Consider the $8\times 8\times 8$ Rubik’s cube below. Each face is painted with a different color, and it is possible to turn any layer, as you can with smaller Rubik’s cubes. Let $X$ denote the move that turns the shaded layer shown (indicated by arrows going from the top to the right of the cube) clockwise by $90$ degrees, about the axis labeled $X$. When move $X$ is performed, the only layer that moves is the shaded layer. Likewise, define move $Y$ to be a clockwise $90$-degree turn about the axis labeled Y, of just the shaded layer shown (indicated by the arrows going from the front to the top, where the front is the side pierced by the $X$ rotation axis). Let $M$ denote the move “perform $X$, then perform $Y$.” [img]https://cdn.artofproblemsolving.com/attachments/e/f/951ea75a3dbbf0ca23c45cd8da372595c2de48.png[/img] Imagine that the cube starts out in “solved” form (so each face has just one color), and we start doing move $M$ repeatedly. What is the least number of repeats of $M$ in order for the cube to be restored to its original colors?

[b]p22.[/b] Consider the series $\{A_n\}^{\infty}_{n=0}$, where $A_0 = 1$ and for every $n > 0$, $$A_n = A_{\left[ \frac{n}{2023}\right]} + A_{\left[ \frac{n}{2023^2}\right]}+A_{\left[ \frac{n}{2023^3}\right]},$$ where $[x]$ denotes the largest integer value smaller than or equal to $x$. Find the $(2023^{3^2}+20)$-th element of the series. [b]p23.[/b] The side lengths of triangle $\vartriangle ABC$ are $5$, $7$ and $8$. Construct equilateral triangles $\vartriangle A_1BC$, $\vartriangle B_1CA$, and $\vartriangle C_1AB$ such that $A_1$,$B_1$,$C_1$ lie outside of $\vartriangle ABC$. Let $A_2$,$B_2$, and $C_2$ be the centers of $\vartriangle A_1BC$, $\vartriangle B_1CA$, and $\vartriangle C_1AB$, respectively. What is the area of $\vartriangle A_2B_2C_2$? [b]p24. [/b]There are $20$ people participating in a random tag game around an $20$-gon. Whenever two people end up at the same vertex, if one of them is a tagger then the other also becomes a tagger. A round consists of everyone moving to a random vertex on the $20$-gon (no matter where they were at the beginning). If there are currently $10$ taggers, let $E$ be the expected number of untagged people at the end of the next round. If $E$ can be written as $\frac{a}{b}$ for $a, b$ relatively prime positive integers, compute $a + b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We have an $n \times n$ board, and in every square there is an integer. The sum of all integers on the board is $0$. We define an action on a square where the integer in the square is decreased by the number of neighbouring squares, and the number inside each of the neighbouring squares is increased by 1. Determine if there exists $n\geq 2$ such that we can turn all the integers into zeros in a finite number of actions.

Fifty students take part in a mathematical competition where a set of $8$ problems is given (same set to each participant). The final result showed that a total of $171$ correct solutions were obtained. Prove that there are $3$ of the given problems that have been correctly solved by the same $3$ students.

Three coins lie on integer points on the number line. A move consists of choosing and moving two coins, the first one $ 1$ unit to the right and the second one $ 1$ unit to the left. Under which initial conditions is it possible to move all coins to one single point?

Located on the plane $\left[ \frac43 n \right]$ rectangles with sides parallel to the coordinate axes. It is known that any rectangle intersects at least n rectangles. Prove that exists a rectangle that intersects all rectangles.

A round-robin tournament is one where each team plays every other team exactly once. Five teams take part in such a tournament getting: $3$ points for a win, $1$ point for a draw and $0$ points for a loss. At the end of the tournament the teams are ranked from first to last according to the number of points. (a) Is it possible that at the end of the tournament, each team has a different number of points, and each team except for the team ranked last has exactly two more points than the next-ranked team? (b) Is this possible if there are six teams in the tournament instead?

Each of the $n^2$ cells of an $n \times n$ grid is colored either black or white. Let $a_i$ denote the number of white cells in the $i$-th row, and let $b_i$ denote the number of black cells in the $i$-th column. Determine the maximum value of $\sum_{i=1}^n a_ib_i$ over all coloring schemes of the grid. [i]Proposed by Alex Zhai[/i]

Let $T$ be the set of all 2-digit numbers whose digits are in $\{1,2,3,4,5,6\}$ and the tens digit is strictly smaller than the units digit. Suppose $S$ is a subset of $T$ such that it contains all six digits and no three numbers in $S$ use all six digits. If the cardinality of $S$ is $n$, find all possible values of $n$.

