Janson found $2025$ dogs on a circle. Janson wants to select some (possibly none) of the dogs to take home, such that no two selected dogs have exactly two dogs (whether selected or not) in between them. Let $S_{1}$ be the number of ways for him to do so. Ivan also found $2025$ cats on a circle. Ivan wants to select some (possibly none) of the cats to take home, such that no two selected cats have exactly five cats (whether selected or not) in between them. Let $S_{2}$ be the number of ways for him to do so. a) Prove that $S_{1}=S_{2}$. b) Prove that $S_{1}$ and $S_{2}$ are both perfect cubes.

Consider some graph $G$ with $2019$ nodes. Let's define [i]inverting[/i] a vertex $v$ the following process: for every other vertex $u$, if there was an edge between $v$ and $u$, it is deleted, and if there wasn't, it is added. We want to minimize the number of edges in the graph by several [i]invertings[/i] (we are allowed to invert the same vertex several times). Find the smallest number $M$ such that we can always make the number of edges in the graph not larger than $M$, for any initial choice of $G$. [i]Proposed by Arsenii Nikolaev, Anton Trygub (Ukraine)[/i]

Determine whether it's possible to cover a $K_{2012}$ with a) 1000 $K_{1006}$'s; b) 1000 $K_{1006,1006}$'s. [i]David Yang.[/i]

100 volleyball teams played a one-round tournament. No two matches held at the same time. It turned out that before each match the teams playing against each other had the same number of wins. Find all possible number of wins for the winner of this tournament.

Arutyun and Amayak show another effective trick. A spectator writes down on a board a sequence of $N$ (decimal) digits. Amayak closes two adjacent digits by a black disc. Then Arutyun comes and says both closed digits (and their order). For which minimal $N$ they may show such a trick? [i]K. Knop, O. Leontieva[/i]

Two players play a game on four piles of pebbles labeled with the numbers $1,2,3,4$ respectively. The players take turns in an alternating fashion. On his or her turn, a player selects integers $m$ and $n$ with $1\leq m<n\leq 4$, removes $m$ pebbles from pile $n$, and places one pebble in each of the piles $n-1,n-2,\dots,n-m$. A player loses the game if he or she cannot make a legal move. For each starting position, determine the player with a winning strategy.

What is the greatest number of positive integers lesser than or equal to 2016 we can choose such that it doesn't have two of them differing by 1,2, or 6?

Let $n$ be a positive integer. Suppose that we have disks of radii $1, 2, . . . , n.$ Of each size there are two disks: a transparent one and an opaque one. In every disk there is a small hole in the centre, with which we can stack the disks using a vertical stick. We want to make stacks of disks that satisfy the following conditions: $i)$ Of each size exactly one disk lies in the stack. $ii)$ If we look at the stack from directly above, we can see the edges of all of the $n$ disks in the stack. (So if there is an opaque disk in the stack,no smaller disks may lie beneath it.) Determine the number of distinct stacks of disks satisfying these conditions. (Two stacks are distinct if they do not use the same set of disks, or, if they do use the same set of disks and the orders in which the disks occur are different.)

A permutation of $\{1, 2 \cdots, n \}$ is [i]magic[/i] if each element $k$ of it has at least $\left\lfloor \frac{k}{2} \right\rfloor$ numbers less to it at the left. For each $n$ find the number of [i]magical[/i] permutations.

The European zoos with at least two types of species are separated in two groups $\hat{A}$ and $\hat{B}$ in such a way that every pair of zoos $(A,B)$ $(A\in\hat{A}, B\in\hat{B})$ have some animal in common. What is the least $k$ for which we can color the cages in the zoos (each cage only has all animals of one species) such that no zoo has cages of only one color (with every animal across all zoos having to be colored in the same color)? For the maximal value of $k$, find all possibilities (zoos and species), for which this maximum is achieved.

A page of an exercise book is painted with $23$ colours, arranged in squares. A pair of colours is called [i]good [/i] if there are neighbouring squares painted with these colours. What is the minimum number of good pairs?

Marjorie is the drum major of the world's largest marching band, with more than one million members. She would like the band members to stand in a square formation. To this end, she determines the smallest integer $n$ such that the band would fit in an $n \times n$ square, and lets the members form rows of $n$ people. However, she is dissatisfied with the result, since some empty positions remain. Therefore, she tells the entire first row of $n$ members to go home and repeats the process with the remaining members. Her aim is to continue it until the band forms a perfect square, but as it happens, she does not succeed until the last members are sent home. Determine the smallest possible number of members in this marching band.

In a round-robin table tennis tournament between $2010$ athletes, where each match ends with a winner and a loser, let $a_1,... , a_{2010}$ denote the number of wins of each athlete, and let $b_1, .., b_{2010}$ denote the number of losses of each athlete. Show that $a^2_1+a^2_2+...+a^2_{2010} =b^2_1 + b^2_2 + ... + b^2_{2010}$.

Some points with the integer coordinates are marked on the coordinate plane. Given a set of nonzero vectors. It is known, that if you apply the beginnings of those vectors to the arbitrary marked point, than there will be more marked ends of the vectors, than not marked. Prove that there is infinite number of marked points.

