Found problems: 1385
(Game) At the beginning of the game the organisers place $4$ piles of paper disks onto the table. The player who is in turn takes away a pile, then divides one of the remaining piles into two nonempty piles. Whoever is unable to move, loses.
[i]Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.[/i]
Two players $A$ and $B$ participate in the following game. Initially we have a pile of 2003 stones. $A$ plays first, and he picks a divisor of 2003 and removes that number of stones from the pile. Then $B$ picks a divisor of the number of remaining stones, and removes that number of stones from the pile, and so forth. The players who removes the last stone loses. Prove that one of the players has a winning strategy and describe it.
A table $10 \times 10$ was filled according to the rules of the game “Bomb Squad”: several cells contain bombs (one bomb per cell) while each of the remaining cells contains a number, equal to the number of bombs in all cells adjacent to it by side or by vertex. Then the table is rearranged in the “reverse” order: bombs are placed in all cells previously occupied with numbers and the remaining cells are filled with numbers according to the same rule. Can it happen that the total sum of the numbers in the table will increase in a result?
Lalo and Sergio play in a regular polygon of $n\geq 4$ sides. In his turn, Lalo paints a diagonal or side of pink, and in his turn Sergio paint a diagonal or side of orange. Wins the game who achieve paint the three sides of a triangle with his color, if none of the players can win, they game tie. Lalo starts playing. Determines all natural numbers $n$ such that one of the players have winning strategy.
A Mathlon is a competition where there are $M$ athletic events. $A, B$ and $C$ were the only participants of a Mathlon. In each event, $p_1$ points were given to the first place, $p_2$ points to the second place and $p_3$ points to third place, with $p_1> p_2> p_3> 0$ where $p_1$, $p_2$ and $p_3$ are integer numbers. The final result was $22$ points for $A$, $9$ for $B$, and $9$ for $C$. $B$ won the $100$ meter dash. Determine $M$ and who was the second in high jump.
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]
The numbers $1, 2, \dots, 1999$ are written on the board. Two players take turn choosing $a,b$ from the board and erasing them then writing one of $ab$, $a+b$, $a-b$. The first player wants the last number on the board to be divisible by $1999$, the second player want to stop him. Determine the winner.
Anna and Ben are playing with a permutation $p$ of length $2020$, initially $p_i = 2021 - i$ for $1\le i \le 2020$. Anna has power $A$, and Ben has power $B$. Players are moving in turns, with Anna moving first.
In his turn player with power $P$ can choose any $P$ elements of the permutation and rearrange them in the way he/she wants.
Ben wants to sort the permutation, and Anna wants to not let this happen. Determine if Ben can make sure that the permutation will be sorted (of form $p_i = i$ for $1\le i \le 2020$) in finitely many turns, if
a) $A = 1000, B = 1000$
b) $A = 1000, B = 1001$
c) $A = 1000, B = 1002$
[i]Anton Trygub[/i]
[b]p1.[/b] Can a number ending in $1999$ be the square of a natural number?
[b]p2.[/b] The Three-Headed Snake Gorynych celebrated his birthday. His heads took turns feasting on birthday cakes and ate two identical cakes in $15$ minutes. It is known that each head ate as much time as it would take the other two to eat the same pie together. In how many minutes would the three heads of the Serpent Gorynych eat one pie together?
[b]p3.[/b] Find the sum of the coefficients of the polynomial obtained after opening the brackets and bringing similar terms into the expression:
a) $(7x - 6)^4 - 1$
b) $(7x - 6)^{1999}-1$
[b]p4.[/b] The general wants to arrange seven anti-aircraft installations so that among any three of them there are two installations, the distance between which is exactly $10$ kilometers. Help the general solve this problem.
[b]p5.[/b] Gulliver, whose height is $999$ millimeters, is building a tower of cubes. The first cube has a height of $1/2$ a lilikilometer, the second - $1/4$ a lilikilometer, the third - $1/8$ a lilikilometer, etc. How many cubes will be in the tower when its height exceeds Gulliver's height. ($1$ lilikilometer is equal to $1000$ lilimeters).
[b]p6.[/b] It is known that in any pentagon you can choose three diagonals from which you can form a triangle. Is there a pentagon in which such diagonals can be chosen in a unique way?
[b]p7.[/b] It is known that for natural numbers $a$ and $b$ the equality $19a = 99b$ holds. Can $a + b$ be a prime number?
[b]p8.[/b] Vitya thought of $5$ integers and told Vanya all their pairwise sums:
$$0, 1, 5, 7, 11, 12, 18, 24, 25, 29.$$
Help Vanya guess the numbers he has in mind.
[b]p9.[/b] In a $3 \times 3$ square, numbers are arranged so that the sum of the numbers in each row, in each column and on each major diagonal is equal to $0$. It is known that the sum of the squares of the numbers in the top row is $n$. What can be the sum of the squares of the numbers in the bottom line?
[b]p10.[/b] $N$ points are marked on a circle. Two players play this game: the first player connects two of these points with a chord, from the end of which the second player draws a chord to one of the remaining points so as not to intersect the already drawn chord. Then the first player makes the same “move” - draws a new chord from the end of the second chord to one of the remaining points so that it does not intersect any of the already drawn ones. The one who cannot make such a “move” loses. Who wins when played correctly? (A chord is a segment whose ends lie on a given circle)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here[/url].
