There are $2023$ guinea pigs placed in a circle, from which everyone except one of them, call it $M$, has a mirror that points towards one of the $2022$ other guinea pigs. $M$ has a lantern that will shoot a light beam towards one of the guinea pigs with a mirror and will reflect to the guinea pig that the mirror is pointing and will keep reflecting with every mirror it reaches. Isaías will re-direct some of the mirrors to point to some other of the $2023$ guinea pigs. In the worst case scenario, what is the least number of mirrors that need to be re-directed, such that the light beam hits $M$ no matter the starting point of the light beam?

Let $P_1P_2\ldots P_n$ be a convex polygon in the plane. We assume that for any arbitrary choice of vertices $P_i,P_j$ there exists a vertex in the polygon $P_k$ distinct from $P_i,P_j$ such that $\angle P_iP_kP_j=60^{\circ}$. Show that $n=3$. [i]Radu Todor[/i]

A fly and $k$ spiders are placed in some vertices of $2012 \times 2012$ lattice. One move consists of following: firstly, fly goes to some adjacent vertex or stays where it is and then every spider goes to some adjacent vertex or stays where it is (more than one spider can be in the same vertex). Spiders and fly knows where are the others all the time. a) Find the smallest $k$ so that spiders can catch the fly in finite number of moves, regardless of their initial position. b) Answer the same question for three-dimensional lattice $2012\times 2012\times 2012$. (Vertices in lattice are adjacent if exactly one coordinate of one vertex is different from the same coordinate of the other vertex, and their difference is equal to $1$. Spider catches a fly if they are in the same vertex.)

[u]Round 1[/u] [b]p1.[/b] This year, the Mathathon consists of $7$ rounds, each with $3$ problems. Another math test, Aspartaime, consists of $3$ rounds, each with $5$ problems. How many more problems are on the Mathathon than on Aspartaime? [b]p2.[/b] Let the solutions to $x^3 + 7x^2 - 242x - 2016 = 0 $be $a, b$, and $c$. Find $a^2 + b^2 + c^2$. (You might find it helpful to know that the roots are all rational.) [b]p3.[/b] For triangle $ABC$, you are given $AB = 8$ and $\angle A = 30^o$ . You are told that $BC$ will be chosen from amongst the integers from $1$ to $10$, inclusive, each with equal probability. What is the probability that once the side length $BC$ is chosen there is exactly one possible triangle $ABC$? [u]Round 2 [/u] [b]p4.[/b] It’s raining! You want to keep your cat warm and dry, so you want to put socks, rain boots, and plastic bags on your cat’s four paws. Note that for each paw, you must put the sock on before the boot, and the boot before the plastic bag. Also, the items on one paw do not affect the items you can put on another paw. How many different orders are there for you to put all twelve items of rain footwear on your cat? [b]p5.[/b] Let $a$ be the square root of the least positive multiple of $2016$ that is a square. Let $b$ be the cube root of the least positive multiple of $2016$ that is a cube. What is $ a - b$? [b]p6.[/b] Hypersomnia Cookies sells cookies in boxes of $6, 9$ or $10$. You can only buy cookies in whole boxes. What is the largest number of cookies you cannot exactly buy? (For example, you couldn’t buy $8$ cookies.) [u]Round 3 [/u] [b]p7.[/b] There is a store that sells each of the $26$ letters. All letters of the same type cost the same amount (i.e. any ‘a’ costs the same as any other ‘a’), but different letters may or may not cost different amounts. For example, the cost of spelling “trade” is the same as the cost of spelling “tread,” even though the cost of using a ‘t’ may be different from the cost of an ‘r.’ If the letters to spell out $1$ cost $\$1001$, the letters to spell out $2$ cost $\$1010$, and the letters to spell out $11$ cost $\$2015$, how much do the letters to spell out $12$ cost? [b]p8.[/b] There is a square $ABCD$ with a point $P$ inside. Given that $PA = 6$, $PB = 9$, $PC = 8$. Calculate $PD$. [b]p9.[/b] How many ordered pairs of positive integers $(x, y)$ are solutions to $x^2 - y^2 = 2016$? [u]Round 4 [/u] [b]p10.[/b] Given a triangle with side lengths $5, 6$ and $7$, calculate the sum of the three heights of the triangle. [b]p11. [/b]There are $6$ people in a room. Each person simultaneously points at a random person in the room that is not him/herself. What is the probability that each person is pointing at someone who is pointing back to them? [b]p12.[/b] Find all $x$ such that $\sum_{i=0}^{\infty} ix^i =\frac34$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782837p24446063]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

