint of $\overline{AB}$, $G$ be the midpoint of $\overline{CD}$, and $H$ be the midpoint of $\overline{AE}$. What is the measure of $\angle FGH$ in degrees? [b]p6.[/b] Water enters at the left end of a pipe at a rate of $1$ liter per $35$ seconds. Some of the water exits the pipe through a leak in the middle. The rest of the water exits from the right end of the pipe at a rate of $1$ liter per $36$ seconds. How many minutes does it take for the pipe to leak a liter of water? [b]p7.[/b] Carson wants to create a wire frame model of a right rectangular prism with a volume of $2022$ cubic centimeters, where strands of wire form the edges of the prism. He wants to use as much wire as possible. If Carson also wants the length, width, and height in centimeters to be distinct whole numbers, how many centimeters of wire does he need to create the prism? [b]p8.[/b] How many ways are there to fill the unit squares of a $3 \times 5$ grid with the digits $1$, $2$, and $3$ such that every pair of squares that share a side differ by exactly $1$? [b]p9.[/b] In pentagon ABCDE, $AB = 54$, $AE = 45$, $DE = 18$, $\angle A = \angle C = \angle E$, $D$ is on line segment $\overline{BE}$, and line $BD$ bisects angle $\angle ABC$, as shown in the diagram below. What is the perimeter of pentagon $ABCDE$? [img]https://cdn.artofproblemsolving.com/attachments/2/0/7c25837bb10b128a1c7a292f6ce8ce3e64b292.png[/img] [b]p10.[/b] If $x$ and $y$ are nonzero real numbers such that $\frac{7}{x} + \frac{8}{y} = 91$ and $\frac{6}{x} + \frac{10}{y} = 89$, what is the value of $x + y$? [b]p11.[/b] Hilda and Marianne play a game with a shued deck of $10$ cards, numbered from $1$ to $10$. Hilda draws five cards, and Marianne picks up the five remaining cards. Hilda observes that she does not have any pair of consecutive cards - that is, no two cards have numbers that differ by exactly $1$. Additionally, the sum of the numbers on Hilda's cards is $1$ less than the sum of the numbers on Marianne's cards. Marianne has exactly one pair of consecutive cards - what is the sum of this pair? [b]p12.[/b] Regular hexagon $AUSTIN$ has side length $2$. Let $M$ be the midpoint of line segment $\overline{ST}$. What is the area of pentagon $MINUS$? [b]p13.[/b] At a collector's store, plushes are either small or large and cost a positive integer number of dollars. All small plushes cost the same price, and all large plushes cost the same price. Two small plushes cost exactly one dollar less than a large plush. During a shopping trip, Isaac buys some plushes from the store for 59 dollars. What is the smallest number of dollars that the small plush could not possibly cost? [b]p14.[/b] Four fair six-sided dice are rolled. What is the probability that the median of the four outcomes is $5$? [b]p15.[/b] Suppose $x_1, x_2,..., x_{2022}$ is a sequence of real numbers such that: $x_1 + x_2 = 1$ $x_2 + x_3 = 2$ $...$ $x_{2021} + x_{2022} = 2021$ If $x_1 + x_{499} + x_{999} + x_{1501} = 222$, then what is the value of $x_{2022}$? [b]p16.[/b] A cone has radius $3$ and height $4$. An infinite number of spheres are placed in the cone in the following way: sphere $C_0$ is placed inside the cone such that it is tangent to the base of the cone and to the curved surface of the cone at more than one point, and for $i \ge 1$, sphere $C_i$ is placed such that it is externally tangent to sphere $C_{i-1}$ and internally tangent to more than one point of the curved surface of the cone. If $V_i$ is the volume of sphere $C_i$, compute $V_0 + V_1 + V_2 + ... $ . [img]https://cdn.artofproblemsolving.com/attachments/b/4/b43e40bb0a5974dd9d656691c14b4ae268b5b5.png[/img] [b]p17.[/b] Call an ordered pair, $(x, y)$, relatable if $x$ and $y$ are positive integers where $y$ divides $3600$, $x$ divides $y$ and $\frac{y}{x}$ is a prime number. For every relatable ordered pair, Leanne wrote down the positive difference of the two terms of the pair. What is the sum of the numbers she wrote down? [b]p18.[/b] Let $r, s$, and $t$ be the three roots of $P(x) = x^3 - 9x - 9$. Compute the value of $(r^3 + r^2 - 10r - 8)(s^3 + s^2 - 10s - 8)(t^3 + t^2 - 10t - 8)$. [b]p19.[/b] Compute the number of ways to color the digits $0, 1, 2, 3, 4, 5, 6, 7, 8$ and $9$ red, blue, or green such that: (a) every prime integer has at least one digit that is not blue, and (b) every composite integer has at least one digit that is not green. Note that $0$ is not composite. For example, since $12$ is composite, either the digit $1$, the digit $2$, or both must be not green. [b]p20.[/b] Pentagon $ABCDE$ has $AB = DE = 4$ and $BC = CD = 9$ with $\angle ABC = \angle CDE = 90^o$, and there exists a circle tangent to all five sides of the pentagon. What is the length of segment $\overline{AE}$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We want to partition the integers $1, 2, 3, . . . , 100$ into several groups such that within each group either any two numbers are coprime or any two are not coprime. At least how many groups are needed for such a partition? [i]We call two integers coprime if they have no common divisor greater than $1$.[/i]