In an $m \times n$ grid, each square is either filled or not filled. For each square, its [i]value[/i] is defined as $0$ if it is filled and is defined as the number of neighbouring filled cells if it is not filled. Here, two squares are neighbouring if they share a common vertex or side. Let $f(m,n)$ be the largest total value of squares in the grid. Determine the minimal real constant $C$ such that $$\frac{f(m,n)}{mn} \le C$$holds for any positive integers $m,n$ [i]CSJL[/i]

[b]p1.[/b] For what integers $n$ is $2^6 + 2^9 + 2^n$ the square of an integer? [b]p2.[/b] Two integers are chosen at random (independently, with repetition allowed) from the set $\{1,2,3,...,N\}$. Show that the probability that the sum of the two integers is even is not less than the probability that the sum is odd. [b]p3.[/b] Let $X$ be a point in the second quadrant of the plane and let $Y$ be a point in the first quadrant. Locate the point $M$ on the $x$-axis such that the angle $XM$ makes with the negative end of the $x$-axis is twice the angle $YM$ makes with the positive end of the $x$-axis. [b]p4.[/b] Let $a,b$ be positive integers such that $a \ge b \sqrt3$. Let $\alpha^n = (a + b\sqrt3)^n = a_n + b_n\sqrt3$ for $n = 1,2,3,...$. i. Prove that $\lim_{n \to + \infty} \frac{a_n}{b_n}$ exists. ii. Evaluate this limit. [b]p5.[/b] Suppose $m$ and $n$ are the hypotenuses are of Pythagorean triangles, i.e,, there are positive integers $a,b,c,d$, so that $m^2 = a^2 + b^2$ and $n^2= c^2 + d^2$. Show than $mn$ is the hypotenuse of at least two distinct Pythagorean triangles. Hint: you may not assume that the pair $(a,b)$ is different from the pair $(c,d)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a set of 102 elements. Is it possible to choose 102 17-element subsets so that the intersection of any two subsets contains no more than 3 elements?

A $m \times n$ table is filled with signs $"+"$ and $"-"$. A table is called irreducible if one cannot reduce it to the table filled with $"+"$, applying the following operations (as many times as one wishes). $a)$ It is allowed to flip all the signs in a row or in a column. Prove that an irreducible table contains an irreducible $2\times 2$ sub table. $b)$ It is allowed to flip all the signs in a row or in a column or on a diagonal (corner cells are diagonals of length $1$). Prove that an irreducible table contains an irreducible $4\times 4$ sub table.

There are $2015$ coins on a table. For $i = 1, 2, \dots , 2015$ in succession, one must turn over exactly $i$ coins. Prove that it is always possible either to make all of the coins face up or to make all of the coins face down, but not both.

Let $a$ and $b$ be positive integers. We tile a rectangle with dimensions $a$ and $b$ using squares whose side-length is a power of $2$, i.e. the tiling may include squares of dimensions $1\times 1, 2\times 2, 4\times 4$ etc. Denote by $M$ the minimal number of squares in such a tiling. Numbers $a$ and $b$ can be uniquely represented as the sum of distinct powers of $2$: $a=2^{a_1}+\cdots+2^{a_k},\; b=2^{b_1}+\cdots +2^{b_\ell}.$ Show that $$M=\sum_{i=1}^k \;\sum_{j=1}^{\ell} 2^{|a_i-b_j|}.$$

Prove The Following identity: $ \sum_{j \equal{} 0}^n \left ({3n \plus{} 2 \minus{} j \choose j}2^j \minus{} {3n \plus{} 1 \minus{} j \choose j \minus{} 1}2^{j \minus{} 1}\right ) \equal{} 2^{3n}$. The Second term on left hand side is to be regarded zero for j=0.

Choose a permutation of$ \{1,2, ..., 20\}$ at random. Let $m$ be the amount of numbers in the permutation larger than all numbers before it. Find the expected value of $2^m$. [i]Proposed by Evan Chang (squareman), USA[/i]

