An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?

3. Plane is divided with horizontal and vertical lines into unit squares. Into each square we write a positive integer so that each positive integer appears exactly once. Determine whether it is possible to write numbers in such a way, that each written number is a divisor of a sum of its four neighbours.

Rosa dos Ventos, Aurora Boreal and Manuela do Norte organized a competition between them last weekend, consisting of several athletics events. The winner in each test obtained $x$ points, the second placed $y$ points and the third placed $z$ points ($x,y,z \in N$ and $x >y>z$). The final result of the competition, obtained by adding up the scores in each event, was Rosa had $22$ points, Manuela had $9$ points, Aurora had $9 $ points. In how many tests did they compete and who came second in the high jump knowing that the Manuela won the $100$ meters and no one gave up in any race? [hide=official wording]Rosa dos Ventos, a Aurora Boreal e a Manuela do Norte organizaram no passado fim de semana uma competi¸c˜ao entre elas, consistindo em v´arias provas de atletismo. A vencedora em cada prova obteve x pontos, a segunda classificada y pontos e a terceira classificada z pontos (x,y,z ∈ IN e x >y>z). O resultado final da competi¸c˜ao, obtido por soma das pontua¸c˜oes em cada prova, foi Rosa 22 pontos Manuela 9 pontos Aurora 9 pontos Em quantas provas competiram e quem ficou em segundo lugar no salto em altura sabendo que a Manuela ganhou os 100 metros e que ningu´em desistiu em nenhuma prova?[/hide]

In some foreign country, there is a secret object, guarded by seven guards. Each guard has a guarding shift of 7 consecutive hours every day, in fixed hours. There is always at least one guard guarding the secret object at any given time. Prove that one of the guards can be fired, and there will still be at least one guard guarding at any given time (without changing the schedule of the other guards).

Consider an equilateral triangular grid $G$ with $20$ points on a side, where each row consists of points spaced $1$ unit apart. More specifically, there is a single point in the first row, two points in the second row, ..., and $20$ points in the last row, for a total of $210$ points. Let $S$ be a closed non-self-intersecting polygon which has $210$ vertices, using each point in $G$ exactly once. Find the sum of all possible values of the area of $S$.

We have 2019 boxes. Initially, they are all empty. At one operation, we can add exactly 100 stones to some 100 boxes and exactly one stone in each of several other (perhaps none) boxes. What is the smallest possible number of moves after which all boxes will have the same (positive) number of stones. [i]Proposed by P. Kozhevnikov[/i]

Compute the number of ways to pick two rectangles in a $5 \times 5$ grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are chosen does not matter.

One million [i]bucks [/i] (i.e. one million male deer) are in different cells of a $1000 \times 1000$ grid. The left and right edges of the grid are then glued together, and the top and bottom edges of the grid are glued together, so that the grid forms a doughnut-shaped torus. Furthermore, some of the bucks are [i]honest bucks[/i], who always tell the truth, and the remaining bucks are [i]dishonest bucks[/i], who never tell the truth. Each of the million [i]bucks [/i] claims that “at most one of my neighboring bucks is an [i]honest buck[/i].” A pair of [i]neighboring bucks[/i] is said to be [i]buckaroo[/i] if exactly one of them is an [i]honest buck[/i] . What is the minimum possible number of [i]buckaroo [/i] pairs in the grid? Note: Two [i]bucks [/i] are considered to be [i]neighboring [/i] if their cells $(x_1, y_1)$ and $(x_2, y_2)$ satisfy either: $x_1 = x_2$ and $y_1 - y_2 \equiv \pm1$ (mod $1000$), or $x_1 - x_2 \equiv \pm 1$ (mod $1000$) and $y_1 = y_2$.

Given a positive integer $n$, determine the maximum number of lattice points in the plane a square of side length $n +\frac{1}{2n+1}$ may cover.

