Every month a forester Ermolay has planted 2000 trees along a fence. On every tree, he has written how many oaks there are among itself and trees at his right and left. This way a sequence of 2000 numbers was created. How many distinct sequences could the forester Ermolay get? (oak is a certain type of tree) [I]Proposed by A. Khrabrov, D.Rostovski[/i]

Consider an infinite square board with an integer written in each square. Assume that for each square the integer in it is equal to the sum of its neighbor to the left and its neighbor above. Assume also that there exists a row $R_0$ in the board such that all numbers in $R_0$ are positive. Denote by $R_1$ the row below $R_0$ , by $R_2$ the row below $R_1$ etc. Show that for each $N \ge 1$ the row $R_N$ cannot contain more than $N$ zeroes.

For each positive integer $ n$, let $ f(n)$ denote the number of ways of representing $ n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $ f(4) \equal{} 4$, because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1. Prove that, for any integer $ n \geq 3$ we have $ 2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}$.

Determine all non empty sets $C_1, C_2, C_3, \cdots $ such that each one of them has a finite number of elements, all their elements are positive integers, and they satisfy the following property: For any positive integers $n$ and $m$, the number of elements in the set $C_n$ plus the number of elements in the set $C_m$ equals the sum of the elements in the set $C_{m + n}$. [i]Note:[/i] We denote $\lvert C_n \lvert$ the number of elements in the set $C_n$, and $S_k$ as the sum of the elements in the set $C_n$ so the problem's condition is that for every $n$ and $m$: \[\lvert C_n \lvert + \lvert C_m \lvert = S_{n + m}\] is satisfied.

Can the numbers $1,2,...,121$ be written in the cells of an $11\times 11$ board in such a way that any two consecutive numbers are in adjacent cells (sharing a side), and all perfect squares are in the same column?

A number of signal lights are equally spaced along a one-way railroad track, labeled in oder $ 1,2, \ldots, N, N \geq 2.$ As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number of trains that can be parked motionless at a signal, one behind the other. (Assume the trains have zero length.) A series of $ K$ freight trains must be driven from Signal 1 to Signal $ N.$ Each train travels at a distinct but constant spped at all times when it is not blocked by the safety rule. Show that, regardless of the order in which the trains are arranged, the same time will elapse between the first train's departure from Signal 1 and the last train's arrival at Signal $ N.$

Find the least number of elements of a finite set $A$ such that there exists a function $f : \left\{1,2,3,\ldots \right\}\rightarrow A$ with the property: if $i$ and $j$ are positive integers and $i-j$ is a prime number, then $f(i)$ and $f(j)$ are distinct elements of $A$.

Assume that the connected graph $G$ has $n$ vertices all with degree at least three. Prove that there exists a spanning tree of $G$ with more than $\frac{2}{9}n$ leaves.

There are $2021$ points on a circle. Kostya marks a point, then marks the adjacent point to the right, then he marks the point two to its right, then three to the next point's right, and so on. Which move will be the first time a point is marked twice? [i]K. Kokhas[/i]

Using squares of side 1, a stair-like figure is formed in stages following the pattern of the drawing. For example, the first stage uses 1 square, the second uses 5, etc. Determine the last stage for which the corresponding figure uses less than 2014 squares. [img]http://www.artofproblemsolving.com/Forum/download/file.php?id=49934[/img]

Baron Munchausen has a set of $50$ coins. The mass of each is a distinct positive integer not exceeding $100$, and the total mass is even. The Baron claims that it is not possible to divide the coins into two piles with equal total mass. Can the Baron be right?

Let $S = \{1, 2, \ldots, n\}$ for some positive integer $n$, and let $A$ be an $n$-by-$n$ matrix having as entries only ones and zeroes. Define an infinite sequence $\{x_i\}_{i \ge 0}$ to be [i]strange[/i] if: [list] [*] $x_i \in S$ for all $i$, [*] $a_{x_kx_{k+1}} = 1$ for all $k$, where $a_{ij}$ denotes the element in the $i^{\text{th}}$ row and $j^{\text{th}}$ column of $A$. [/list] Prove that the set of strange sequences is empty if and only if $A$ is nilpotent, i.e. $A^m = 0$ for some integer $m$.

Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of one of its elements.

There are $2015$ points on a plane and no two distances between them are equal. We call the closest $22$ points to a point its $neighbours$. If $k$ points share the same neighbour, what is the maximum value of $k$?

Each face of a regular tetrahedron with edge length $2011$ is divided into $2011^2$ equilateral triangles of side length $1$, created by drawing lines parallel to each edge. Bruno and Mariano take turns marking one of the unit triangles. Except for the first move, every triangle marked must share at least one point with the triangle marked in the previous move. Bruno plays first. The game ends when a player cannot make a move, and that player loses. Determine which of the two players has a winning strategy and describe the strategy.