[u]Round 1[/u] [b]p1.[/b] What is the smallest number equal to its cube? [b]p2.[/b] Fhomas has $5$ red spaghetti and $5$ blue spaghetti, where spaghetti are indistinguishable except for color. In how many different ways can Fhomas eat $6$ spaghetti, one after the other? (Two ways are considered the same if the sequence of colors are identical) [b]p3.[/b] Jocelyn labels the three corners of a triangle with three consecutive natural numbers. She then labels each edge with the sum of the two numbers on the vertices it touches, and labels the center with the sum of all three edges. If the total sum of all labels on her triangle is $120$, what is the value of the smallest label? [u]Round 2[/u] [b]p4.[/b] Adam cooks a pie in the shape of a regular hexagon with side length $12$, and wants to cut it into right triangular pieces with angles $30^o$, $60^o$, and $90^o$, each with shortest side $3$. What is the maximum number of such pieces he can make? [b]p5.[/b] If $f(x) =\frac{1}{2-x}$ and $g(x) = 1-\frac{1}{x}$ , what is the value of $f(g(f(g(... f(g(f(2019))) ...))))$, where there are $2019$ functions total, counting both $f$ and $g$? [b]p6.[/b] Fhomas is buying spaghetti again, which is only sold in two types of boxes: a $200$ gram box and a $500$ gram box, each with a fixed price. If Fhomas wants to buy exactly $800$ grams, he must spend $\$8:80$, but if he wants to buy exactly 900 grams, he only needs to spend $\$7:90$! In dollars, how much more does the $500$ gram box cost than the $200$ gram box? [u]Round 3[/u] [b]p7.[/b] Given that $$\begin{cases} a + 5b + 9c = 1 \\ 4a + 2b + 3c = 2 \\ 7a + 8b + 6c = 9\end{cases}$$ what is $741a + 825b + 639c$? [b]p8.[/b] Hexagon $JAMESU$ has line of symmetry $MU$ (i.e., quadrilaterals $JAMU$ and $SEMU$ are reflections of each other), and $JA = AM = ME = ES = 1$. If all angles of $JAMESU$ are $135$ degrees except for right angles at $A$ and $E$, find the length of side $US$. [b]p9.[/b] Max is parked at the $11$ mile mark on a highway, when his pet cheetah, Min, leaps out of the car and starts running up the highway at its maximum speed. At the same time, Max starts his car and starts driving down the highway at $\frac12$ his maximum speed, driving all the way to the $10$ mile mark before realizing that his cheetah is gone! Max then immediately reverses directions and starts driving back up the highway at his maximum speed, nally catching up to Min at the $20$ mile mark. What is the ratio between Max's max speed and Min's max speed? [u]Round 4[/u] [b]p10.[/b] Kevin owns three non-adjacent square plots of land, each with side length an integer number of meters, whose total area is $2019$ m$^2$. What is the minimum sum of the perimeters of his three plots, in meters? [b]p11.[/b] Given a $5\times 5$ array of lattice points, how many squares are there with vertices all lying on these points? [b]p12.[/b] Let right triangle $ABC$ have $\angle A = 90^o$, $AB = 6$, and $AC = 8$. Let points $D,E$ be on side $AC$ such that $AD = EC = 2$, and let points $F,G$ be on side $BC$ such that $BF = FG = 3$. Find the area of quadrilateral $FGED$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2949413p26408203]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $k\ge 3$ be an integer. Prove that if $n>\binom k3$, then for any $3n$ pairwise different real numbers $a_i,b_i,c_i$ ($1\le i\le n$), among the numbers $a_i+b_i$, $a_i+c_i$, $b_i+c_i$, one can find at least $k+1$ pairwise different numbers. Show that this is not always the case when $n=\binom k3$.

Find all $(m,n) \in \mathbb{Z}^2$ that we can color each unit square of $m \times n$ with the colors black and white that for each unit square number of unit squares that have the same color with it and have at least one common vertex (including itself) is even.

In ancient times there was a Celtic tribe consisting of several families. Many of these families were at odds with each other, so that their chiefs would not shake hands. At some point at the annual meeting of the chiefs they found it even impossible to assemble four or more of them in a circle with each of them being willing to shake his neighbour's hand. To emphasize the gravity of the situation, the Druid collected three pieces of gold from each family. The Druid then let all those chiefs shake hands who were willing to. For each handshake of two chiefs he paid each of them a piece of gold as a reward. Show that the number of pieces of gold collected by the Druid exceeds the number of pieces paid out by at least three.

