Draw all the diagonals of a convex polygon of $10$ sides. They divide their angles into $80$ parts. It is known that at least $59$ of those parts are equal. Determine the largest number of distinct values ​​among the $ 80$ angles of division and how many times each of those values ​​occurs.

Six thousand points are marked on a circle, and they are colored using 10 colors in such a way that within every group of 100 consecutive points all the colors are used. Determine the least positive integer $ k$ with the following property: In every coloring satisfying the condition above, it is possible to find a group of $ k$ consecutive points in which all the colors are used.

[b]p1.[/b] There is a string of numbers $1234567891023...910134 ...91012...$ that concatenates the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, then $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, then $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, $2$, and so on. After $10$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, the string will be concatenated with $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$ again. What is the $2021$st digit? [b]p2.[/b] Bob really likes eating rice. Bob starts eating at the rate of $1$ bowl of rice per minute. Every minute, the number of bowls of rice Bob eats per minute increases by $1$. Given there are $78$ bowls of rice, find number of minutes Bob needs to finish all the rice. [b]p3.[/b] Suppose John has $4$ fair coins, one red, one blue, one yellow, one green. If John flips all $4$ coins at once, the probability he will land exactly $3$ heads and land heads on both the blue and red coins can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, Find $a + b$. [b]p4.[/b] Three of the sides of an isosceles trapezoid have lengths $1$, $10$, $20$ Find the sum of all possible values of the fourth side. [b]p5.[/b] An number two-three-delightful if and only if it can be expressed as the product of $2$ consecutive integers larger than $1$ and as the product of $3$ consecutive integers larger than $1$. What is the smallest two-three-delightful number? [b]p6.[/b] There are $3$ students total in Justin's online chemistry class. On a $100$ point test, Justin's two classmates scored $4$ and $7$ points. The teacher notices that the class median score is equal to $gcd(x, 42)$, where the positive integer $x$ is Justin's score. Find the sum of all possible values of Justin's score. [b]p7.[/b] Eddie's gym class of $10$ students decides to play ping pong. However, there are only $4$ tables and only $2$ people can play at a table. If $8$ students are randomly selected to play and randomly assigned a partner to play against at a table, the probability that Eddie plays against Allen is $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, Find $a + b$. [b]p8.[/b] Let $S$ be the set of integers $k$ consisting of nonzero digits, such that $300 < k < 400$ and $k - 300$ is not divisible by $11$. For each $k$ in $S$, let $A(k)$ denote the set of integers in $S$ not equal to $k$ that can be formed by permuting the digits of $k$. Find the number of integers $k$ in $S$ such that $k$ is relatively prime to all elements of $A(k)$. [b]p9.[/b] In $\vartriangle ABC$, $AB = 6$ and $BC = 5$. Point $D$ is on side $AC$ such that $BD$ bisects angle $\angle ABC$. Let $E$ be the foot of the altitude from $D$ to $AB$. Given $BE = 4$, find $AC^2$. [b]p10.[/b] For each integer $1 \le n \le 10$, Abe writes the number $2^n + 1$ on a blackboard. Each minute, he takes two numbers $a$ and $b$, erases them, and writes $\frac{ab-1}{a+b-2}$ instead. After $9$ minutes, there is one number $C$ left on the board. The minimum possible value of $C$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$. [b]p11.[/b] Estimation (Tiebreaker) Let $A$ and $B$ be the proportions of contestants that correctly answered Questions $9$ and $10$ of this round, respectively. Estimate $\left \lfloor \dfrac{1}{(AB)^2} \right \rfloor$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There is a cube of $3 \times 3 \times 3$ formed by the union of $27$ cubes of $1 \times 1 \times 1$. Some cubes are removed in such a way that those that remain continue to form a solid made up of cubes that are united by at least one facing the rest of the solid. When a cube is removed, those that remain do so in the same place they were. What is the maximum number of cubes that can be removed so that the area of the resulting solid is equal to the area of the original cube?