On a table, there are $10\,000$ matches, two of which are inside a box. Ana and Beto take turns playing the following game. On each turn, a player adds to the box a number of matches equal to a proper divisor of the current number of matches in the box. The game ends when, for the first time, there are more than $2024$ matches in the box and the person who played the last turn is the winner. If Ana starts the game, determine who has a winning strategy.

A domino is a rectangle whose length is twice its width. Any square can be divided into seven dominoes, for example as shown in the figure below. [img]https://cdn.artofproblemsolving.com/attachments/7/6/c055d8d2f6b7c24d38ded7305446721e193203.png[/img] a) Show that you can divide a square into $n$ dominoes for all $n \ge 5$. b) Show that you cannot divide a square into three or four dominoes.

On each cell of a $3\times 6$ the board lies one coin. It is known that some two coins lying on adjacent cells are fake. They have the same weigh, but are lighter than the real ones. All the other coins are real. How can one find both counterfeit coins in three weightings on a double-pan balance, without using weights? [i]Proposed by K. Knop[/i]

Find the number of pairs of sets $(A, B)$ satisfying $A \subseteq B \subseteq \{1, 2, ...,10\}$

In a group of people, there are some mutually friendly pairs. For positive integer $k\ge3$ we say the group is $k$-great, if every (unordered) $k$-tuple of people from the group can be seated around a round table it the way that all pairs of neighbors are mutually friendly. [i](Since this was the 67th year of CZE/SVK MO,)[/i] show that if the group is 6-great, then it is 7-great as well. [b]Bonus[/b] (not included in the competition): Determine all positive integers $k\ge3$ for which, if the group is $k$-great, then it is $(k+1)$-great as well.

A one-player game is played on a $m \times n$ table with $m \times n$ nuts. One of the nuts' sides is black, and the other side of them is white. In the beginning of the game, there is one nut in each cell of the table and all nuts have their white side upwards except one cell in one corner of the table which has the black side upwards. In each move, we should remove a nut which has its black side upwards from the table and reverse all nuts in adjacent cells (i.e. the cells which share a common side with the removed nut's cell). Find all pairs $(m,n)$ for which we can remove all nuts from the table.

Lalo and Sergio play in a regular polygon of $n\geq 4$ sides. In his turn, Lalo paints a diagonal or side of pink, and in his turn Sergio paint a diagonal or side of orange. Wins the game who achieve paint the three sides of a triangle with his color, if none of the players can win, they game tie. Lalo starts playing. Determines all natural numbers $n$ such that one of the players have winning strategy.

At the end of a soccer tournament in which any pair of teams played between them exactly once, and in which there were not draws, it was observed that for any three teams $A, B$ and C, if $A$ defeated $B$ and $B$ defeated $C$, then $A$ defeated $C$. Any team calculated the difference (positive) between the number of games that it won and the number of games it lost. The sum of all these differences was $5000$. How many teams played in the tournament? Find all possible answers.

For any 2 convex polygons $S$ and $T$, if all the vertices of $S$ are vertices of $T$, call $S$ a sub-polygon of $T$. [b]I. [/b]Prove that for an odd number $n \geq 5$, there exists $m$ sub-polygons of a convex $n$-gon such that they do not share any edges, and every edge and diagonal of the $n$-gon are edges of the $m$ sub-polygons. [b]II.[/b] Find the smallest possible value of $m$.

Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.