[b]p1.[/b] Minseo writes all of the divisors of $1,000,000$ on the whiteboard. She then erases all of the numbers which have the digit $0$ in their decimal representation. How many numbers are left? [b]p2.[/b] $n < 100$ is an odd integer and can be expressed as $3k - 2$ and $5m + 1$ for positive integers $k$ and $m$. Find the sum of all possible values of $n$. [b]p3.[/b] Mr. Pascal is a math teacher who has the license plate $SQUARE$. However, at night, a naughty student scrambles Mr. Pascal’s license plate to $UQRSEA$. The math teacher luckily has an unscrambler that is able to move license plate letters. The unscrambler swaps the positions of any two adjacent letters. What is the minimum number of times Mr. Pascal must use the unscrambler to restore his original license plate? [b]p4.[/b] Find the number of distinct real numbers $x$ which satisfy $x^2 + 4 \lfloor x \rfloor + 4 = 0$. [b]p5.[/b] All four faces of tetrahedron $ABCD$ are acute. The distances from point $D$ to $\overline{BC}$, $\overline{CA}$ and $\overline{AB}$ are all $7$, and the distance from point $D$ to face $ABC$ is $5$. Given that the volume of tetrahedron $ABCD$ is $60$, find the surface area of tetrahedron $ABCD$. [b]p6.[/b] Forrest has a rectangular piece of paper with a width of $5$ inches and a height of $3$ inches. He wants to cut the paper into five rectangular pieces, each of which has a width of $1$ inch and a distinct integer height between $1$ and $5$ inches, inclusive. How many ways can he do so? (One possible way is shown below.) [img]https://cdn.artofproblemsolving.com/attachments/7/3/205afe28276f9df1c6bcb45fff6313c6c7250f.png[/img] [b]p7.[/b] In convex quadrilateral $ABCD$, $AB = CD = 5$, $BC = 4$ and $AD = 8$. If diagonal $\overline{AC}$ bisects $\angle DAB$, find the area of quadrilateral $ABCD$. [b]p8.[/b] Let $x$ and $y$ be real numbers such that $$x + y = x^3 + y^3 + \frac34 = \frac{1}{8xy}.$$ Find the value of $x + y$. [b]p9.[/b] Four blue squares and four red parallelograms are joined edge-to-edge alternately to form a ring of quadrilateral as shown. The areas of three of the red parallelograms are shown. Find the area of the fourth red parallelogram. [img]https://cdn.artofproblemsolving.com/attachments/9/c/911a8d53604f639e2f9bd72b59c7f50e43e258.png[/img] [b]p10.[/b] Define $f(x, n) =\sum_{d|n}\frac{x^n-1}{x^d-1}$ . For how many integers $n$ between $1$ and $2023$ inclusive is $f(3, n)$ an odd integer? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For $i = 1,2, \ldots, 1991$, we choose $n_i$ points and write number $i$ on them (each point has only written one number on it). A set of chords are drawn such that: (i) They are pairwise non-intersecting. (ii) The endpoints of each chord have distinct numbers. If for all possible assignments of numbers the operation can always be done, find the necessary and sufficient condition the numbers $n_1, n_2, \ldots, n_{1991}$ must satisfy for this to be possible.

Half the cells of a $2m \times n$ board are colored black and the other half are colored white. The cells at the opposite ends of the main diagonal are different colors. The center of each black cell is connected to the center of every other black cell by a straight line segment, and similarly for the white cells. Show that we can place an arrow on each segment so that it becomes a vector and the vectors sum to zero.

A positive integer $ n$ is given. In an association consisting of $ n$ members work $ 6$ commissions. Each commission contains at least $ \large \frac{n}{4}$ persons. Prove that there exist two commissions containing at least $ \large \frac{n}{30}$ persons in common.

Jane writes down $2024$ natural numbers around the perimeter of a circle. She wants the $2024$ products of adjacent pairs of numbers to be exactly the set $\{ 1!, 2!, \ldots, 2024! \}.$ Can she accomplish this?

