Ababa plays with a word made up of the letters of his name and has set certain rules: If you find an $A$ followed immediately by a $B$, you can substitute $BAA$ for them. If you find two consecutive $B$'s, you can delete them. If you find three consecutive $A$'s, you can delete them. Ababa begins with the word $ABABABAABAAB$. With the above rules, how many letters do you have the shortest word you can come up with? Why can't you come up with one more word shorter?

Ana writes a three-digit code, and Beto has to guess it. To do so, he can ask about a sequence of three digits, and Ana will respond "warm" if the sequence Beto proposes has at least one correct digit in the correct position, and she will respond "cold" if none of the digits are correct. For example, if the correct code is $014$, then if Beto asks $099$ or $014$, he receives the answer "warm", and if he asks $140$ or $322$, he receives the answer "cold". Determine the minimum number of questions Beto needs to ask in order to know the correct code with certainty.

Monica and Bogdan are playing a game, depending on given integers $n, k$. First, Monica writes some $k$ positive numbers. Bogdan wins, if he is able to find $n$ points on the plane with the following property: for any number $m$ written by Monica, there are some two points chosen by Bogdan with distance exactly $m$ between them. Otherwise, Monica wins. Determine who has a winning strategy depending on $n, k$. [i](Proposed by Fedir Yudin)[/i]

The sides of an equilateral triangle with sides of length $n$ have been divided into equal parts, each of length $1$, and lines have been drawn through the points of division parallel to the sides of the triangle, thus dividing the large triangle into many small triangles. Nils has a pile of rhombic tiles, each of side $1$ and angles $60^\circ$ and $120^\circ$, and wants to tile most of the triangle using these, so that each tile covers two small triangles with no overlap. In the picture, three tiles are placed somewhat arbitrarily as an illustration. How many tiles can Nils fit inside the triangle? [asy] /* original code by fedja: https://artofproblemsolving.com/community/c68h207503p1220868 modified by Klaus-Anton: https://artofproblemsolving.com/community/c2083h3267391_draw_me_a_grid_of_regular_triangles */ size(5cm); int n=6; pair A=(1,0), B=dir(60); path P=A--B--(0,0)--cycle; path Pp=A--shift(A)*B--B--cycle; /* label("$A$",A,S); label("$B$",B,dir(120)); label("$(0,0)$",(0,0),dir(210)); fill(shift(2*A-1+2*B-1)*P,yellow+white); fill(shift(2*A-1+2*B-0)*P,yellow+white); fill(shift(2*A-1+2*B+1)*P,yellow+white); fill(shift(2*A-1+2*B+2)*P,yellow+white); fill(shift(1*A-1+1*B)*P,blue+white); fill(shift(2*A-1+1*B)*P,blue+white); fill(shift(3*A-1+1*B)*P,blue+white); fill(shift(4*A-1+1*B)*P,blue+white); fill(shift(5*A-1+1*B)*P,blue+white); fill(shift(0*A+0*B)*P,green+white); fill(shift(0*A+1+0*B)*P,green+white); fill(shift(0*A+2+0*B)*P,green+white); fill(shift(0*A+3+0*B)*P,green+white); fill(shift(0*A+4+0*B)*P,green+white); fill(shift(0*A+5+0*B)*P,green+white); fill(shift(2*A-1+3*B-1)*P,magenta+white); fill(shift(3*A-1+3*B-1)*P,magenta+white); fill(shift(4*A-1+3*B-1)*P,magenta+white); fill(shift(5*A+5*B-5)*P,heavyred+white); fill(shift(4*A+4*B-4)*P,palered+white); fill(shift(4*A+4*B-3)*P,palered+white); fill(shift(0*A+0*B)*Pp,gray); fill(shift(0*A+1+0*B)*Pp,gray); fill(shift(0*A+2+0*B)*Pp,gray); fill(shift(0*A+3+0*B)*Pp,gray); fill(shift(0*A+4+0*B)*Pp,gray); fill(shift(1*A+1*B-1)*Pp,lightgray); fill(shift(1*A+1*B-0)*Pp,lightgray); fill(shift(1*A+1*B+1)*Pp,lightgray); fill(shift(1*A+1*B+2)*Pp,lightgray); fill(shift(2*A+2*B-2)*Pp,red); fill(shift(2*A+2*B-1)*Pp,red); fill(shift(2*A+2*B-0)*Pp,red); fill(shift(3*A+3*B-2)*Pp,blue); fill(shift(3*A+3*B-3)*Pp,blue); fill(shift(4*A+4*B-4)*Pp,cyan); fill(shift(0*A+1+0*B)*Pp,gray); fill(shift(0*A+2+0*B)*Pp,gray); fill(shift(0*A+3+0*B)*Pp,gray); fill(shift(0*A+4+0*B)*Pp,gray); */ fill(Pp, rgb(244, 215, 158)); fill(shift(dir(60))*P, rgb(244, 215, 158)); fill(shift(1.5,(-sqrt(3)/2))*shift(2*dir(60))*Pp, rgb(244, 215, 158)); fill(shift(1.5,(-sqrt(3)/2))*shift(2*dir(60))*P, rgb(244, 215, 158)); fill(shift(-.5,(-sqrt(3)/2))*shift(4*dir(60))*Pp, rgb(244, 215, 158)); fill(shift(.5,(-sqrt(3)/2))*shift(4*dir(60))*P, rgb(244, 215, 158)); for(int i=0;i<n;++i){ for(int j;j<n-i;++j) {draw(shift(i*A+j*B)*P);}} shipout(bbox(2mm,Fill(white))); [/asy]