A chessboard $1000 \times 1000$ is covered by dominoes $1 \times 10$ that can be rotated. We don't know which is the cover, but we are looking for it. For this reason, we choose a few $N$ cells of the chessboard, for which we know the position of the dominoes that cover them. Which is the minimum $N$ such that after the choice of $N$ and knowing the dominoed that cover them, we can be sure and for the rest of the cover? (Bulgaria)

A cloakroom in a cinema works with some breaks. The total time cloakroom worked today is 8 hours. The schedule of the cloakroom is such that it is possible to show any film of duration at most 12 hours such that the cloakroom will be open at least one hour before and after the film (the films are shown without breaks). Find the minimal possible amount of breaks in the schedule of cloakroom. [i]A. Voidelevich[/i]

IMONST = [b]I[/b]nternational [b]M[/b]athematical [b]O[/b]lympiad [b]N[/b]ational [b]S[/b]election [b]T[/b]est Malaysia 2021 Round 1 Juniors Time: 2.5 hours [hide=Rules] $\bullet$ For each problem you have to submit the answer only. The answer to each problem is a non-negative integer. $\bullet$ No mark is deducted for a wrong answer. $\bullet$ The maximum number of points is (1 + 2 + 3 + 4) x 5 = 50 points.[/hide] [b]Part A[/b] (1 point each) p1. Adam draws $7$ circles on a paper, with radii $ 1$ cm, $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm, and $7$ cm. The circles do not intersect each other. He colors some circles completely red, and the rest of the circles completely blue. What is the minimum possible difference (in cm$^2$) between the total area of the red circles and the total area of the blue circles? p2. The number $2021$ has a special property that the sum of any two neighboring digits in the number is a prime number ($2 + 0 = 2$, $0 + 2 = 2$, and $2 + 1 = 3$ are all prime numbers). Among numbers from $2021$ to $2041$, how many of them have this property? p3. Clarissa opens a pet shop that sells three types of pets: goldshes, hamsters, and parrots. The pets inside the shop together have a total of $14$ wings, $24$ heads, and $62$ legs. How many goldshes are there inside Clarissa's shop? p4. A positive integer $n$ is called special if $n$ is divisible by $4$, $n+1$ is divisible by $5$, and $n + 2$ is divisible by $6$. How many special integers smaller than $1000$ are there? p5. Suppose that this decade begins on $ 1$ January $2020$ (which is a Wednesday) and the next decade begins on $ 1$ January $2030$. How many Wednesdays are there in this decade? [b]Part B[/b] (2 points each) p6. Given an isosceles triangle $ABC$ with $AB = AC$. Let D be a point on $AB$ such that $CD$ is the bisector of $\angle ACB$. If $CB = CD$, what is $\angle ADC$, in degrees? p7. Determine the number of isosceles triangles with the following properties: all the sides have integer lengths (in cm), and the longest side has length $21$ cm. p8. Haz marks $k$ points on the circumference of a circle. He connects every point to every other point with straight lines. If there are $210$ lines formed, what is $k$? p9. What is the smallest positive multiple of $24$ that can be written using digits $4$ and $5$ only? p10. In a mathematical competition, there are $2021$ participants. Gold, silver, and bronze medals are awarded to the winners as follows: (i) the number of silver medals is at least twice the number of gold medals, (ii) the number of bronze medals is at least twice the number of silver medals, (iii) the number of all medals is not more than $40\%$ of the number of participants. The competition director wants to maximize the number of gold medals to be awarded based on the given conditions. In this case, what is the maximum number of bronze medals that can be awarded? [b]Part C[/b] (3 points each) p11. Dinesh has several squares and regular pentagons, all with side length $ 1$. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so? [img]https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg [/img] p12. If $x +\frac{1}{x} = 5$, what is the value of $x^3 +\frac{1}{x^3} $ ? p13. There are $10$ girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue? p14. The two diagonals of a rhombus have lengths with ratio $3 : 4$ and sum $56$. What is the perimeter of the rhombus? p15. How many integers $n$ (with $1 \le n \le 2021$) have the property that $8n + 1$ is a perfect square? [b]Part D[/b] (4 points each) p16. Given a segment of a circle, consisting of a straight edge and an arc. The length of the straight edge is $24$. The length between the midpoint of the straight edge and the midpoint of the arc is $6$. Find the radius of the circle. p17. Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$. How many passcodes satisfy these conditions? p18. A tree grows in the following manner. On the first day, one branch grows out of the ground. On the second day, a leaf grows on the branch and the branch tip splits up into two new branches. On each subsequent day, a new leaf grows on every existing branch, and each branch tip splits up into two new branches. How many leaves does the tree have at the end of the tenth day? p19. Find the sum of (decimal) digits of the number $(10^{2021} + 2021)^2$? p20. Determine the number of integer solutions $(x, y, z)$, with $0 \le x, y, z \le 100$, for the equation$$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$

There are five dots arranged in a line from left to right. Each of the dots is colored from one of five colors so that no $3$ consecutive dots are all the same color. How many ways are there to color the dots?