On a plane there is an invisible [color=#eee]rabbit[/color] (rabbit) hiding on a lattice point. We want to put $n$ hunters on some lattice points to catch the rabbit. In a turn each hunter can choose to shoot to left/right or top/bottom direction. On the $i$th turn there will be these steps in order 1. The rabbit jumps to an adjacent lattice point on the top, bottom, left, or right. 2. item Each hunter moves to an adjacent lattice point on the top, bottom, left or right (each hunter can move to different direction). Then they shoot a bullet which travels $\frac{334111214}{334111213}i$ units on the directions they chose. If a bullet hits the rabbit then it is caught. Find the smallest number $n$ such that the rabbit would definitely be caught in a finite number of turns. [i]Proposed by tob8y[/i]

Consider infinite diagrams [asy] import graph; size(90); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; label("$n_{00} \ n_{01} \ n_{02} \ldots$", (1.14,1.38), SE*lsf); label("$n_{10} \ n_{11} \ n_{12} \ldots$", (1.2,1.8), SE*lsf); label("$n_{20} \ n_{21} \ n_{22} \ldots$", (1.2,2.2), SE*lsf); label("$\vdots \quad \vdots \qquad \vdots $", (1.32,2.72), SE*lsf); draw((1,1)--(3,1)); draw((1,1)--(1.02,2.62)); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle); [/asy] where all but a finite number of the integers $n_{ij} , i = 0, 1, 2, \ldots, j = 0, 1, 2, \ldots ,$ are equal to $0$. Three elements of a diagram are called [i]adjacent[/i] if there are integers $i$ and $j$ with $i \geq 0$ and $j \geq 0$ such that the three elements are [b](i)[/b] $n_{ij}, n_{i,j+1}, n_{i,j+2},$ or [b](ii)[/b] $n_{ij}, n_{i+1,j}, n_{i+2,j} ,$ or [b](iii)[/b] $n_{i+2,j}, n_{i+1,j+1}, n_{i,j+2}.$ An elementary operation on a diagram is an operation by which three [i]adjacent[/i] elements $n_{ij}$ are changed into $n_{ij}'$ in such a way that $|n_{ij}-n_{ij}'|=1.$ Two diagrams are called equivalent if one of them can be changed into the other by a finite sequence of elementary operations. How many inequivalent diagrams exist?

a) A certain committee has gathered $40$ times. There were $10$ members on every meeting. Not a single couple has met on the meetings twice. Prove that there were no less then $60$ members in the committee. b) Prove that you can not construct more then $30$ subcommittees of $5$ members from the committee of $25$ members, with no couple of subcommittees having more than one common member.

On a circle there are $n$ red and $n$ blue arcs given in such a way that each red arc intersects each blue one. Prove that some point is contained by at least $n$ of the given coloured arcs.

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: [list=1] [*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell. [/list] At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. [i]Proposed by Warut Suksompong, Thailand[/i]

In an urn we have red and blue balls. A person has invented the next game: he extracts balls until he realises for the first time that the number of blue balls is equal to the number of red balls. After a such game he finds out that he has extracted 10 balls, and that there does not exist 3 consecutive balls of the same color. Prove that the fifth and the sixth balls have different collors.

Given two heaps of checkers. the bigger contains $m$ checkers, the smaller -- $n$ ($m>n$). Two players are taking checkers in turn from the arbitrary heap. The players are allowed to take from the heap a number of checkers (not zero) divisible by the number of checkers in another heap. The player that takes the last checker in any heap wins. a) Prove that if $m > 2n$, than the first can always win. b) Find all $x$ such that if $m > xn$, than the first can always win.

We will call a rectangular table filled with natural numbers [i]“good”[/i], if for each two rows, there exist a column for which its two cells that are also in these two rows, contain numbers of different parity. Prove that for $\forall$ $n>2$ we can erase a column from a [i]good[/i] $n$ x $n$ table so that the remaining $n$ x $(n-1)$ table is also [i]good[/i].

There are $n{}$ wise men in a hall and everyone sees each other. Each man will wear a black or white hat. The wise men should simultaneously write down on their piece of paper a guess about the color of their hat. If at least one does not guess, they will all be executed. The wise men can discuss a strategy before the test and they know that the layout of the hats will be chosen randomly from the set of all $2^n$ layouts. They want to choose their strategy so that the number of layouts for which everyone guesses correctly is as high as possible. What is this number equal to?

[list] [*] A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this? [*] Previous problem for the grid of $5\times 5$ lattice.[/list]