Let $n \geq k$ be positive integers and let $a_1, \dots, a_n$ be a non-increasing list of positive real numbers. Prove that there exists $k$ sets $B_1, \dots, B_k$ which partition the set $\{1, 2, \dots, n\}$ such that $$\min_{1 \le j \le k} \left(\sum_{i \in B_j} a_i \right) \geq \min_{1 \le j \le k} \left(\frac{1}{2k+1-2j} \cdot \sum^n_{i=j} a_i\right).$$ [i]Proposed by Navid Safaei[/i]

Alice and Bob are playing a game. First, Alice chooses a partition $\mathcal{C}$ of the positive integers into a (not necessarily finite) set of sets, such that each positive integer is in exactly one of the sets in $\mathcal{C}$. Then Bob does the following operation a finite number of times. Choose a set $S \in \mathcal{C}$ not previously chosen, and let $D$ be the set of all positive integers dividing at least one element in $S$. Then add the set $D \setminus S$ (possibly the empty set) to $\mathcal{C}$. Bob wins if there are two equal sets in $\mathcal{C}$ after he has done all his moves, otherwise, Alice wins. Determine which player has a winning strategy.

The chess piece [i]sick rook[/i] can move along rows and columns as a regular rook, but at most by $2$ fields. We can place [i]sick rooks[/i] on a square board in such a way that no two of them attack each other and no field is attacked by more than one [i]sick rook[/i]. a) Prove that on $30\times 30$ board, we cannot place more than $100$ [i]sick rooks[/i]. b) Find the maximum number of [i]sick rooks[/i] which can be placed on $8\times 8$ board. c) Prove that on $32\times 32$ board, we cannot place more than $120$ [i]sick rooks[/i].

