The following picture illustrates the model of the Tháp Rùa (The Central Tower in Hanoi), which consists of $3$ levels. For the first and second levels, each has $10$ doorways among which $3$ doorways are located at the front, $3$ at the back, $2$ on the right side and $2$ on the left side. The third level is on the top of the tower model and has no doorways. The front of the tower model is signified by a circle symbol on the top level (Figure). We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements: (a) The top level is painted with only one color. (b) The $3$ doorways at the front on the second level are painted with the same color. (c) The $3$ doorways at the front on the first level are painted with the same color. (d) Each of the remaining $14$ doorways is painted with one of the three colors in such a way that any two adjacent doorways with a common side on the same level, including the pairs at the same corners, are painted with different colors. How many ways are there to paint the first level? How many ways are there to paint the entire tower model? [img]https://cdn.artofproblemsolving.com/attachments/f/9/2249f8595a8efe711680f3dfb8ff959c140a21.png[/img]

Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?

On the table there are $2016$ coins. Two players play the following game making alternating moves. In one move it is allowed to take $1, 2$ or $3$ coins. The player who takes the last coin wins. Which player has a winning strategy?

For positive integers $m, n$ ($m>n$), $a_{n+1}, a_{n+2}, ..., a_m$ are non-negative integers that satisfy the following inequality. $$ 2> \frac{a_{n+1}}{n+1} \ge \frac{a_{n+2}}{n+2} \ge \cdots \ge \frac{a_m}{m}$$ Find the number of pair $(a_{n+1}, a_{n+2}, \cdots, a_m)$.

[b]7.1 / 6.1[/b] There are $8$ rooks on the chessboard such that no two of them they don't hit each other. Prove that the black squares contain an even number of rooks. [b]7.2[/b] The sides of triangle $ABC$ are extended as shown in the figure. At this $AA' = 3 AB$,, $BB' = 5BC$ , $CC'= 8 CA$. How many times is the area of the triangle $ABC$ less than the area of the triangle $A'B'C' $? [img]https://cdn.artofproblemsolving.com/attachments/9/f/06795292291cd234bf2469e8311f55897552f6.png[/img] [url=https://artofproblemsolving.com/community/c893771h1860178p12579333]7.3[/url] Prove the equality $$\frac{2}{x^2-1}+\frac{4}{x^2-4} +\frac{6}{x^2-9}+...+\frac{20}{x^2-100} =\frac{11}{(x-1)(x+10)}+\frac{11}{(x-2)(x+9)}+...+\frac{11}{(x-10)(x+1)}$$ [url=https://artofproblemsolving.com/community/c893771h1861966p12597273]7.4* / 8.4 *[/url] (asterisk problems in separate posts) [b]7.5 [/b]. The collective farm consists of $4$ villages located in the peaks of square with side $10$ km. It has the means to conctruct 28 kilometers of roads . Can a collective farm build such a road system so that was it possible to get from any village to any other? [b]7.6 / 6.6[/b] Two brilliant mathematicians were told in natural terms number and were told that these numbers differ by one. After that they take turns asking each other the same question: “Do you know my number?" Prove that sooner or later one of them will answer positively. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here[/url].