There are $3$ unit squares in a row as shown in the figure below. Each side of this figure is painted by one of the three colors: Blue, Green or Red. It is known that for any square, all the three colors are used and no two adjacent sides have the same color. Find the number of possible colorings. [img]https://cdn.artofproblemsolving.com/attachments/e/c/8963e6716b7d9b23479dd7e106b4bd9a3267c1.png[/img] A. $48$ B. $96$ C. $108$ D. $192$ E. $216$

In an $8 \times 8$ grid of squares, each square was colored black or white so that no $2\times 2$ square has all its squares in the same color. A sequence of distinct squares $x_1,\dots, x_m$ is called a [b]snake of length $m$[/b] if for each $1\leq i <m$ the squares $x_i, x_{i+1}$ are adjacent and are of different colors. What is the maximum $m$ for which there must exist a snake of length $m$?

The geometric mean (G.M.) of $k$ positive integers $a_1$, $a_2$, $\dots$, $a_k$ is defined to be the (positive) $k$-th root of their product. For example, the G.M. of 3, 4, 18 is 6. Show that the G.M. of a set $S$ of $n$ positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of $S$.

We color each number in the set $S = \{1, 2, ..., 61\}$ with one of $25$ given colors, where it is not necessary that every color gets used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$?

Given a positive integer $N$. There are three squirrels that each have an integer. It is known that the largest integer and the least one differ by exactly $N$. Each time, the squirrel with the second largest integer looks at the squirrel with the largest integer. If the integers they have are different, then the squirrel with the second largest integer would be unhappy and attack the squirrel with the largest one, making its integer decrease by two times the difference between the two integers. If the second largest integer is the same as the least integer, only of the squirrels would attack the squirrel with the largest integer. The attack continues until the largest integer becomes the same as the second largest integer. What is the maximum total number of attacks these squirrels make? Proposed by USJL, ST.

Consider all the ways of writing exactly ten times each of the numbers $0, 1, 2, \ldots , 9$ in the squares of a $10 \times 10$ board. Find the greatest integer $n$ with the property that there is always a row or a column with $n$ different numbers.

Seyed has 998 white coins, a red coin, and an unusual coin with one red side and one white side. He can not see the color of the coins instead he has a scanner which checks if all of the coin sides touching the scanner glass are white. Is there any algorithm to find the red coin by using the scanner at most 17 times? [i]Proposed by Seyed Reza Hosseini[/i]

There are $N{}$ mess-loving clerks in the office. Each of them has some rubbish on the desk. The mess-loving clerks leave the office for lunch one at a time (after return of the preceding one). At that moment all those remaining put half of rubbish from their desks on the desk of the one who left. Can it so happen that after all of them have had lunch the amount of rubbish at the desk of each one will be the same as before lunch if a) $N = 2{}$ and b) $N = 10$? [i]Alexey Zaslavsky[/i]

Once upon a time there are $n$ pairs of princes and princesses who are in love with each other. One day a witch comes along and turns all the princes into frogs; the frogs can be distinguished by sight but the princesses cannot tell which frog corresponds to which prince. The witch tells the princesses that if any of them kisses the frog that corresponds to the prince very that she loves then that frog will immediately transform back into a prince. If each princess can stand kissing at most $k$ frogs, what is the maximum number of princes they can be sure to save? (The princesses may take turns kissing in any order, communicate with each other and vary their strategy for future kisses depending on information gained from past kisses.)