In a certain Zoom meeting, there are $4$ students. How many ways are there to split them into any number of distinguishable breakout rooms, each with at least $ 1$ student?

On the infinite surface of the sea floats a black and bounded oil slick. After every minute the slick and the sea change according to the following law: at each point $P$ of the sea (or of the slick), a disk $D$ of radius $1$ is considered centered on $ P$. If more than half of the area inside the disk $D$ is black, the $P$ point will remain black for the next minute. If more than half of the area inside the disk $D$ is dark blue, the point $P$ will be dark blue for the next minute. In the event that both the clean and the contaminated area within the disk $D$ are the same, its center $P$ will not change color. Can that stain "live" forever or will it disappear at some point?

$2022$ teams participated in an underwater polo tournament, each two teams played exactly once against each other. Team receives $2, 1, 0$ points for win, draw, and loss correspondingly. It turned out that all teams got distinct numbers of points. In the final standings the teams were ordered by the total number of points. A few days later, organizers realized that the results in the final standings were wrong due to technical issues: in fact, each match that ended with a draw according to them in fact had a winner, and each match with a winner in fact ended with a draw. It turned out that all teams still had distinct number of points! They corrected the standings, and ordered them by the total number of points. Could the correct order turn out to be the reversed initial order? [i](Proposed by Fedir Yudin)[/i]

The $m \times n$ chessboard is colored by black and white. In one step, two neighbouring squares are selected (squares with a common side) and their color changes according to the follwing way: - white becomes black, - black become red, - Red becomes white. For which $m$ and $n$, these steps can change the colors of all the initial squares from white to black and from black to white?

Let $n>1$ be a positive integer. Find the number of binary strings $(a_1, a_2, \ldots, a_n)$, such that the number of indices $1\leq i \leq n-1$ such that $a_i=a_{i+1}=0$ is equal to the number of indices $1 \leq i \leq n-1$, such that $a_i=a_{i+1}=1$.

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

Let $n$ be a positive integer and let vectors $v_1$, $v_2$, $\ldots$, $v_n$ be given in the plain. A flea originally sitting in the origin moves according to the following rule: in the $i$th minute (for $i=1,2,\ldots,n$) it will stay where it is with probability $1/2$, moves with vector $v_i$ with probability $1/4$, and moves with vector $-v_i$ with probability $1/4$. Prove that after the $n$th minute there exists no point which is occupied by the flea with greater probability than the origin. [i]Proposed by Péter Pál Pach, Budapest[/i]

Let $N$ and $d$ be two positive integers with $N\geq d+2$. There are $N$ countries connected via two-way direct flights, where each country is connected to exactly $d$ other countries. It is known that for any two different countries, it is possible to go from one to another via several flights. A country is \emph{important} if after removing it and all the $d$ countries it is connected to, there exist two other countries that are no longer connected via several flights. Show that if every country is important, then one can choose two countries so that more than $2d/3$ countries are connected to both of them via direct flights. [i]Proposed by usjl[/i]

prove that if graph $G$ is a tree, then there is a vertex that is common between all of the longest paths. [i]proposed by Sina Rezayi[/i]

Alice and Bob play the following game: Alice picks a set $A = \{1, 2, ..., n \}$ for some natural number $n \ge 2$. Then, starting from Bob, they alternatively choose one number from the set $A$, according to the following conditions: initially Bob chooses any number he wants, afterwards the number chosen at each step should be distinct from all the already chosen numbers and should differ by $1$ from an already chosen number. The game ends when all numbers from the set $A$ are chosen. Alice wins if the sum of all the numbers that she has chosen is composite. Otherwise Bob wins. Decide which player has a winning strategy. Proposed by [i]Demetres Christofides, Cyprus[/i]

You are given a balance and one copy of each of ten weights of $1, 2, 4, 8, 16, 32, 64, 128, 256$ and $512$ grams. An object weighing $M$ grams, where $M$ is a positive integer, is put on one of the pans and may be balanced in different ways by placing various combinations of the given weights on either pan of the balance. (a) Prove that no object may be balanced in more than $89$ ways. (b) Find a value of $M$ such that an object weighing $M$ grams can be balanced in $89$ ways. (A Shapovalov, A Kulakov)

Let $N$ and $K$ be positive integers satisfying $1 \leq K \leq N$. A deck of $N$ different playing cards is shuffled by repeating the operation of reversing the order of $K$ topmost cards and moving these to the bottom of the deck. Prove that the deck will be back in its initial order after a number of operations not greater than $(2N/K)^2$.