[b]p1.[/b] Given any four digit number $X = \underline{ABCD}$, consider the quantity $Y(X) = 2 \cdot \underline{AB}+\underline{CD}$. For example, if $X = 1234$, then $Y(X) = 2 \cdot 12+34 = 58$. Find the sum of all natural numbers $n \le 10000$ such that over all four digit numbers $X$, the number $n$ divides $X$ if and only if it also divides $Y(X)$. [b]p2.[/b] A sink has a red faucet, a blue faucet, and a drain. The two faucets release water into the sink at constant but different rates when turned on, and the drain removes water from the sink at a constant rate when opened. It takes $5$ minutes to fill the sink (from empty to full) when the drain is open and only the red faucet is on, it takes $10$ minutes to fill the sink when the drain is open and only the blue faucet is on, and it takes $15$ seconds to fill the sink when both faucets are on and the drain is closed. Suppose that the sink is currently one-thirds full of water, and the drain is opened. Rounded to the nearest integer, how many seconds will elapse before the sink is emptied (keeping the two faucets closed)? [b]p3.[/b] One of the bases of a right triangular prism is a triangle $XYZ$ with side lengths $XY = 13$, $YZ = 14$, $ZX = 15$. Suppose that a sphere may be positioned to touch each of the five faces of the prism at exactly one point. A plane parallel to the rectangular face of the prism containing $\overline{YZ}$ cuts the prism and the sphere, giving rise to a cross-section of area $A$ for the prism and area $15\pi$ for the sphere. Find the sum of all possible values of $A$. [b]p4.[/b] Albert, Brian, and Christine are hanging out by a magical tree. This tree gives each of them a stick, each of which have a non-negative real length. Say that Albert gets a branch of length $x$, Brian a branch of length $y$, and Christine a branch of length $z$, and the lengths follow the condition that $x+y+z = 2$. Let $m$ and $n$ be the minimum and maximum possible values of $xy+yz+xz-xyz$, respectively. What is $m+n$? [b]p5.[/b] Let $S := MATHEMATICSMATHEMATICSMATHE...$ be the sequence where $7$ copies of the word $MATHEMATICS$ are concatenated together. How many ways are there to delete all but five letters of $S$ such that the resulting subsequence is $CHMMC$? [b]p6.[/b] Consider two sequences of integers $a_n$ and $b_n$ such that $a_1 = a_2 = 1$, $b_1 = b_2 = 1$ and that the following recursive relations are satisfied for integers $n > 2$: $$a_n = a_{n-1}a_{n-2}-b_{n-1}b_{n-2},$$ $$b_n = b_{n-1}a_{n-2}+a_{n-1}b_{n-2}.$$ Determine the value of $$\sum_{1\le n\le2023,b_n \ne 0} \frac{a_n}{b_n}.$$ [b]p7.[/b] Suppose $ABC$ is a triangle with circumcenter $O$. Let $A'$ be the reflection of $A$ across $\overline{BC}$. If $BC =12$, $\angle BAC = 60^o$, and the perimeter of $ABC$ is $30$, then find $A'O$. [b]p8.[/b] A class of $10$ students wants to determine the class president by drawing slips of paper from a box. One of the students, Bob, puts a slip of paper with his name into the box. Each other student has a $\frac12$ probability of putting a slip of paper with their own name into the box and a $\frac12$ probability of not doing so. Later, one slip is randomly selected from the box. Given that Bob’s slip is selected, find the expected number of slips of paper in the box before the slip is selected. [b]p9.[/b] Let $a$ and $b$ be positive integers, $a > b$, such that $6! \cdot 11$ divides $x^a -x^b$ for all positive integers $x$. What is the minimum possible value of $a+b$? [b]p10.[/b] Find the number of pairs of positive integers $(m,n)$ such that $n < m \le 100$ and the polynomial $x^m+x^n+1$ has a root on the unit circle. [b]p11.[/b] Let $ABC$ be a triangle and let $\omega$ be the circle passing through $A$, $B$, $C$ with center $O$. Lines $\ell_A$, $\ell_B$, $\ell_C$ are drawn tangent to $\omega$ at $A$, $B$, $C$ respectively. The intersections of these lines form a triangle $XYZ$ where $X$ is the intersection of $\ell_B$ and $\ell_C$, $Y$ is the intersection of $\ell_C$ and $\ell_A$, and $Z$ is the intersection of $\ell_A$ and $\ell_B$. Let $P$ be the intersection of lines $\overline{OX}$ and $\overline{YZ}$. Given $\angle ACB = \frac32 \angle ABC$ and $\frac{AC}{AB} = \frac{15}{16}$ , find $\frac{ZP}{YP}$. [b]p12.[/b] Compute the remainder when $$\sum_{1\le a,k\le 2021} a^k$$ is divided by $2022$ (in the above summation $a,k$ are integers). [b]p13.[/b] Consider a $7\times 2$ grid of squares, each of which is equally likely to be colored either red or blue. Madeline would like to visit every square on the grid exactly once, starting on one of the top two squares and ending on one of the bottom two squares. She can move between two squares if they are adjacent or diagonally adjacent. What is the probability that Madeline may visit the squares of the grid in this way such that the sequence of colors she visits is alternating (i.e., red, blue, red,... or blue, red, blue,... )? [b]p14.[/b] Let $ABC$ be a triangle with $AB = 8$, $BC = 10$, and $CA = 12$. Denote by $\Omega_A$ the $A$-excircle of $ABC$, and suppose that $\Omega_A$ is tangent to $\overline{AB}$ and $\overline{AC}$ at $F$ and $E$, respectively. Line $\ell \ne \overline{BC}$ is tangent to $\Omega_A$ and passes through the midpoint of $\overline{BC}$. Let $T$ be the intersection of $\overline{EF}$ and $\ell$. Compute the area of triangle $ATB$. [b]p15.[/b] For any positive integer $n$, let $D_n$ be the set of ordered pairs of positive integers $(m,d)$ such that $d$ divides $n$ and gcd$(m,n) = 1$, $1 \le m \le n$. For any positive integers $a$, $b$, let $r(a,b)$ be the non-negative remainder when $a$ is divided by $b$. Denote by $S_n$ the sum $$S_n = \sum_{(m,d)\in D_n} r(m,d).$$ Determine the value of $S_{396}$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that if we erase $n-3$ diagonals of a regular $n$-gon, then we may still choose $n-3$ of the remaining diagonals such that they don't intersect inside the $n$-gon; but it is possible to erase $n-2$ diagonals such that this statement doesn't hold.

