Consider an infinite plane divided into unit squares by horizontal and vertical lines. A coloring of some cells in this grid is called a $\emph{net coloring}$ if the centers of the colored squares coincide with the intersection points of an infinite family of equally spaced parallel lines and another directed and equally spaced infinite family of lines. The distance between the centers of the nearest colored squares is called the size of the $\emph{net coloring}.$ Determine all natural numbers \(N\) for which it is possible to color all these unit squares using \(N\) colors such that the following conditions are met: $\bullet$ Each color is used to color at least one square. $\bullet$The coloring for every color forms a $\emph{net coloring}.$ $\bullet$ The sizes of each of the \(N\) $\emph{net colorings}$ are equal. [i]Proposed by Aleksandre Saatashvili, Georgia [/i]

Let $k$ be a positive integer. A sequence of integers $a_1, a_2, \cdots$ is called $k$-pop if the following holds: for every $n \in \mathbb{N}$, $a_n$ is equal to the number of distinct elements in the set $\{a_1, \cdots , a_{n+k} \}$. Determine, as a function of $k$, how many $k$-pop sequences there are. [i]Proposed by Sutanay Bhattacharya[/i]

Divide a circle into $2n$ equal sections. We call a circle [i]filled[/i] if it is filled with the numbers $0,1,2,\dots,n-1$. We call a filled circle [i] good[/i] if it has the following properties: $i$. Each number $0 \leq a \leq n-1$ is used exactly twice $ii$. For any $a$ we have that there are exactly $a$ sections between the two sections that have the number $a$ in them. Here is an example of a good filling for $n=5$ (View attachment) Prove that there doesn’t exist a good filling for $n=1399$

$N$ boxes are arranged in a circle and are numbered $1,2,3,.....N$ In a clockwise direction. A ball is assigned a number from${1,2,3,....N}$ and is placed in one of the boxes.A round consist of the following; if the current number on the ball is $n$, the ball is moved $n$ boxes in the clockwise direction and the number on the ball is changed to $n+1$ if $n<N$ and to $1$ if $n=N$. Is it possible to choose $N$, the initial number on the ball, and the first position of the ball in such a way that the ball gets back to the same box with the same number on it for the first time after exactly $2020$ rounds

In a group of $n$ people, each one had a different ball. They performed a sequence of swaps, in each swap, two people swapped the ball they had at that moment. Each pair of people performed at least one swap. In the end each person had the ball he/she had at the start. Find the least possible number of swaps, if: a) $n = 5$, b) $n = 6$.

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 2020 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. The number $N$ is the smallest positive integer with the sum of its digits equal to $2020$. What is the first (leftmost) digit of $N$? p2. At a food stall, the price of $16$ banana fritters is $k$ RM , and the price of $k$ banana fritters is $ 1$ RM . What is the price of one banana fritter, in sen? Note: $1$ RM is equal to $100$ sen. p3. Given a trapezium $ABCD$ with $AD \parallel$ to $BC$, and $\angle A = \angle B = 90^o$. It is known that the area of the trapezium is 3 times the area of $\vartriangle ABD$. Find $$\frac{area \,\, of \,\, \vartriangle ABC}{area \,\, of \,\, \vartriangle ABD}.$$ p4. Each $\vartriangle$ symbol in the expression below can be substituted either with $+$ or $-$: $$\vartriangle 1 \vartriangle 2 \vartriangle 3 \vartriangle 4.$$ How many possible values are there for the resulting arithmetic expression? Note: One possible value is $-2$, which equals $-1 - 2 - 3 + 4$. p5. How many $3$-digit numbers have its sum of digits equal to $4$? [b]Part B[/b] (2 points each) p6. Find the value of $$+1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 +... - 2020$$ where the sign alternates between $+$ and $-$ after every three numbers. p7. If Natalie cuts a round pizza with $4$ straight cuts, what is the maximum number of pieces that she can get? Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting. p8. Given a square with area $ A$. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area $ B$ lies inside the circle, such that all its vertices lie on the circle. Find the value of $A/B$. p9. This sequence lists the perfect squares in increasing order: $$0, 1, 4, 9, 16, ... ,a, 10^8, b, ...$$ Determine the value of $b - a$. p10. Determine the last digit of $5^5 + 6^6 + 7^7 + 8^8 + 9^9$. [b]Part C[/b] (3 points each) p11. Find the sum of all integers between $-\sqrt{1442}$ and $\sqrt{2020}$. p12. Three brothers own a painting company called Tiga Abdul Enterprise. They are hired to paint a building. Wahab says, "I can paint this building in $3$ months if I work alone". Wahib says, "I can paint this building in $2$ months if I work alone". Wahub says, "I can paint this building in $k$ months if I work alone". If they work together, they can finish painting the building in $1$ month only. What is $k$? p13. Given a rectangle $ABCD$ with a point $P$ inside it. It is known that $PA = 17$, $PB = 15$, and $PC = 6$. What is the length of $PD$? p14. What is the smallest positive multiple of $225$ that can be written using digits $0$ and $ 1$ only? p15. Given positive integers $a, b$, and $c$ with $a + b + c = 20$. Determine the number of possible integer values for $\frac{a + b}{c}$. [b]Part D[/b] (4 points each) p16. If we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest possible value of $p$. p17. A football is made by sewing together some black and white leather patches. The black patches are regular pentagons of the same size. The white patches are regular hexagons of the same size. Each pentagon is bordered by $5$ hexagons. Each hexagons is bordered by $3$ pentagons and $3$ hexagons. We need $12$ pentagons to make one football. How many hexagons are needed to make one football? p18. Given a right-angled triangle with perimeter $18$. The sum of the squares of the three side lengths is $128$. What is the area of the triangle? p19. A perfect square ends with the same two digits. How many possible values of this digit are there? p20. Find the sum of all integers $n$ that fulfill the equation $2^n(6 - n) = 8n$.

