Let $ABC$ be an equilateral triangle of side $1$. There are three grasshoppers sitting in $A$, $B$, $C$. At any point of time for any two grasshoppers separated by a distance $d$ one of them can jump over other one so that distance between them becomes $2kd$, $k,d$ are nonfixed positive integers. Let $M$, $N$ be points on rays $AB$, $AC$ such that $AM=AN=l$, $l$ is fixed positive integer. In a finite number of jumps all of grasshoppers end up sitting inside the triangle $AMN$. Find, in terms of $l$, the number of final positions of the grasshoppers. (Grasshoppers can leave the triangle $AMN$ during their jumps.)

In the game of [i]Ring Mafia[/i], there are $2019$ counters arranged in a circle. $673$ of these counters are mafia, and the remaining $1346$ counters are town. Two players, Tony and Madeline, take turns with Tony going first. Tony does not know which counters are mafia but Madeline does. On Tony’s turn, he selects any subset of the counters (possibly the empty set) and removes all counters in that set. On Madeline’s turn, she selects a town counter which is adjacent to a mafia counter and removes it. Whenever counters are removed, the remaining counters are brought closer together without changing their order so that they still form a circle. The game ends when either all mafia counters have been removed, or all town counters have been removed. Is there a strategy for Tony that guarantees, no matter where the mafia counters are placed and what Madeline does, that at least one town counter remains at the end of the game? [i]Proposed by Andrew Gu[/i]

The points of this lattice $4\times 4 = 16$ points can be vertices of squares. [asy] unitsize(1 cm); int i, j; for (i = 0; i <= 3; ++i) { draw((i,0)--(i,3)); draw((0,i)--(3,i)); } draw((1,1)--(2,2)--(1,3)--(0,2)--cycle); for (i = 0; i <= 3; ++i) { for (j = 0; j <= 3; ++j) { dot((i,j)); }} [/asy] Calculate the number of different squares that can be formed in a lattice of $100\times 100$ points.

On a circle, Alina draws $2019$ chords, the endpoints of which are all different. A point is considered [i]marked[/i] if it is either $\text{(i)}$ one of the $4038$ endpoints of a chord; or $\text{(ii)}$ an intersection point of at least two chords. Alina labels each marked point. Of the $4038$ points meeting criterion $\text{(i)}$, Alina labels $2019$ points with a $0$ and the other $2019$ points with a $1$. She labels each point meeting criterion $\text{(ii)}$ with an arbitrary integer (not necessarily positive). Along each chord, Alina considers the segments connecting two consecutive marked points. (A chord with $k$ marked points has $k-1$ such segments.) She labels each such segment in yellow with the sum of the labels of its two endpoints and in blue with the absolute value of their difference. Alina finds that the $N + 1$ yellow labels take each value $0, 1, . . . , N$ exactly once. Show that at least one blue label is a multiple of $3$. (A chord is a line segment joining two different points on a circle.)

Imagine a square scheme consisting of $n\times n$ fields with edge length $1$, where $n$ is an arbitrary positive integer. What is the maximum possible length of a route you can follow along the edges of the fields from point $A$ in the lower left corner to point $B$ in the upper right corner if you must never return to one point where you have been before? (The figure shows for $n = 5$ an example of a permitted route and an example of a not permitted route). [img]https://cdn.artofproblemsolving.com/attachments/6/e/92931d87f11b9fb3120b8dccc2c37c35a04456.png[/img]

In a room, there are $n$ lights numbered with positive integers $1, 2, \ldots, n$. At the beginning of the game subsets $S_1, S_2,\ldots,S_k$ of $\{1,\ldots, n\}$ can be chosen. For every integer $1\le i\le k$, there is a button that turns on the lights corresponding to the elements of $S_i$ and also a button that turns off all the lights corresponding to the elements of $S_i$. For any positive integer $n$, determine the smallest $k$ for which it is possible to choose the sets $S_1, S_2, \ldots, S_n$ in such a way that allows any combination of the $n$ lights to be turned on, starting from the state where all the lights are off. [i]Proposed by Kristóf Zólomy, Budapest[/i]

Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a $ 14\times 16$ rectangle. What maximum number of knights can this rectangle contain?