We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A [i]bicoloured triangulation[/i] is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which share an edge have different colours. We call a positive integer $n \ge 4$ [i]triangulable[/i] if every regular $n$-gon has a bicoloured triangulation such that for each vertex $A$ of the $n$-gon the number of black triangles of which $A$ is a vertex is greater than the number of white triangles of which $A$ is a vertex. Find all triangulable numbers.

A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.

Let $p$ be a prime number and $a_1, \ldots, a_{p-1}$ be not necessarily distinct nonzero elements of the $p$-element $\mathbb Z_p \pmod{p}$ group. Prove that each element of $\mathbb Z_p$ equals a sum of some of the $a_i$'s (the empty sum is $0$). (translated by Miklós Maróti)

In a country with $2015$ cities there is exactly one two-way flight between each city. The three flights made between three cities belong to at most two different airline companies. No matter how the flights are shared between some number of companies, if there is always a city in which $k$ flights belong to the same airline, what is the maximum value of $k$?

There are a million numbered chairs at a large round table. The Sultan has seated a million wise men on them. Each of them sees the thousand people following him in clockwise order. Each of them was given a cap of black or white color, and they must simultaneously write down on their own piece of paper a guess about the color of their cap. Those who do not guess will be executed. The wise men had the opportunity to agree on a strategy before the test. What is the largest number of survivors that they can guarantee?

A sequence $a_1, a_2, \ldots $ consisting of $1$'s and $0$'s satisfies for all $k>2016$ that \[ a_k=0 \quad \Longleftrightarrow \quad a_{k-1}+a_{k-2}+\cdots+a_{k-2016}>23. \] Prove that there exist positive integers $N$ and $T$ such that $a_k=a_{k+T}$ for all $k>N$.

Suppose we have a $8\times8$ chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into $1\times2$ domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers. (Day 1, 3rd problem author: Michal Rolínek)

There are three straight roads. Three pedestrians are moving along those roads, and they are NOT on one line in the initial moment. Prove that they will be one line not more than twice

Some students of a group were friends of some others. One day all students of the group take part in a picnic. During the picnic some friends had a quarrel with each other, but some other students became friends. After the picnic, the number of friends for each student changed by $1$. Prove that the number of students in the group was even.