Let $n$ be a positive integer. Find the number of sequences $a_0,a_1,a_2,\dots,a_{2n}$ of integers in the range $[0,n]$ such that for all integers $0\leq k\leq n$ and all nonnegative integers $m$, there exists an integer $k\leq i\leq 2k$ such that $\lfloor k/2^m\rfloor=a_i.$ [i]Andrew Carratu[/i]

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Prove that any finite set $H$ of lattice points on the plane has a subset $K$ with the following properties: [list] [*]any vertical or horizontal line in the plane cuts $K$ in at most $2$ points, [*]any point of $H\setminus K$ is contained by a segment with endpoints from $K$.[/list]

Pairwise distinct real numbers are arranged into an $m \times n$ rectangular array. In each row the entries are arranged increasingly from left to right. Each column is then rearranged in decreasing order from top to bottom. Prove that in the reorganized array, the rows remain arranged increasingly.

A square has been divided into $2022$ rectangles with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles?

In the Faculty of Cybernetics football championship, \( n \geq 3 \) teams participated. The competition was held in a round-robin format, meaning that each team played against every other team exactly once. For a win, a team earns 3 points, for a loss no points are awarded, and for a draw, both teams receive 1 point each. It turned out that the winning team scored strictly more points than any other team and had at most as many wins as losses. What is the smallest \( n \) for which this could happen? [i]Proposed by Bogdan Rublov[/i]

A jeweler makes necklaces with round stones, four emeralds (green) and four rubies (red), arranged at equal distances from each other. One day, they decide to give away some necklaces. How many necklaces can they give away without the risk of two friends ending up with the same necklace? (*Observation: The necklace is completely symmetrical except for the type of stone, meaning there is not a unique way to form it. Consider this while solving the problem.*)