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Compute $\frac{2024}{2 + 0 \times 2 - 4}.$ [b]p2.[/b] Find the smallest integer that can be written as the product of three distinct positive odd integers. [b]p3.[/b] Bryan’s physics test score is a two-digit number. When Bryan reverses its digits and adds the tens digit of his test score, he once again obtains his test score. Determine Bryan’s physics test score. [b]p4.[/b] Grant took four classes today. He spent $70$ minutes in math class. Had his math class been $40$ minutes instead, he would have spent $15\%$ less total time in class today. Find how many minutes he spent in his other classes combined. [b]p5.[/b] Albert’s favorite number is a nonnegative integer. The square of Albert’s favorite number has $9$ digits. Find the number of digits in Albert’s favorite number. [b]p6.[/b] Two semicircular arcs are drawn in a rectangle, splitting it into four regions as shown below. Given the areas of two of the regions, find the area of the entire rectangle. [img]https://cdn.artofproblemsolving.com/attachments/1/a/22109b346c7bdadeaf901d62155de4c506b33c.png[/img] [b]p7.[/b] Daria is buying a tomato and a banana. She has a $20\%$-off coupon which she may use on one of the two items. If she uses it on the tomato, she will spend $\$1.21$ total, and if she uses it on the banana, she will spend $\$1.31$ total. In cents, find the absolute difference between the price of a tomato and the price of a banana. [b]p8.[/b] Celine takes an $8\times 8$ checkerboard of alternating black and white unit squares and cuts it along a line, creating two rectangles with integer side lengths, each of which contains at least $9$ black squares. Find the number of ways Celine can do this. (Rotations and reflections of the cut are considered distinct.) [b]p9.[/b] Each of the nine panes of glass in the circular window shown below has an area of $\pi$, eight of which are congruent. Find the perimeter of one of the non-circular panes. [img]https://cdn.artofproblemsolving.com/attachments/b/c/0d3644dde33b68f186ba1ff0602e08ce7996f5.png[/img] [b]p10.[/b] In Alan’s favorite book, pages are numbered with consecutive integers starting with $1$. The average of the page numbers in Chapter Five is $95$ and the average of the page numbers in Chapter Six is $114$. Find the number of pages in Chapters Five and Six combined. [b]p11.[/b] Find the number of ordered pairs $(a, b)$ of positive integers such that $a + b = 2024$ and $$\frac{a}{b}>\frac{1000}{1025}.$$ [b]p12.[/b] A square is split into three smaller rectangles $A$, $B$, and $C$. The area of $A$ is $80$, $B$ is a square, and the area of $C$ is $30$. Compute the area of $B$. [img]https://cdn.artofproblemsolving.com/attachments/d/5/43109b964eacaddefd410ddb8bf4e4354a068b.png[/img] [b]p13.[/b] A knight on a chessboard moves two spaces horizontally and one space vertically, or two spaces vertically and one space horizontally. Two knights attack each other if each knight can move onto the other knight’s square. Find the number of ways to place a white knight and a black knight on an $8 \times 8$ chessboard so that the two knights attack each other. One such possible configuration is shown below. [img]https://cdn.artofproblemsolving.com/attachments/2/2/b4a83fbbab7e54dda81ac5805728d268b6db9f.png[/img] [b]p14.[/b] Find the sum of all positive integers $N$ for which the median of the positive divisors of $N$ is $9$. [b]p15.[/b] Let $x$, $y$, and $z$ be nonzero real numbers such that $$\begin{cases} 20x + 24y = yz \\ 20y + 24x = xz \end{cases}$$ Find the sum of all possible values of $z$. [b]p16.[/b] Ava glues together $9$ standard six-sided dice in a $3 \times 3$ grid so that any two touching faces have the same number of dots. Find the number of dots visible on the surface of the resulting shape. (On a standard six-sided die, opposite faces sum to $7$.) [img]https://cdn.artofproblemsolving.com/attachments/5/5/bc71dac9b8ae52a4456154000afde2c89fd83a.png[/img] [b]p17.[/b] Harini has a regular octahedron of volume $1$. She cuts off its $6$ vertices, turning the triangular faces into regular hexagons. Find the volume of the resulting solid. [b]p18.[/b] Each second, Oron types either $O$ or $P$ with equal probability, forming a growing sequence of letters. Find the probability he types out $POP$ before $OOP$. [b]p19.[/b] For an integer $n \ge 10$, define $f(n)$ to be the number formed after removing the first digit from $n$ (and removing any leading zeros) and define $g(n)$ to be the number formed after removing the last digit from $n$. Find the sum of the solutions to the equation $f(n) + g(n) = 2024$. [b]p20.[/b] In convex trapezoid $ABCD$ with $\overline{AB} \parallel \overline{CD}$ and $AD = BC$, let $M$ be the midpoint of $\overline{BC}$. If $\angle AMB = 24^o$ and $\angle CMD = 66^o$, find $\angle ABC$, in degrees. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Peter has a deck of $1001$ cards, and with a blue pen he has written the numbers $1,2,\ldots,1001$ on the cards (one number on each card). He replaced cards in a circle so that blue numbers were on the bottom side of the card. Then, for each card $C$, he took $500$ consecutive cards following $C$ (clockwise order), and denoted by $f(C)$ the number of blue numbers written on those $500$ cards that are greater than the blue number written on $C$ itself. After all, he wrote this $f(C)$ number on the top side of the card $C$ with a red pen. Prove that Peter's friend Basil, who sees all the red numbers on these cards, can determine the blue number on each card.

Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees? [i]Proposed by Seyed Reza Hosseini[/i]

Let $\ell = 1$, $M = 23$, $N = 45$, and $u = 67$. Compute the number of ordered pairs of nonnegative integers $(X, Y)$ with $X \leq M - \ell$ and $Y \leq N + u$ such that the sum \[ \sum_{k=\ell}^{u} \binom{X + k}{M}\cdot\binom{Y - k}{N} \] is divisible by $89$ (for integers $a$ and $b$, define the binomial coefficient $\tbinom{a}{b}$ to be the number of $b$-element subsets of any given $a$-element set, which is $0$ when $a < 0$, $b < 0$, or $b > a$).

