Grisha wrote $n$ different natural numbers, the sum of which does not exceed $S$. The saboteur added to each of them a number from the half-interval $[0, 1)$. The sabotage is successful if there exists two subsets, the sums of the numbers in which differ by no more than $1$. At what minimum $S$ can Grisha ensure that the sabotage will definitely not be succeeded?

A quarter of an checkered plane is given, infinite to the right and up. All its rows and columns are numbered starting from $0$. All cells with coordinates $(2n, n)$, were cut out from this figure, starting from $n = 1$. In each of the remaining cells they wrote a number, the number of paths from the corner cell to this one (you can only walk up and to the right and you cannot pass through the removed cells). Prove that for each removed cell the numbers to the left and below it differ by exactly $2$.

Given the nine-sided regular polygon $ A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $ \{A_1,A_2,...A_9\}$? $ \textbf{(A)} \ 30 \qquad \textbf{(B)} \ 36 \qquad \textbf{(C)} \ 63 \qquad \textbf{(D)} \ 66 \qquad \textbf{(E)} \ 72$

A unit square is removed from the corner of an $n\times n$ grid where $n \geq 2$. Prove that the remainder can be covered by copies of the "L-shapes" consisting of $3$ or $5$ unit square, as depicted in the figure below. Every square must be covered once and the L-shapes must not go over the bounds of the grid. [asy] import geometry; draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle); draw((-3.5,1)--(-2.5,1)--(-2.5,0)); draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle); draw((1.5,0)--(1.5,1)); draw((2.5,0)--(2.5,1)); draw((0.5,1)--(1.5,1)); draw((0.5,2)--(1.5,2)); [/asy][i]Estonian Olympiad, 2009[/i]

We number both the rows and the columns of an $8 \times 8$ chessboard with the numbers $1$ to $8$. Some grains of rice are placed on each square, in such a way that the number of grains on each square is equal to the product of the row and column numbers of the square. How many grains of rice are there on the entire chessboard?

On the $1$ km long ridge of Mount SPAM, there are $2006$ lemmings. In the beginning, each of them walks along the ridge in one of the two possible directions with speed $1$ m/s . When two lemmings meet, they both reverse the directions they walk but keep their walking speed. When some lemming reaches the end of the ridge, he falls down and dies. Find the least upper bound for the time it can take until all the lemmings are dead.

A set of positive integers $X$ is called [i]connected[/i] if $|X|\ge 2$ and there exist two distinct elements $m$ and $n$ of $X$ such that $m$ is a divisor of $n$. Determine the number of connected subsets of the set $\{1,2,\ldots,10\}$.

Let $S = \lbrace 1, 2, \ldots, 15 \rbrace$. Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions: [b]I.[/b] $|A_i| = 7, i = 1, 2, \ldots, n$; [b]II.[/b] $|A_i \cap A_j| \leq 3, 1 \leq i < j \leq n$ [b]III.[/b] For any 3-element subset $M$ of $S$, there exists $A_k$ such that $M \subset A_k$. Find the smallest possible value of $n$.

A cube side $5$ is made up of unit cubes. Two small cubes are [i]adjacent [/i] if they have a common face. Can we start at a cube adjacent to a corner cube and move through all the cubes just once? (The path must always move from a cube to an adjacent cube).

(a) The numbers $1, 2,... , 100$ are divided into two groups so that the sum of all numbers in one group is equal to that in the other. Prove that one can remove two numbers from each group so that the sums of all numbers in each group are still the same. (b) The numbers $1, 2 , ... , n$ are divided into two groups so that the sum of all numbers in one group is equal to that in the other . Is it true that for every such$ n > 4$ one can remove two numbers from each group so that the sums of all numbers in each group are still the same? (A Shapovalov) [(a) for Juniors, (a)+(b) for Seniors]

X-wich has a vibrant club-life. For every pair of inhabitants there is exactly one club to which they both belong. For every pair of clubs there is exactly one person who is a member of both. No club has fewer than $3$ members, and at least one club has $17$ members. How many people live in X-wich?