[b]6.1 / 7.1[/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]6.2 [/b] The natural numbers are arranged in a $3 \times 3$ table. Kolya and Petya crossed out 4 numbers each. It turned out that the sum of the numbers crossed out by Petya is three times the sum numbers crossed out by Kolya. What number is left uncrossed? $$\begin{tabular}{|c|c|c|}\hline 4 & 12 & 8 \\ \hline 13 & 24 & 14 \\ \hline 7 & 5 & 23 \\ \hline \end{tabular} $$ [b]6.3 [/b] Misha and Sasha left at noon on bicycles from city A to city B. At the same time, I left from B to A Vanya. All three travel at constant but different speeds. At one o'clock Sasha was exactly in the middle between Misha and Vanya, and at half past one Vanya was in the middle between Misha and Sasha. When Misha will be exactly in the middle between Sasha and Vanya? [b]6.4[/b] There are $35$ piles of nuts on the table. Allowed to add one nut at a time to any $23$ piles. Prove that by repeating this operation, you can equalize all the heaps. [b]6.5[/b] There are $64$ vertical stripes on the round drum, and each stripe you need to write down a six-digit number from digits $1$ and $2$ so that all the numbers were different and any two adjacent ones differed in exactly one discharge. How to do this? [b]6.6 / 7.6[/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].

Mr. Murgatroyd decides to throw his class a pizza party, but he's going to make them hunt for it first. He chooses eleven locations in the school, which we'll call $1, 2, \ldots, 11$. His plan is to tell students to start at location $1$, and at each location $n$ from $1$ to $10$, they will find a message directing them to go to location $n+1$; at location $11$, there's pizza! Mr. Murgatroyd sends his teaching assistant to post the ten messages in locations $1$ to $10$. Unfortunately, the assistant jumbles up the message cards at random before posting them. If the students begin at location $1$ as planned and follow the directions at each location, show that they will still get to the pizza.

The edges of $K_{2017}$ are each labeled with $1,2,$ or $3$ such that any triangle has sum of labels at least $5.$ Determine the minimum possible average of all $\dbinom{2017}{2}$ labels. (Here $K_{2017}$ is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.) [i]Proposed by Michael Ma[/i]

On a board, Ana and Bob start writing $0$s and $1$s alternatively until each of them has written $2021$ digits. Ana starts this procedure and each of them always adds a digit to the right of the already existing ones. Ana wins the game if, after they stop writing, the resulting number (in binary) can be written as the sum of two squares. Otherwise, Bob wins. Determine who has a winning strategy.

Ana draws a checkered board that has at least 20 rows and at least 24 columns. Then, Beto must completely cover that board, without holes or overlaps, using only pieces of the following two types: Each piece must cover exactly 4 or 3 squares of the board, as shown in the figure, without leaving the board. It is permitted to rotate the pieces and it is not necessary to use all types of pieces. Explain why, regardless of how many rows and how many columns Ana's board has, Beto can always complete his task.

Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \] are pairwise disjoint.