On an infinite “squared” sheet six squares are shaded as in the diagram. On some squares there are pieces. It is possible to transform the positions of the pieces according to the following rule: if the neighbour squares to the right and above a given piece are free, it is possible to remove this piece and put pieces on these free squares. The goal is to have all the shaded squares free of pieces. Is it possible to reach this goal if (a) In the initial position there are $6$ pieces and they are placed on the $6$ shaded squares? (b) In the initial position there is only one piece, located in the bottom left shaded square? [img]https://cdn.artofproblemsolving.com/attachments/2/d/0d5cbc159125e2a84edd6ac6aca5206bf8d83b.png[/img] (M Kontsevich, Moscow)

Let \(p\) a prime number, and \(N\) the number of matrices \(p \times p\) \[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1p}\\ a_{21} & a_{22} & \ldots & a_{2p}\\ \vdots & \vdots & \ddots & \vdots \\ a_{p1} & a_{p2} & \ldots & a_{pp} \end{array}\] such that \(a_{ij} \in \{0,1,2,\ldots,p\} \) and if \(i \leq i^\prime\) and \(j \leq j^\prime\), then \(a_{ij} \leq a_{i^\prime j^\prime}\). Find \(N \pmod{p}\).

Into how many parts can a convex $n$-gon be divided by its diagonals if no three diagonals meet at one point?

Determine the smallest number $n \in Z_+$, which can be written as $n = \Sigma_{a\in A}a^2$, where $A$ is a finite set of positive integers and $\Sigma_{a\in A}a= 2014$. In other words: what is the smallest positive number which can be written as a sum of squares of different positive integers summing to $2014$?

Each point in space is colored either blue or red. Show that there exists a unit square having exactly $0, 1$ or $4$ blue vertices.

The following is a code and is meant to be broken. 2 707 156 377 38 2 328 17 185 2 713 73 566 1130 328 73 38 259 471 38 17 566 2 134 707 38 274 377 328 38 1130 40 377 566 73 820 566 566 134 11 2 328 38 185 2 713 566 134 328 2 918 134 11 713 134 274 707 713 73 38 1130 17 134 707 11 820 707 707 38 17 713 73 38 134 566 40 2 918 377 566 134 713 38 328 820 274 4 38 566 707 156 377 38 707 40 2 918 377 566 134 713 38 328 820 274 4 38 566 134 707 713 73 38 2 328 707 991 38 566 713 377 713 73 38 707 38 918 38 328 713 73 707 73 377 566 713 2 328 707 991 38 566 532 820 38 707 713 134 377 328 377 328 713 73 134 707 713 38 707 713 185 2 713 73 566 1130 328 707 40 2 918 377 566 134 713 38 328 820 274 4 38 566 134 707 713 73 38 2 328 707 991 38 566 713 377 713 73 38 707 38 11 377 328 17 259 377 328 79 2 328 707 991 38 566 532 820 38 707 713 134 377 328 377 328 713 73 134 707 713 38 707 713 991 73 2 713 134 707 713 73 38 707 820 274 377 40 713 73 38 134 566 40 2 918 377 566 134 713 38 328 820 274 4 38 566 707

The familiar $3$-dimensional cube has $6$ $2$-dimensional faces, $12$ $1$-dimensional edges, and $8$ $0$-dimensional vertices. Find the number of $9$-dimensional sub-subfaces in a $12$-dimensional cube.

We color some unit squares in a $ 99\times 99 $ square grid with one of $ 5 $ given distinct colors, such that each color appears the same number of times. On each row and on each column there are no differently colored unit squares. Find the maximum possible number of colored unit squares.

Let $k$ be a positive integer. The little one and the magician on the skywalk play a game. Initially, there are $N = 2^k$ distinct balls line up in a row, with each of the ball covered by a cup. On each turn, the little one chooses two cups, then the magician can either swap the balls in the two cups, or do a fake move so that the balls in the two cups stay the same. The little one cannot distinguish whether the magician fakes a move on not, nor can she observe the balls inside the cups. After $M = k \times 2^{k-1}$ turns, the magician opens all cups so the little one can check the ball in each of the cups. If the little one can identify whether the magician fakes a move or not for each of the $M$ turns, then the little one win. Prove that the little one has a winning strategy. [i] Proposed by usjl[/i]