A $20 \times 20$ rectangular grid has been given. It is known that one of the grid's unit squares contains a hidden treasure. To find the treasure, we have been given an opportunity to order several scientific studies at the same time, results of which will be known only after some time. For each study we must choose one $1 \times 4$ rectangle, and the study will tell whether the rectangle contains the treasure. The $1 \times 4$ rectangle can be either horizontal or vertical, and it can extend over a side of the $20 \times 20$ grid, coming back in at the opposite side (you can think of the $20 \times 20$ grid as a torus - the opposite sides are connected). What is the minimal amount of studies that have to ordered for us to precisely determine the unit square containing the treasure?

Assume that $M \subset N$ has the property that every two numbers $m,n$ of $M$ satisfy $|m-n| \ge mn/25$. Prove that the set $M$ contains no more than $9$ elements. Decide whether there exists such set M.

Given several matrices of the same size. Given a positive integer $N$, let's say that a matrix is $N$-balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to $N$. (i) Show that every $2N$-balanced matrix can be written as a sum of two $N$-balanced matrices. (ii) Show that every $3N$-balanced matrix can be written as a sum of three $N$-balanced matrices.

Consider a finite collection of 3-element sets $A_i$, no two of which share more than one element, whose union has cardinality 2017. Show that the elements of this union can be coloured with two colors, blue and red, so that at least 64 elements are blue and each $A_i$ has at least one red element.

In a certain language there are only two letters, $A$ and $B$. We know that (i) There are no words of length $1$, and the only words of length $2$ are $AB$ and $BB$. (ii) A segment of length $n > 2$ is a word if and only if it can be obtained from a word of length less than $n$ by replacing each letter $B$ by some (not necessarily the same) word. Prove that the number of words of length $n$ is equal to $\frac{2^n +2\cdot (-1)^n}{3}$

A $100 \times100$ chessboard has a non-negative real number in each of its cells. A chessboard is [b]balanced[/b] if and only if the numbers sum up to one for each column of cells as well as each row of cells. Find the largest positive real number $x$ so that, for any balanced chessboard, we can find $100$ cells of it so that these cells all have number greater or equal to $x$, and no two of these cells are on the same column or row. [i]Proposed by CSJL.[/i]

Let $S$ be a set of $N \ge 3$ points in the plane. Assume that no $3$ points in $S$ are collinear. The segments with both endpoints in $S$ are colored in two colors. Prove that there is a set of $N - 1$ segments of the same color which don't intersect except in their endpoints such that no subset of them forms a polygon with positive area.

Let $n \ge 3$ points be given in the plane, no three of which lie on the same line. Determine whether it is always possible to draw an $n$-gon whose vertices are the given points and whose sides do not intersect. [i]Remark.[/i] The $n$-gon can be concave.

Daniel has an unlimited supply of tiles labeled “$2$” and “$n$” where $n$ is an integer. Find (with proof) all the values of $n$ that allow Daniel to fill an $8 \times 10$ grid with these tiles such that the sum of the values of the tiles in each row or column is divisible by $11$.

Daniel chooses a positive integer $n$ and tells Ana. With this information, Ana chooses a positive integer $k$ and tells Daniel. Daniel draws $n$ circles on a piece of paper and chooses $k$ different points on the condition that each of them belongs to one of the circles he drew. Then he deletes the circles, and only the $k$ points marked are visible. From these points, Ana must reconstruct at least one of the circumferences that Daniel drew. Determine which is the lowest value of $k$ that allows Ana to achieve her goal regardless of how Daniel chose the $n$ circumferences and the $k$ points.

Let $ n\geq 3 $ be an integer and let $ p\geq 2n-3 $ be a prime number. For a set $ M $ of $ n $ points in the plane, no 3 collinear, let $ f: M\to \{0,1,\ldots, p-1\} $ be a function such that (i) exactly one point of $ M $ maps to 0, (ii) if a circle $ \mathcal{C} $ passes through 3 distinct points of $ A,B,C\in M $ then $ \sum_{P\in M\cap \mathcal{C}} f(P) \equiv 0 \pmod p $. Prove that all the points in $ M $ lie on a circle.