The picture shows a triangle divided into $25$ smaller triangles, numbered $1$ to $25$. Is it possible to place the same numbers in the square cells 5$\times 5$ so that any two numbers written in adjacent triangles were are also written in adjacent cells of the square? (The cells of a square are considered adjacent if they have a common side.) [img]https://cdn.artofproblemsolving.com/attachments/4/3/758fe5531ab3e576ef4712c095b393f8dff397.png[/img]

Cities $P_1,...,P_{1025}$ are connected to each other by airlines $A_1,...,A_{10}$ so that for any two distinct cities $P_k$ and $P_m$ there is an airline offering a direct flight between them. Prove that one of the airlines can offer a round trip with an odd number of flights.

A finite set $\mathcal{S}$ of positive integers is called cardinal if $\mathcal{S}$ contains the integer $|\mathcal{S}|$ where $|\mathcal{S}|$ denotes the number of distinct elements in $\mathcal{S}$. Let $f$ be a function from the set of positive integers to itself such that for any cardinal set $\mathcal{S}$, the set $f(\mathcal{S})$ is also cardinal. Here $f(\mathcal{S})$ denotes the set of all integers that can be expressed as $f(a)$ where $a \in \mathcal{S}$. Find all possible values of $f(2024)$ $\quad$ Proposed by Sutanay Bhattacharya

The numbers $ 1, 2,\ldots, 50 $ are written on a blackboard. Each minute any two numbers are erased and their positive difference is written instead. At the end one number remains. Which values can take this number?

What is the largest possible number of edges in a graph on $2n$ nodes, if there exists exactly one way to split its nodes into $n$ pairs so that the nodes from each pair are connected by an edge? [i]Proposed by Anton Trygub[/i]

A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off. [i]Proposed by Australia[/i]

Represent $1957$ as the sum of $12$ positive integer summands $a_1, a_2, ... , a_{12}$ for which the number $a_1! \cdot a_2! \cdot a_3! \cdot ... \cdot a_{12}!$ is minimal.

How many $n\times n$ matrices $A$, with all entries from the set $\{0, 1, 2\}$, are there, such that for all $i=1,2,\cdots,n$ $A_{ii} > \displaystyle{\sum \limits_{j=1 j\neq i}^n} A_{ij}$ [Where $A_{ij}$ is the $(i,j)$th element of the matrix $A$]

There is a ruble coin in each cell of the board with $2\times 200$. Dasha and Sonya play, taking turns making moves, Dasha starts. In one move, it is allowed to select any coin and move it: Dasha moves the coin to a diagonally adjacent cell, Sonya is to the side adjacent. If two coins end up in the same cell, one of them is immediately removed from the board and goes to Sonya. Sonya can stop the game at any time and take all the coins she has received. What is the biggest win she can get, no matter how she plays Dasha? [i]A. Kuznetsov[/i]