For each five distinct variables from the set $x_1,..., x_{10}$ there is a single card on which their product is written. Peter and Basil play the following game. At each move, each player chooses a card, starting with Peter. When all cards have been taken, Basil assigns values to the variables as he wants, so that $0 \le x_1 \le ... \le x_10$. Can Basil ensure that the sum of the products on his cards is greater than the sum of the products on Peter's cards? (Ilya Bogdanov)

For $n\geq3$ define $S_n=\{1, 2, ..., n\}$. $A_1, A_{2}, ..., A_{n}$ are given subsets of $S_n$, each having an even number of elements. Prove that there exists a set $\{i_1, i_2, ..., i_t\}$, a nonempty subset of $S_n$ such that $$A_{i_1} \Delta A_{i_2} \Delta \ldots \Delta A_{i_t}=\emptyset$$ (For two sets $A, B$, we define $\Delta$ as $A \Delta B=(A\cup B)-(A\cap B)$)

Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$. [i]Proposed by Calvin Deng[/i]

Let $n \geq 1$ be an integer, and let $S$ be a set of integer pairs $(a,b)$ with $1 \leq a < b \leq 2^n$. Assume $|S| > n \cdot 2^{n+1}$. Prove that there exists four integers $a < b < c < d$ such that $S$ contains all three pairs $(a,c)$, $(b,d)$ and $(a,d)$.

Let $f(n)$ denote the number of permutations $(a_{1}, a_{2}, \ldots ,a_{n})$ of the set $\{1,2,\ldots,n\}$, which satisfy the conditions: $a_{1}=1$ and $|a_{i}-a_{i+1}|\leq2$, for any $i=1,2,\ldots,n-1$. Prove that $f(2006)$ is divisible by 3.

The sides of a convex regular polygon of $L + M + N$ sides are to be given draw in three colors: $L$ of them with a red stroke, $M$ with a yellow stroke, and $N$ with a blue. Express, through inequalities, the necessary and sufficient conditions so that there is a solution (several, in general) to the problem of doing it without leaving two adjacent sides drawn with the same color.

There is a closed polyline with $n$ edges on the plane. We build a new polyline which edges connect the midpoints of two adjacent edges of the previous polyline. Then we erase previous polyline and start over and over. Also we know that each polyline satisfy that all vertices are different and not all of them are collinear. For which $n$ we can get a polyline that is a сonvex polygon?

Ten boxes are arranged in a circle. Each box initially contains a positive number of golf balls. A move consists of taking all of the golf balls from one of the boxes and placing them into the boxes that follow it in a counterclockwise direction, putting one ball into each box. Prove that if the next move always starts with the box where the last ball of the previous move was placed, then after some number of moves, we get back to the initial distribution of golf balls in the boxes.

[b]p1.[/b] Anne purchased yesterday at WalMart in Puerto Rico $6$ identical notebooks, $8$ identical pens and $7$ identical erasers. Anne remembers that each eraser costs $73$ cents. She did not buy anything else. Anne told her mother that she spent $12$ dollars and $76$ cents at Walmart. Can she be right? Note that in Puerto Rico there is no sales tax. [b]p2.[/b] Two men ski one after the other first in a flat field and then uphill. In the field the men run with the same velocity $12$ kilometers/hour. Uphill their velocity drops to $8$ kilometers/hour. When both skiers enter the uphill trail segment the distance between them is $300$ meters less than the initial distance in the field. What was the initial distance between skiers? (There are $1000$ meters in 1 kilometer.) [b]p3.[/b] In the equality $** + **** = ****$ all the digits are replaced by $*$. Restore the equality if it is known that any numbers in the equality does not change if we write all its digits in the opposite order. [b]p4.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies. Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!” The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!” A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!” Then a stripped polyleg started: ”-None of you has $8$ legs. Only I have 8 legs!” Which polyleg has exactly $8$ legs? [b]p5.[/b] Cut the figure shown below in two equal pieces. (Both the area and the form of the pieces must be the same.) [img]https://cdn.artofproblemsolving.com/attachments/e/4/778678c1e8748e213ffc94ba71b1f3cc26c028.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a triple $(a, b, c)$ of pairwise distinct positive integers satisfying $a + b + c = 2013$. A step consists of replacing the triple $(x, y, z)$ by the triple $(y + z - x,z + x - y,x + y - z)$. Prove that, starting from the given triple $(a, b,c)$, after $10$ steps we obtain a triple containing at least one negative number.

Given $3n$ points in the plane, no three collinear, is it always possible to form $n$ triangles (with vertices at the points), so that no point in the plane lies in more than one triangle?

Tanya and Serezha have a heap of $2016$ candies. They make moves in turn, Tanya moves first. At each move a player can eat either one candy or (if the number of candies is even at the moment) exactly half of all candies. The player that cannot move loses. Which of the players has a winning strategy?