Does the set $\{1,2,3,...,3000\}$ contain a subset $ A$ consisting of 2000 numbers that $x\in A$ implies $2x \notin A$ ?!! :?:

[u]Round 6[/u] 16. If you roll four fair $6$-sided dice, what is the probability that at least three of them will show the same value. 17. In a tetrahedron with volume $1$, four congruent speres are placed each tangent to three walls and three other spheres. What is the radii of each of the spheres. 18. let $f(x)$ be twice the number of letters in $x$. What is the sum of the unique $x,y$ such that $x \ne y$ and $f(x)=y$ and $f(y)=x$. [u]Round 7[/u] [b]p19.[/b] How many $4$ digit numbers with distinct digits $ABCD$ with $A$ not equal to $0$ are divisible by $11$? [b]p20.[/b] How many ($2$-dimensional) faces does a $2014$-dimensional hypercube have? [b]p21.[/b] How many subsets of the numbers $1,2,3,4...2^{2014}$ have a sum of $2014$ mod $2^{2014}$? [u]Round 8[/u] [b]p22.[/b] Two diagonals of a dodecagon measure $1$ unit and $2$ units. What is the area of this dodecagon? [b]p23.[/b] Square $ABCD$ has point $X$ on AB and $Y$ on $BC$ such that angle $ADX = 15$ degrees and angle $CDY = 30$ degrees. what is the degree measure of angle $DXY$? [b]p24.[/b] A $4\times 4$ grid has the numbers $1$ through $16$ placed in it, $1$ per cell, such that no adjacent boxes have cells adding to a number divisible by $3$. In how many ways is this possible? [u]Round 9[/u] [b]p25.[/b] Let $B$ and $C$ be the answers to $26$ and $27$ respectively.If $S(x)$ is the sum of the digits in $x$, what is the unique integer $A$ such that $S(A), S(B), S(C) \subset A,B,C$. [b]p26.[/b] Let $A$ and $C$ be the answers to $25$ and $27$ respectively. What is the third angle of a triangle with two of its angles equal to $A$ and $C$ degrees. [b]p27.[/b] Let $A$ and $B$ be the answers to $25$ and $26$ respectively. How many ways are there to put $A$ people in a line, with exactly $B$ places where a girl and a boy are next to each other. [u]Round 10[/u] [b]p28.[/b] What is the sum of all the squares of the digits to answers to problems on the individual, team, and theme rounds of this years LMT? If the correct answer is $N$ and you submit $M$, you will recieve $\lfloor 15 - 10  \times \log (M - N) \rfloor $. [b]p29.[/b] How many primes have all distinct digits, like $2$ or $109$ for example, but not $101$. If the correct answer is $N$ and you submit $M$, you will recieve $\left\lfloor 15 \min \left( \frac{M}{N} , \frac{N}{M} \right)\right\rfloor $. [b]p30.[/b] For this problem, you can use any $10$ mathematical symbols that you want, to try to achieve the highest possible finite number. (So "Twenty-one", " $\frac{12}{100} +843$" and "$\sum^{10}_{i=0} i^2 +1$" are all valid submissions.) If your team has the $N$th highest number, you will recieve $\max (16 - N, 0)$. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3156859p28695035]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Points $A_1, A_2, \ldots, A_{2022}$ are chosen on a plane so that no three of them are collinear. Consider all angles $A_iA_jA_k$ for distinct points $A_i, A_j, A_k$. What largest possible number of these angles can be equal to $90^\circ$? [i]Proposed by Anton Trygub[/i]

The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$. The students are initially divided up into $20$ groups of $101$, with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these $20$ students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as $\frac{a}{b}$, where $a, b$ are relatively prime integers, determine $a + b$.

Pål has more chickens than he can manage to keep track of. Therefore, he keeps an index card for each chicken. He keeps the cards in ten boxes, each of which has room for $2021$ cards. Unfortunately, Pål is quite disorganized, so he may lose some of his boxes. Therefore, he makes several copies of each card and distributes them among different boxes, so that even if he can only find seven boxes, no matter which seven, these seven boxes taken together will contain at least one card for each of his chickens. What is the largest number of chickens Pål can keep track of using this system?

Given a sequence $1,1,2,2,3,3,\ldots,1986,1986$, determine, with proof, if we can rearrange the sequence so that for any integer $1\le k \le 1986$ there are exactly $k$ numbers between the two “$k$”s.

The numbers from 1 to $n^2$ are randomly arranged in the cells of a $n \times n$ square ($n \geq 2$). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated. Let us call the [b]characteristic[/b] of the arrangement the smallest of these $n^2\left(n-1\right)$ fractions. What is the highest possible value of the characteristic ?