[u]Round 6 [/u] [b]p16.[/b] Triangle $ABC$ with $AB < AC$ is inscribed in a circle. Point $D$ lies on the circle and point $E$ lies on side $AC$ such that $ABDE$ is a rhombus. Given that $CD = 4$ and $CE = 3$, compute $AD^2$. [b]p17.[/b] Wam and Sang are walking on the coordinate plane. Both start at the origin. Sang walks to the right at a constant rate of $1$ m/s. At any positive time $t$ (in seconds),Wam walks with a speed of $1$ m/s with a direction of $t$ radians clockwise of the positive $x$-axis. Evaluate the square of the distance betweenWamand Sang in meters after exactly $5\pi$ seconds. [b]p18.[/b] Mawile is playing a game against Salamance. Every turn,Mawile chooses one of two moves: Sucker Punch or IronHead, and Salamance chooses one of two moves: Dragon Dance or Earthquake. Mawile wins if the moves used are Sucker Punch and Earthquake, or Iron Head and Dragon Dance. Salamance wins if the moves used are Iron Head and Earthquake. If the moves used are Sucker Punch and Dragon Dance, nothing happens and a new turn begins. Mawile can only use Sucker Punch up to $8$ times. All other moves can be used indefinitely. Assuming bothMawile and Salamance play optimally, find the probability thatMawile wins. [u]Round 7 [/u] [b]p19.[/b] Ephram is attempting to organize what rounds everyone is doing for the NEAML competition. There are $4$ rounds, of which everyone must attend exactly $2$. Additionally, of the 6 people on the team(Ephram,Wam, Billiam, Hacooba,Matata, and Derke), exactly $3$ must attend every round. In how many different ways can Ephram organize the teams like this? [b]p20.[/b] For some $4$th degree polynomial $f (x)$, the following is true: $\bullet$ $f (-1) = 1$. $\bullet$ $f (0) = 2$. $\bullet$ $f (1) = 4$. $\bullet$ $f (-2) = f (2) = f (3)$. Find $f (4)$. [b]p21.[/b] Find the minimum value of the expression $\sqrt{5x^2-16x +16}+\sqrt{5x^2-18x +29}$ over all real $x$. [u]Round 8 [/u] [b]p22.[/b] Let $O$ and $I$ be the circumcenter and incenter, respectively, of $\vartriangle ABC$ with $AB = 15$, $BC = 17$, and $C A = 16$. Let $X \ne A$ be the intersection of line $AI$ and the circumcircle of $\vartriangle ABC$. Find the area of $\vartriangle IOX$. [b]p23.[/b] Find the sum of all integers $x$ such that there exist integers $y$ and $z$ such that $$x^2 + y^2 = 3(2016^z )+77.$$ [b]p24.[/b] Evaluate $$ \left \lfloor \sum^{2022}_{i=1} \frac{1}{\sqrt{i}} \right \rfloor = \left \lfloor \frac{1}{\sqrt{1}} +\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+ \frac{1}{\sqrt{2022}}\right \rfloor$$ [u]Round 9[/u] [b]p25.[/b]Either: 1. Submit $-2$ as your answer and you’ll be rewarded with two points OR 2. Estimate the number of teams that choose the first option. If your answer is within $1$ of the correct answer, you’ll be rewarded with three points, and if you are correct, you’ll receive ten points. [b]p26.[/b] Jeff is playing a turn-based game that starts with a positive integer $n$. Each turn, if the current number is $n$, Jeff must choose one of the following: 1. The number becomes the nearest perfect square to $n$ 2. The number becomes $n-a$, where $a$ is the largest digit in $n$ Let $S(k)$ be the least number of turns Jeff needs to get from the starting number $k$ to $0$. Estimate $$\sum^{2023}_{k=1}S(k).$$ If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{6000} \right| \right \rfloor , 0 \right)$ points. [b]p27.[/b] Estimate the smallest positive integer n such that if $N$ is the area of the $n$-sided regular polygon with circumradius $100$, $10000\pi -N < 1$ is true. If your estimation is $E$ and the actual answer is $A$, you will receive $ \max \left \lfloor \left( 10 - \left| 10 \cdot \log_3 \left( \frac{A}{E}\right) \right|\right| ,0\right \rfloor.$ points. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167360p28825713]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

At the start of the PUMaC opening ceremony in McCosh auditorium, the speaker counts $90$ people in the audience. Every minute afterwards, either one person enters the auditorium (due to waking up late) or leaves (in order to take a dreadful math contest). The speaker observes that in this time, exactly $100$ people enter the auditorium, $100$ leave, and $100$ was the largest audience size he saw. Find the largest integer $m$ such that $2^m$ divides the number of different possible sequences of entries and exits given the above information.

(a) Decide whether the fields of the $8 \times 8$ chessboard can be numbered by the numbers $1, 2, \dots , 64$ in such a way that the sum of the four numbers in each of its parts of one of the forms [list][img]http://www.artofproblemsolving.com/Forum/download/file.php?id=28446[/img][/list] is divisible by four. (b) Solve the analogous problem for [list][img]http://www.artofproblemsolving.com/Forum/download/file.php?id=28447[/img][/list]

It is known that each vertex of the convex polyhedron $P$ belongs to three different faces, and each vertex of $P$ can be dyed black and white, so that the two endpoints of each edge of $P$ are different colors. Proof: The interior of each edge of $P$ can be dyed red, yellow, and blue, so that the colors of the three edges connected to each vertex are different, and each face contains two colors of edges. [i]Created by Liang Xiao[/i]

10 vertices of a regular 100-gon are coloured red and ten other (distinct) vertices are coloured blue. Prove that there is at least one connection edge (segment) of two red which is as long as the connection edge of two blue points. [hide="Hint"]Possible approaches are pigeon hole principle, proof by contradiction, consider turns (bijective congruent mappings) which maps red in blue points. [/hide]