Q.A bug travels in the co-ordinate plane moving along only the lines that are parallel to the $X$ and $Y$ axes.Let $A=(-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$.How many lattice points lie on at least one of these paths. My answer ($87$)

Let $ABC$ be any triangle with $\angle BAC \le \angle ACB \le \angle CBA$. Let $D, E$ and $F$ be the midpoints of $BC, CA$ and $AB$, respectively, and let $\epsilon$ be a positive real number. Suppose there is an ant (represented by a point $T$ ) and two spiders (represented by points $P_1$ and $P_2$, respectively) walking on the sides $BC, CA, AB, EF, FD$ and $DE$. The ant and the spiders may vary their speeds, turn at an intersection point, stand still, or turn back at any point; moreover, they are aware of their and the others’ positions at all time. Assume that the ant’s speed does not exceed $1$ mm/s, the first spider’s speed does not exceed $\frac{\sin A}{2 \sin A+\sin B}$ mm/s, and the second spider’s speed does not exceed $\epsilon$ mm/s. Show that the spiders always have a strategy to catch the ant regardless of the starting points of the ant and the spiders. Note: the two spiders can discuss a plan before the hunt starts and after seeing all three starting points, but cannot communicate during the hunt.

Let $n$ be a positive integer. Find the greatest possible integer $m$, in terms of $n$, with the following property: a table with $m$ rows and $n$ columns can be filled with real numbers in such a manner that for any two different rows $\left[ {{a_1},{a_2},\ldots,{a_n}}\right]$ and $\left[ {{b_1},{b_2},\ldots,{b_n}} \right]$ the following holds: \[\max\left( {\left| {{a_1} - {b_1}} \right|,\left| {{a_2} - {b_2}} \right|,...,\left| {{a_n} - {b_n}} \right|} \right) = 1\] [i]Poland (Tomasz Kobos)[/i]

An infinite fairytale is a book with pages numbered $1,2,3,\ldots$ where all natural numbers appear. An author wants to write an infinite fairytale such that a new dwarf is introduced on each page. Afterward, the page contains several discussions between groups of at least two of the already introduced dwarfs. The publisher wants to make the book more exciting and thus requests the following condition: Every infinite set of dwarfs contains a group of at least two dwarfs, who formed a discussion group at some point as well as a group of the same size for which this is not true. Can the author fulfill this condition?

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

International Mathematical Olympiad National Selection Test Malaysia 2020 Round 1 Primary 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. Annie asks his brother four questions, "What is $20$ plus $20$? What is $20$ minus $20$? What is $20$ times $20$? What is $20$ divided by $20$?". His brother adds the answers to these four questions, and then takes the (positive) square root of the result. What number does he get? p2. A broken watch moves slower than a regular watch. In every $7$ hours, the broken watch lags behind a regular watch by $10$ minutes. In one week, how many hours does the broken watch lags behind a regular watch? p3. Given a square $ABCD$. A point $P$ is chosen outside the square so that triangle $BCP$ is equilateral. Find $\angle APC$, in degrees. p4. Hussein throws 4 dice simultaneously, and then adds the number of dots facing up on all $4$ dice. How many possible sums can Hussein get? Note: For example, he can get sum $14$, by throwing $4$, $6$, $3$, and $ 1$. Assume these are regular dice, with $1$ to $6$ dots on the faces. p5. Mrs. Sheila says, "I have $5$ children. They were born one by one every $3$ years. The age of my oldest child is $7$ times the age of my youngest child." What is the age of her third child? [b]Part B [/b](2 points each) p6. The number $N$ is the smallest positive integer with the sum of its digits equal to $2020$. What is the first (leftmost) digit of $N$? p7. 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. p8. 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$$\frac{area \,\, of \,\, \vartriangle ABC}{area \,\, of \,\, \vartriangle ABD}.$$ p9. 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 $-1 - 2 - 3 + 4$. p10. How many $3$-digit numbers have its sum of digits equal to $4$? [b]Part C[/b] (3 points each) p11. 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. p12. 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. p13. 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 $A/B$. p14. This sequence lists the perfect squares in increasing order:$$0, 1, 4, 9, 16, ... ,a, 10^8, b, ...$$Determine the value of $b - a$. p15. Determine the last digit of $5^5 + 6^6 + 7^7 + 8^8 + 9^9$ [b]Part D[/b] (4 points each) p16. Find the sum of all integers between $-\sqrt{1442}$ and $\sqrt{2020}$. p17. 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 $k$? p18. 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 $PD$? p19. What is the smallest positive multiple of $225$ that can be written using digits $0$ and $ 1$ only? p20. Given positive integers $a, b$, and $c$ with $a + b + c = 20$. Determine the number of possible integer values for $\frac{a + b}{c}$. PS. Problems 6-20 were also used in [url=https://artofproblemsolving.com/community/c4h2675966p23194287]Juniors [/url]as 1-15. Problems 11-20 were also used in Seniors 1-10.