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]D.16 / L.6[/b] Alex has $100$ Bluffy Funnies in some order, which he wants to sort in order of height. They’re already almost in order: each Bluffy Funny is at most $1$ spot off from where it should be. Alex can only swap pairs of adjacent Bluffy Funnies. What is the maximum possible number of swaps necessary for Alex to sort them? [b]D.17[/b] I start with the number $1$ in my pocket. On each round, I flip a coin. If the coin lands heads heads, I double the number in my pocket. If it lands tails, I divide it by two. After five rounds, what is the expected value of the number in my pocket? [b]D.18 / L.12[/b] Point $P$ inside square $ABCD$ is connected to each corner of the square, splitting the square into four triangles. If three of these triangles have area $25$, $25$, and $15$, what are all the possible values for the area of the fourth triangle? [b]D.19[/b] Mr. Stein and Mr. Schwartz are playing a yelling game. The teachers alternate yelling. Each yell is louder than the previous and is also relatively prime to the previous. If any teacher yells at $100$ or more decibels, then they lose the game. Mr. Stein yells first, at $88$ decibels. What volume, in decibels, should Mr. Schwartz yell at to guarantee that he will win? [b]D.20 / L.15[/b] A semicircle of radius $1$ has line $\ell$ along its base and is tangent to line $m$. Let $r$ be the radius of the largest circle tangent to $\ell$, $m$, and the semicircle. As the point of tangency on the semicircle varies, the range of possible values of $r$ is the interval $[a, b]$. Find $b - a$. [u]Set 5[/u] [b]D.21 / L.14[/b] Hungryman starts at the tile labeled “$S$”. On each move, he moves $1$ unit horizontally or vertically and eats the tile he arrives at. He cannot move to a tile he already ate, and he stops when the sum of the numbers on all eaten tiles is a multiple of nine. Find the minimum number of tiles that Hungryman eats. [img]https://cdn.artofproblemsolving.com/attachments/e/7/c2ecc2a872af6c4a07907613c412d3b86cd7bc.png [/img] [b]D.22 / L.11[/b] How many triples of nonnegative integers $(x, y, z)$ satisfy the equation $6x + 10y +15z = 300$? [b]D.23 / L.16[/b] Anson, Billiam, and Connor are looking at a $3D$ figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a $5 \times 5$ square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure. [b]D.24 / L.13[/b] Tse and Cho are playing a game. Cho chooses a number $x \in [0, 1]$ uniformly at random, and Tse guesses the value of $x(1 - x)$. Tse wins if his guess is at most $\frac{1}{50}$ away from the correct value. Given that Tse plays optimally, what is the probability that Tse wins? [b]D.25 / L.20[/b] Find the largest solution to the equation $$2019(x^{2019x^{2019}-2019^2+2019})^{2019}) = 2019^{x^{2019}+1}.$$ [u]Set 6[/u] [i]This round is an estimation round. No one is expected to get an exact answer to any of these questions, but unlike other rounds, you will get points for being close. In the interest of transparency, the formulas for determining the number of points you will receive are located on the answer sheet, but they aren’t very important when solving these problems.[/i] [b]D.26 / L.26[/b] What is the sum over all MBMT volunteers of the number of times that volunteer has attended MBMT (as a contestant or as a volunteer, including this year)? Last year there were $47$ volunteers; this is the fifth MBMT. [b]D.27 / L.27[/b] William is sharing a chocolate bar with Naveen and Kevin. He first randomly picks a point along the bar and splits the bar at that point. He then takes the smaller piece, randomly picks a point along it, splits the piece at that point, and gives the smaller resulting piece to Kevin. Estimate the probability that Kevin gets less than $10\%$ of the entire chocolate bar. [b]D.28 / L.28[/b] Let $x$ be the positive solution to the equation $x^{x^{x^x}}= 1.1$. Estimate $\frac{1}{x-1}$. [b]D.29 / L.29[/b] Estimate the number of dots in the following box: [img]https://cdn.artofproblemsolving.com/attachments/8/6/416ba6379d7dfe0b6302b42eff7de61b3ec0f1.png[/img] It may be useful to know that this image was produced by plotting $(4\sqrt{x}, y)$ some number of times, where x, y are random numbers chosen uniformly randomly and independently from the interval $[0, 1]$. [b]D.30 / L.30[/b] For a positive integer $n$, let $f(n)$ be the smallest prime greater than or equal to $n$. Estimate $$(f(1) - 1) + (f(2) - 2) + (f(3) - 3) + ...+ (f(10000) - 10000).$$ For $26 \le i \le 30$, let $E_i$ be your team’s answer to problem $i$ and let $A_i$ be the actual answer to problem $i$. Your score $S_i$ for problem $i$ is given by $S_{26} = \max(0, 12 - |E_{26} - A_{26}|/5)$ $S_{27} = \max(0, 12 - 100|E_{27} - A_{27}|)$ $S_{28} = \max(0, 12 - 5|E_{28} - A_{28}|))$ $S_{29} = 12 \max \left(0, 1 - 3 \frac{|E_{29} - A_{29}|}{A_{29}} \right)$ $S_{30} = \max (0, 12 - |E_{30} - A_{30}|/2000)$ PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We call a set of cells on a checkered plane [i]rook-connected[/i] if from any of its cells one can get to any other by moving along the cells of this set by moving the rook (the rook is allowed to fly through fields that do not belong to our set). Prove that a [i]rook-connected[/i] set of $100$ cells can be divided into pairs of cells, lying in one row or in one column.

Let $n$ be a positive integer and set $S=\{n,n+1,n+2,...,5n\}$ $a)$ If set $S$ is divided into two disjoint sets , prove that there exist three numbers $x$, $y$ and $z$(possibly equal) which belong to same subset of $S$ and $x+y=z$ $b)$ Does $a)$ hold for set $S=\{n,n+1,n+2,...,5n-1\}$

Anya, Borya, and Vasya listed words that could be formed from a given set of letters. They each listed a different number of words : Anya listed the most, Vasya the least . They were awarded points as follows. Each word listed by only one of them scored $2$ points for this child. Each word listed by two of them scored $1$ point for each of these two children. Words listed by all three of them scored $0$ points. Is it possible that Vasya got the highest score, and Anya the lowest? (A Shapovalov)