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

[b]p1.[/b] Compute $21 \cdot 21 - 20 \cdot 20$. [b]p2.[/b] A square has side length $2$. If the square is scaled by a factor of $n$, the perimeter of the new square is equal to the area of the original square. Find $10n$. [b]p3.[/b] Kevin has $2$ red marbles and $2$ blue marbles in a box. He randomly grabs two marbles. The probability that they are the same color can be expressed as $\frac{a}{b}$ for relatively prime integers $a$ and $b$. Find $a +b$. [b]p4.[/b] In a classroom, if the teacher splits the students into groups of $3$ or $4$, there is one student left out. If the students formgroups of $5$, every student is in a group. What is the fewest possible number of students in this classroom? [b]p5.[/b] Find the sum of all positive integer values of $x$ such that $\lfloor \sqrt{x!} \rfloor = x$. [b]p6.[/b] Find the number of positive integer factors of $2021^{(2^0+2^1)} \cdot 1202^{(1^2+0^2)}$. [b]p7.[/b] Let $n$ be the number of days over a $13$ year span. Find the difference between the greatest and least possible values of $n$. Note: All years divisible by $4$ are leap years unless they are divisible by 100 but not $400$. For example, $2000$ and $2004$ are leap years, but $1900$ is not. [b]p8.[/b] In isosceles $\vartriangle ABC$, $AB = AC$, and $\angle ABC = 72^o$. The bisector of $\angle ABC$ intersects $AC$ at $D$. Given that $BC = 30$, find $AD$. [b]p9.[/b] For an arbitrary positive value of $x$, let $h$ be the area of a regular hexagon with side length $x$ and let $s$ be the area of a square with side length $x$. Find the value of $\left \lfloor \frac{10h}{s} \right \rfloor$. [b]p10.[/b] There is a half-full tub of water with a base of $4$ inches by $5$ inches and a height of $8$ inches. When an infinitely long stick with base $1$ inch by $1$ inch is inserted vertically into the bottom of the tub, the number of inches the water level rises by can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p11.[/b] Find the sum of all $4$-digit numbers with digits that are a permutation of the digits in $2021$. Note that positive integers cannot have first digit $0$. [b]p12.[/b] A $10$-digit base $8$ integer is chosen at random. The probability that it has $30$ digits when written in base $2$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p13.[/b] Call a natural number sus if it can be expressed as $k^2 +k +1$ for some positive integer $k$. Find the sum of all sus integers less than $2021$. [b]p14.[/b] In isosceles triangle $ABC$, $D$ is the intersection of $AB$ and the perpendicular to $BC$ through $C$. Given that $CD = 5$ and $AB = BC = 1$, find $\sec^2 \angle ABC$. [b]p15.[/b] Every so often, the minute and hour hands of a clock point in the same direction. The second time this happens after 1:00 is a b minutes later, where a and b are relatively prime positive integers. Find a +b. [b]p16.[/b] The $999$-digit number $N = 123123...123$ is composed of $333$ iterations of the number $123$. Find the least nonnegative integerm such that $N +m$ is a multiple of $101$. [b]p17.[/b] The sum of the reciprocals of the divisors of $2520$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p18.[/b] Duncan, Paul, and $6$ Atreides guards are boarding three helicopters. Duncan, Paul, and the guards enter the helicopters at random, with the condition that Duncan and Paul do not enter the same helicopter. Note that not all helicoptersmust be occupied. The probability that Paul has more guards with him in his helicopter than Duncan does can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p19.[/b] Let the minimum possible distance from the origin to the parabola $y = x^2 -2021$ be $d$. The value of d2 can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p20.[/b] In quadrilateral $ABCD$ with interior point $E$ and area $49 \sqrt3$, $\frac{BE}{CE}= 2 \sqrt3$, $\angle ABC = \angle BCD = 90^o$, and $\vartriangle ABC \sim \vartriangle BCD \sim \vartriangle BEC$. The length of $AD$ can be expressed aspn where $n$ is a positive integer. Find $n$. [b]p21.[/b] Find the value of $$\sum^{\infty}_{i=1}\left( \frac{i^2}{2^{i-1}}+\frac{i^2}{2^{i}}+\frac{i^2}{2^{i+1}}\right)=\left( \frac{1^2}{2^{0}}+\frac{1^2}{2^{1}}+\frac{1^2}{2^{2}}\right)+\left( \frac{2^2}{2^{1}}+\frac{2^2}{2^{2}}+\frac{2^2}{2^{3}}\right)+\left( \frac{3^2}{2^{2}}+\frac{2^2}{2^{3}}+\frac{2^2}{2^{4}}\right)+...$$ [b]p22.[/b] Five not necessarily distinct digits are randomly chosen in some order. Let the probability that they form a nondecreasing sequence be $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a +b$ is divided by$ 1000$. [b]p23.[/b] Real numbers $a$, $b$, $c$, and d satisfy $$ac -bd = 33$$ $$ad +bc = 56.$$ Given that $a^2 +b^2 = 5$, find the sum of all possible values of $c^2 +d^2$. [b]p24.[/b] Jeff has a fair tetrahedral die with sides labeled $0$, $1$, $2$, and $3$. He continuously rolls the die and record the numbers rolled in that order. For example, if he rolls a $1$, then rolls a $2$, and then rolls a $3$, he writes down $123$. He keeps rolling the die until he writes the substring $2021$. What is the expected number of times he rolls the die? [b]p25.[/b] In triangle $ABC$, $BC = 2\sqrt3$, and $AB = AC = 4\sqrt3$. Circle $\omega$ with center $O$ is tangent to segment $AB$ at $T$ , and $\omega$ is also tangent to ray $CB$ past $B$ at another point. Points $O, T$ , and $C$ are collinear. Let $r$ be the radius of $\omega$. Given that $r^2 = \frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a +b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In country there are two capitals ($A$ and $B$) and finite number of towns. Some towns (or town with one of capital) connected with roads (one-way). (between every two towns or capital and town there are arbitrary number of roads) such that exist at least one way from $A$ to $B$. Given, that any two ways from $A$ to $B$ have at least one common road. Prove, that exist one road, such that all ways from $A$ to $B$ pass through this road.