The plane is colored with two colors so that the following property holds: for each real $a>0$ there is an equilateral triangle of side length $a$ whose $3$ vertices are of the same color. Prove that for any three numbers $a,b,c>0$ for which the sum of any two is greater than the third there is a triangle with sides $a$, $b$, and $c$ whose $3$ vertices are of the same color.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Compute $(2 + 0)(2 + 2)(2 + 0)(2 + 2)$. [b]p2.[/b] Given that $25\%$ of $x$ is $120\%$ of $30\%$ of $200$, find $x$. [b]p3.[/b] Jacob had taken a nap. Given that he fell asleep at $4:30$ PM and woke up at $6:23$ PM later that same day, for how many minutes was he asleep? [b]p4.[/b] Kevin is painting a cardboard cube with side length $12$ meters. Given that he needs exactly one can of paint to cover the surface of a rectangular prism that is $2$ meters long, $3$ meters wide, and $6$ meters tall, how many cans of paint does he need to paint the surface of his cube? [b]p5.[/b] How many nonzero digits does $200 \times 25 \times 8 \times 125 \times 3$ have? [b]p6.[/b] Given two real numbers $x$ and $y$, define $x \# y = xy + 7x - y$. Compute the absolute value of $0 \# (1 \# (2 \# (3 \# 4)))$. [b]p7.[/b] A $3$-by-$5$ rectangle is partitioned into several squares of integer side length. What is the fewest number of such squares? Squares in this partition must not overlap and must be contained within the rectangle. [b]p8.[/b] Points $A$ and $B$ lie in the plane so that $AB = 24$. Given that $C$ is the midpoint of $AB$, $D$ is the midpoint of $BC$, $E$ is the midpoint of $AD$, and $F$ is the midpoint of $BD$, find the length of segment $EF$. [b]p9.[/b] Vincent the Bug and Achyuta the Anteater are climbing an infinitely tall vertical bamboo stalk. Achyuta begins at the bottom of the stalk and climbs up at a rate of $5$ inches per second, while Vincent begins somewhere along the length of the stalk and climbs up at a rate of $3$ inches per second. After climbing for $t$ seconds, Achyuta is half as high above the ground as Vincent. Given that Achyuta catches up to Vincent after another $160$ seconds, compute $t$. [b]p10.[/b] What is the minimum possible value of $|x - 2022| + |x - 20|$ over all real numbers $x$? [b]p11.[/b] Let $ABCD$ be a rectangle. Lines $\ell_1$ and $\ell_2$ divide $ABCD$ into four regions such that $\ell_1$ is parallel to $AB$ and line $\ell_2$ is parallel to $AD$. Given that three of the regions have area $6$, $8$, and $12$, compute the sum of all possible areas of the fourth region. [b]p12.[/b] A diverse number is a positive integer that has two or more distinct prime factors. How many diverse numbers are less than $50$? [b]p13.[/b] Let $x$, $y$, and $z$ be real numbers so that $(x+y)(y +z) = 36$ and $(x+z)(x+y) = 4$. Compute $y^2 -x^2$. [b]p14.[/b] What is the remainder when $ 1^{10} + 3^{10} + 7^{10}$ is divided by $58$? [b]p15.[/b] Let $A = (0, 1)$, $B = (3, 5)$, $C = (1, 4)$, and $D = (3, 4)$ be four points in the plane. Find the minimum possible value of $AP + BP + CP + DP$ over all points $P$ in the plane. [b]p16.[/b] In trapezoid $ABCD$, points $E$ and $F$ lie on sides $BC$ and $AD$, respectively, such that $AB \parallel CD \parallel EF$. Given that $AB = 3$, $EF = 5$, and $CD = 6$, the ratio $\frac{[ABEF]}{[CDFE]}$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. (Note: $[F]$ denotes the area of $F$.) [b]p17.[/b] For sets $X$ and $Y$ , let $|X \cap Y |$ denote the number of elements in both $X$ and $Y$ and $|X \cup Y|$ denote the number of elements in at least one of $X$ or $Y$ . How many ordered pairs of subsets $(A,B)$ of $\{1, 2, 3,..., 8\}$ are there such that $|A \cap B| = 2$ and $|A \cup B| = 5$? [b]p18.[/b] A tetromino is a polygon composed of four unit squares connected orthogonally (that is, sharing a edge). A tri-tetromino is a polygon formed by three orthogonally connected tetrominoes. What is the maximum possible perimeter of a tri-tetromino? [b]p19.[/b] The numbers from $1$ through $2022$, inclusive, are written on a whiteboard. Every day, Hermione erases two numbers $a$ and $b$ and replaces them with $ab+a+b$. After some number of days, there is only one number $N$ remaining on the whiteboard. If $N$ has $k$ trailing nines in its decimal representation, what is the maximum possible value of $k$? [b]p20.[/b] Evaluate $5(2^2 + 3^2) + 7(3^2 + 4^2) + 9(4^2 + 5^2) + ... + 199(99^2 + 100^2)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A square grid $2n \times 2n$ is constructed of matches (each match is a segment of length 1). By one move Peter can choose a vertex which (at this moment) is the endpoint of 3 or 4 matches and delete two matches whose union is a segment of length 2. Find the least possible number of matches that could remain after a number of Peter's moves.

Let there be a $n\times n$ board. Write down $0$ or $1$ in all $n^2$ squares. For $1 \le k \le n$, let $A_k$ be the product of all numbers in the $k$th row. How many ways are there to write down the numbers so that $A_1 + A_2 + ... + A_n$ is even?

Let $ n \geq 2$ be a positive integer and $ \lambda$ a positive real number. Initially there are $ n$ fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points $ A$ and $ B$, with $ A$ to the left of $ B$, and letting the flea from $ A$ jump over the flea from $ B$ to the point $ C$ so that $ \frac {BC}{AB} \equal{} \lambda$. Determine all values of $ \lambda$ such that, for any point $ M$ on the line and for any initial position of the $ n$ fleas, there exists a sequence of moves that will take them all to the position right of $ M$.