For $n \geq 3$, a [i]special n-triangle[/i] is a triangle with $n$ distinct numbers on each side such that the sum of the numbers on a side is the same for all sides. For instance, because $41 + 23 + 43 = 43 + 17 + 47 = 47 + 19 + 41$, the following is a special $3$-triangle: $$41$$ $$23\text{ }\text{ }\text{ }\text{ }\text{ }19$$ $$43\text{ }\text{ }\text{ }\text{ }\text{ }17\text{ }\text{ }\text{ }\text{ }\text{ }47$$ Note that a special $n$-triangle contains $3(n - 1)$ numbers. An infinite set $A$ of positive integers is a [i]special set[/i] if, for each $n \geq 3$, the smallest $3(n - 1)$ numbers of $A$ can be used to form a special $n$-triangle. Show that the set of positive integers that are not multiples of $2023$ is a special set.

Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$

Suppose two polygons may be glued together at an edge if and only if corresponding edges of the same length are made to coincide. A $3\times 4$ rectangle is cut into $n$ pieces by making straight line cuts. What is the minimum value of $n$ so that it’s possible to cut the pieces in such a way that they may be glued together two at a time into a polygon with perimeter at least $2021$?

Given the number $720$, Juan must choose $4$ numbers that are divisors of $720$. He wins if none of the four chosen numbers is a divisor of the product of the other three. Decide whether Juan can win.

A rabbit is placed on a $2n\times 2n$ chessboard. Every time the rabbit moves to one of the adjacent squares. (Adjacent means sharing an edge). It is known that the rabbit went through every square and came back to the place where the rabbit started, and the path of the rabbit form a polygon $\mathcal{P}$. Find the maximum possible number of the vertices of $\mathcal{P}$. For example the answer for the case $n=2$ would be $12$. [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(2cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -11.3, xmax = 27.16, ymin = -11.99, ymax = 10.79; /* image dimensions */ /* draw figures */ draw((5.14,3.19)--(8.43,3.22), linewidth(1)); draw((8.43,3.22)--(11.72,3.25), linewidth(1)); draw((11.72,3.25)--(11.75,-0.04), linewidth(1)); draw((11.75,-0.04)--(11.78,-3.33), linewidth(1)); draw((11.78,-3.33)--(8.49,-3.36), linewidth(1)); draw((8.49,-3.36)--(5.2,-3.39), linewidth(1)); draw((5.2,-3.39)--(5.17,-0.1), linewidth(1)); draw((5.17,-0.1)--(5.14,3.19), linewidth(1)); draw((6.785,3.205)--(6.845,-3.375), linewidth(1)); draw((8.43,3.22)--(8.49,-3.36), linewidth(1)); draw((10.075,3.235)--(10.135,-3.345), linewidth(1)); draw((5.155,1.545)--(11.735,1.605), linewidth(1)); draw((5.17,-0.1)--(11.75,-0.04), linewidth(1)); draw((11.765,-1.685)--(5.185,-1.745), linewidth(1)); draw((5.97,2.375)--(10.905,2.42), linewidth(1)); draw((10.905,2.42)--(10.92,0.775), linewidth(1)); draw((10.92,0.775)--(9.275,0.76), linewidth(1)); draw((9.275,0.76)--(9.29,-0.885), linewidth(1)); draw((9.29,-0.885)--(10.935,-0.87), linewidth(1)); draw((10.935,-0.87)--(10.95,-2.515), linewidth(1)); draw((10.95,-2.515)--(6.015,-2.56), linewidth(1)); draw((6.015,-2.56)--(6,-0.915), linewidth(1)); draw((6,-0.915)--(7.645,-0.9), linewidth(1)); draw((7.645,-0.9)--(7.63,0.745), linewidth(1)); draw((7.63,0.745)--(5.985,0.73), linewidth(1)); draw((5.985,0.73)--(5.97,2.375), linewidth(1)); /* dots and labels */ dot((5.97,2.375),linewidth(4pt) + dotstyle); dot((5.985,0.73),linewidth(4pt) + dotstyle); dot((6,-0.915),linewidth(4pt) + dotstyle); dot((6.015,-2.56),linewidth(4pt) + dotstyle); dot((7.645,-0.9),linewidth(4pt) + dotstyle); dot((7.63,0.745),linewidth(4pt) + dotstyle); dot((9.275,0.76),linewidth(4pt) + dotstyle); dot((9.29,-0.885),linewidth(4pt) + dotstyle); dot((10.95,-2.515),linewidth(4pt) + dotstyle); dot((10.935,-0.87),linewidth(4pt) + dotstyle); dot((10.92,0.775),linewidth(4pt) + dotstyle); dot((10.905,2.42),linewidth(4pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

Some coins are placed on a $20 \times 13$ board. Two coins are called [i]neighbours[/i] if they are in the same row or column and no other coins lie between them. What is the largest number of coins that can be placed on the board if no coin is allowed to have more than two neighbours?