A row of $1000$ numbers is written on the blackboard. We write a new row, below the first according to the rule: We write under every number $a$ the natural number, indicating how many times the number $a$ is encountered in the first line. Then we write down the third line: under every number $b$ -- the natural number, indicating how many times the number $b$ is encountered in the second line, and so on. a) Prove that there is a line that coincides with the preceding one. b) Prove that the eleventh line coincides with the twelfth. c) Give an example of the initial line such, that the tenth row differs from the eleventh.

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

Let $m,n\geq 2$. One needs to cover the table $m \times n$ using only the following tiles: Tile 1 - A square $2 \times 2$. Tile 2 - A L-shaped tile with five cells, in other words, the square $3 \times 3$ [b]without[/b] the upper right square $2 \times 2$. Each tile 1 covers exactly $4$ cells and each tile 2 covers exactly $5$ cells. Rotation is allowed. Determine all pairs $(m,n)$, such that the covering is possible.

A [i]simple polygon[/i] is a polygon whose perimeter does not self-intersect. Suppose a simple polygon $\mathcal P$ can be tiled with a finite number of parallelograms. Prove that regardless of the tiling, the sum of the areas of all rectangles in the tiling is fixed.\\ [i]Note:[/i] Points will be awarded depending on the generality of the polygons for which the result is proven.

Circle is divided into $n$ arcs by $n$ marked points on the circle. After that circle rotate an angle $ 2\pi k/n $ (for some positive integer $ k $), marked points moved to $n$ [i] new points [/i], dividing the circle into $ n $ [i] new arcs[/i]. Prove that there is a new arc that lies entirely in the one of the old arсs. (It is believed that the endpoints of arcs belong to it.) [i]I. Mitrophanov[/i]

Find the number of binary sequences $S$ of length $2015$ such that for any two segments $I_1, I_2$ of $S$ of the same length, we have • The sum of digits of $I_1$ differs from the sum of digits of $I_2$ by at most $1$, • If $I_1$ begins on the left end of S then the sum of digits of $I_1$ is not greater than the sum of digits of $I_2$, • If $I_2$ ends on the right end of S then the sum of digits of $I_2$ is not less than the sum of digits of $I_1$. Lê Anh Vinh

Let $n$ be a positive integer. Daniel and Merlijn are playing a game. Daniel has $k$ sheets of paper lying next to each other on a table, where $k$ is a positive integer. On each of the sheets, he writes some of the numbers from $1$ up to $n$ (he is allowed to write no number at all, or all numbers). On the back of each of the sheets, he writes down the remaining numbers. Once Daniel is finished, Merlijn can flip some of the sheets of paper (he is allowed to flip no sheet at all, or all sheets). If Merlijn succeeds in making all of the numbers from $1$ up to n visible at least once, then he wins. Determine the smallest $k$ for which Merlijn can always win, regardless of Daniel’s actions.

For a positive integer $n$, if $n$ is a product of two different primes and $n \equiv 2 \pmod 3$, then $n$ is called "special number." For example, $14, 26, 35, 38$ is only special numbers among positive integers $1$ to $50$. Prove that for any finite set $S$ with special numbers, there exist two sets $A, B$ such that [list] [*] $A \cap B = \emptyset, A \cup B = S$ [*] $||A| - |B|| \leq 1$ [*] For all primes $p$, the difference between number of elements in $A$ which is multiple of $p$ and number of elements in $B$ which is multiple of $p$ is less than or equal to $1$. [/list]

For a fixed $k$ with $4 \le k \le 9$ consider the set of all positive integers with $k$ decimal digits such that each of the digits from $1$ to $k$ occurs exactly once. Show that it is possible to partition this set into two disjoint subsets such that the sum of the cubes of the numbers in the first set is equal to the sum of the cubes in the second set.

$32$ competitors participate in a tournament. No two of them are equal and in a one against one match the better always wins. Show that the gold, silver, and bronze medal winners can be found in $39$ matches.

On the board are written $n$ natural numbers, $n\in \mathbb{N}$. In one move it is possible to choose two equal written numbers and increase one by $1$ and decrease the other by $1$. Prove that in this the game cannot be played more than $\frac{n^3}{6}$ moves.

Thirty numbers are placed on a circle. For every number $A$ we have: $A$ equals the absolute value of $(B- C)$, where $B$ and $C$ follow $A$ clockwise. The total sum of the numbers equals $1$. Find all the numbers. (Folklore)