a) Given a big square consisting of $7\times 7$ squares. You should mark the centres of $k$ points in such a way, that no quadruple of the marked points will be the vertices of a rectangle with the sides parallel to the sides of the given squares. What is the greatest $k$ such that the problem has solution? b) The same problem for $13\times 13$ square.

Does there exist a set $S$ of positive integers satisfying the following conditions? $\text{(i)}$ $S$ contains $2020$ distinct elements; $\text{(ii)}$ the number of distinct primes in the set $\{\gcd(a, b) : a, b \in S, a \neq b\}$ is exactly $2019$; and $\text{(iii)}$ for any subset $A$ of $S$ containing at least two elements, $\sum\limits_{a,b\in A; a<b} ab$ is not a prime power.

Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?

On a straight road, points $1, 2, \ldots, n$ are marked. The distance between any two adjacent points is 1. A "placement" refers to the arrangement of $n$ cars, numbered with the same numbers, at the marked points (there can be multiple cars at one point). The "distance" between two placements is defined as the minimum total length of sections that need to be paved so that cars from the first placement can drive on the asphalt, forming the second one (cars can change places on the road). Prove that for any $\alpha<1$, there exists an integer number $n$ for which there are $100^n$ placements, the pairwise distances between which are greater than $\alpha n$. [i]Proposed by Ilya Bogdanov[/i]

There are $2000$ components in a circuit, every two of which were initially joined by a wire. The hooligans Vasya and Petya cut the wires one after another. Vasya, who starts, cuts one wire on his turn, while Petya cuts one or three. The hooligan who cuts the last wire from some component loses. Who has the winning strategy?

A magician should determine the area of a hidden convex $ 2008$-gon $ A_{1}A_{2}\cdots A_{2008}$. In each step he chooses two points on the perimeter, whereas the chosen points can be vertices or points dividing selected sides in selected ratios. Then his helper divides the polygon into two parts by the line through these two points and announces the area of the smaller of the two parts. Show that the magician can find the area of the polygon in $ 2006$ steps.

$m$ and $n$ are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows: Starting from one of the hexagons, the Donkey moves $m$ cells in one of the $6$ directions, then it turns $60$ degrees clockwise and after that moves $n$ cells in this new direction until it reaches it's final cell. At most how many cells are in the Philosopher's chessboard such that one cannot go from anyone of them to the other with a finite number of movements of the Donkey? [i]Proposed by Shayan Dashmiz[/i]

A regular hexagon is split into $6$ congruent equilateral triangles by drawing in the $3$ main diagonals. Each triangle is colored $1$ of $4$ distinct colors. Rotations and reflections of the figure are considered nondistinct. Find the number of possible distinct colorings.