Madoka chooses $4$ random numbers $a, b, c, d$ between $0$ and $1$. She notices that $a+b+c = 1$. If the probability that $d > a, b, c$ can be written in simplest form as $\frac{m}{n}$, find $m + n$.

Let $n$ be a positive integer, and let $A$ be a subset of $\{ 1,\cdots ,n\}$. An $A$-partition of $n$ into $k$ parts is a representation of n as a sum $n = a_1 + \cdots + a_k$, where the parts $a_1 , \cdots , a_k $ belong to $A$ and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set $\{ a_1 , a_2 , \cdots , a_k \} $. We say that an $A$-partition of $n$ into $k$ parts is optimal if there is no $A$-partition of $n$ into $r$ parts with $r<k$. Prove that any optimal $A$-partition of $n$ contains at most $\sqrt[3]{6n}$ different parts.

A function $f: \mathbb{R}\to \mathbb{R}$ is [i]essentially increasing[/i] if $f(s)\leq f(t)$ holds whenever $s\leq t$ are real numbers such that $f(s)\neq 0$ and $f(t)\neq 0$. Find the smallest integer $k$ such that for any 2022 real numbers $x_1,x_2,\ldots , x_{2022},$ there exist $k$ essentially increasing functions $f_1,\ldots, f_k$ such that \[f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.\]

Find the largest positive integer $n$, such that there exists a finite set $A$ of $n$ reals, such that for any two distinct elements of $A$, there exists another element from $A$, so that the arithmetic mean of two of these three elements equals the third one.

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2016$ good partitions. PS. [url=https://artofproblemsolving.com/community/c6h1268855p6622233]2015 ISL C3 [/url] has 2015 instead of 2016

Let $S = \{1, 2, 3, \ldots , 2046, 2047, 2048\}$. Two subsets $A$ and $B$ of $S$ are said to be [i]friends[/i] if the following conditions are true: [list] [*] They do not share any elements. [*] They both have the same number of elements. [*] The product of all elements from $A$ equals the product of all elements from $B$. [/list] Prove that there are two subsets of $S$ that are [i]friends[/i] such that each one of them contains at least $738$ elements.

You and your friend play a game on a $ 7 \times 7$ grid of buckets. Your friend chooses $5$ “lucky” buckets by marking an “$X$” on the bottom that you cannot see. However, he tells you that they either form a vertical, or horizontal line of length $5$. To clarify, he will select either of the following sets of buckets: either $\{(a, b),(a, b + 1),(a, b + 2),(a, b + 3),(a, b + 4)\}$, or $\{(b, a),(b + 1, a),(b + 2, a),(b + 3, a),(b + 4, a)\}$, with $1\le a \le 7$, and $1 \le b \le 3$. Your friend lets you pick up at most $n$ buckets, and you win if one of the buckets you picked was a “lucky” bucket. What is the minimum possible value of $n$ such that, if you pick your buckets optimally, you can guarantee that at least one is “lucky”?

Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.

Aisling and Brendan take alternate moves in the following game. Before the game starts, the number $x = 2023$ is written on a piece of paper. Aisling makes the first move. A move from a positive integer $x$ consists of replacing $x$ either with $x + 1$ or with $x/p$ where $p$ is a prime factor of $x$. The winner is the first player to write $x = 1$. Determine whether Aisling or Brendan has a winning strategy for this game.

Cards numbered from 1 to $2^n$ are distributed among $k$ children, $1\leq k\leq 2^n$, so that each child gets at least one card. Prove that the number of ways to do that is divisible by $2^{k-1}$ but not by $2^k$. [i] M. Ivanov [/i]

Let $A_1,\ldots,A_m$ be three-element subsets of an $n$-element set $X$ such that $|A_i\cup A_j|\le1$ whenever $i\ne j$. Prove that there exists a subset $A$ of $X$ with $|A|\ge2\sqrt n$ such that it does not contain any of the $A_i$.

On a connected graph $G$, one may perform the following operations: [list] [*]choose a vertice $v$, and add a vertice $v'$ such that $v'$ is connected to $v$ and all of its neighbours [*] choose a vertice $v$ with odd degree and delete it [/list] Show that for any connected graph $G$, we may perform a finite number of operations such that the resulting graph is a clique. Proposed by [i]idonthaveanaopsaccount[/i]

Joey is playing with a $2$-by-$2$-by-$2$ Rubik’s cube made up of $ 8$ $1$-by-$1$-by-$1$ cubes (with two of these smaller cubes along each of the sides of the bigger cubes). Each face of the Rubik’s cube is distinct color. However, one day, Joey accidentally breaks the cube! He decides to put the cube back together into its solved state, placing each of the pieces one by one. However, due to the nature of the cube, he is only able to put in a cube if it is adjacent to a cube he already placed. If different orderings of the ways he chooses the cubes are considered distinct, determine the number of ways he can reassemble the cube.

There are two bowls on a table, one white and one black. In the white bowl there $2019$ balls. Players $A$ and $B$ play a game where they make every other move ($A$ begins). One move consists is $\bullet$ to move one or your balls from one bowl to the other, or $\bullet$ to remove a ball from the white bowl, with the condition that the resulting position (that is, the number of bullets in the two bowls) have not occurred before. The player who has no valid move to make loses. Can any of the players be sure to win? If so, which one?

Sunay is in the bottom-left square of a checkerboard which is $5$ squares wide (the left-right direction) and $3$ squares tall (the up-down direction). From any square, he may move one square up, one square down, or one square to the right, provided that he does not fall off the checkerboard and provided that he does not revisit a square. How many paths are there for Sunay from the bottom-left square to the top-right square?

A rectangular board, where in each square there is a symbol, is said to be [i]magnificent [/i] if, for each line$ L$ and for each pair of columns $C$ and $D$, there is on the board another line $M$ exactly equal to $L$, except in columns $C$ and $D$, where $M$ has symbols different from those of $L$. What is the smallest possible number of rows on a magnificent board with $2023$ columns?

Determine all graphs $G$ with the following two properties: $\bullet$ G contains at least one Hamilton path. $\bullet$ For any pair of vertices, $u, v \in G$, if there is a Hamilton path from $u$ to $v$ then the edge $uv$ is in the graph $G$

In a $m\times{n}$ grid are there are token. Every token [i]dominates [/i] every square on its same row ($\leftrightarrow$), its same column ($\updownarrow$), and diagonal ($\searrow\hspace{-4.45mm}\nwarrow$)(Note that the token does not \emph{dominate} the diagonal ($\nearrow\hspace{-4.45mm}\swarrow$), determine the lowest number of tokens that must be on the board to [i]dominate [/i] all the squares on the board.

[list] [*] There are $128$ coins of two different weights, $64$ each. How can one always find two coins of different weights by performing no more than $7$ weightings on a regular balance? [*] There are $8$ coins of two different weights, $4$ each. How can one always find two coins of different weights by performing two weightings on a regular balance?[/list]