There are 2009 boxes numbered from 1 to 2009, some of which contain stones. Two players, $ A$ and $ B$, play alternately, starting with $ A$. A move consists in selecting a non-empty box $ i$, taking one or more stones from that box and putting them in box $ i \plus{} 1$. If $ i \equal{} 2009$, the selected stones are eliminated. The player who removes the last stone wins
a) If there are 2009 stones in the box 2 and the others are empty, find a winning strategy for either player.
b) If there is exactly one stone in each box, find a winning strategy for either player.
Given an $m \times n$ table consisting of $mn$ unit cells. Alice and Bob play the following game: Alice goes first and the one who moves colors one of the empty cells with one of the given three colors. Alice wins if there is a figure, such as the ones below, having three different colors. Otherwise Bob is the winner. Determine the winner for all cases of $m$
and $n$ where $m, n \ge 3$.
Proposed by [i]Toghrul Abbasov, Azerbaijan[/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$$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$
On an infinite checkerboard two players alternately mark one unmarked cell. One of them uses $\times$, the other $\circ$. The first who fills a $2\times 2$ square with his symbols wins. Can the player who starts always win?
The number $10^{2007}$ is written on the blackboard. Anne and Berit play a two player game in which the player in turn performs one of the following operations:
1) replace a number $x$ on the blackboard with two integers $a,b>1$ such that $ab=x$.
2) strike off one or both of two equal numbers on the blackboard.
The person who cannot perform any operation loses.
Who has the winning strategy if Anne starts?
Kobar and Borah are playing on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board, either from $P$ and $Q$ to $2P-Q$ and $2Q-P$, or from $P$ and $Q$ to $5P-4Q$ and $5Q-4P$. The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner.
At the beginning of the game, the two numbers on the board are $2024$ and $A$. If it is known that Kobar does not lose on his first move, determine the largest possible value of $A$ so that Borah can win this game.
We are given a natural number $k$. Let us consider the following game on an infinite onedimensional board. At the start of the game, we distrubute $n$ coins on the fields of the given board (one field can have multiple coins on itself). After that, we have two choices for the following moves:
$(i)$ We choose two nonempty fields next to each other, and we transfer all the coins from one of the fields to the other.
$(ii)$ We choose a field with at least $2$ coins on it, and we transfer one coin from the chosen field to the $k-\mathrm{th}$ field on the left , and one coin from the chosen field to the $k-\mathrm{th}$ field on the right.
$\mathbf{(a)}$ If $n\leq k+1$, prove that we can play only finitely many moves.
$\mathbf{(b)}$ For which values of $k$ we can choose a natural number $n$ and distribute $n$ coins on the given board such that we can play infinitely many moves.
The number $2$ is written on the board. Ana and Bruno play alternately. Ana begins. Each one, in their turn, replaces the number written by the one obtained by applying exactly one of these operations: multiply the number by $2$, multiply the number by $3$ or add $1$ to the number. The first player to get a number greater than or equal to $2011$ wins. Find which of the two players has a winning strategy and describe it.
We are given a set $P$ of points and a set $L$ of straight lines. At the beginning there are 4 points, no three of which are collinear, and $L=\emptyset $. Two players are taking turns adding one or two lines to $L$, where each of these lines has to pass through at least two of the points in $P$. After that all intersection points of the lines in $L$ are added to $P$, if they are not already part of it. A player wins, if after his turn there are three collinear points from $P$, which lie on a line that isn’t from $L$. Find who of the two players has a winning strategy.
Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card.
Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$.
[i]Proposed by Carl Schildkraut and Colin Tang[/i]
Ada and Charles play the following game:at the beginning, an integer n>1 is written on the blackboard.In turn, Ada and Charles remove the number k that they find on the blackboard.In turn Ad and Charles remove the number k that they find on the blackboard and they replace it :
1 -either with a positive divisor k different from 1 and k
2- or with k+1
At the beginning each players have a thousand points each.When a player choses move 1, he/she gains one point;when a player choses move 2, he/she loses one point.The game ends when one of the tho players is left with zero points and this player loses the game.Ada moves first.For what values Chares has a winning strategy?
On a chessboard, Po controls a white queen and plays, in alternate turns, against an invisible black king (there are only those two pieces on the board). The king cannot move to a square where he would be in check, neither capture the queen. Every time the king makes a move, Po receives a message from beyond that tells which direction the king has moved (up, right, up-right, etc). His goal is to make the king unable to make a movement.
Can Po reach his goal with at most $150$ moves, regardless the starting position of the pieces?
[u]Part 1[/u]
[b]p1.[/b] Two kids $A$ and $B$ play a game as follows: From a box containing $n$ marbles ($n > 1$), they alternately take some marbles for themselves, such that:
1. $A$ goes first.
2. The number of marbles taken by $A$ in his first turn, denoted by $k$, must be between $1$ and $n$, inclusive.
3. The number of marbles taken in a turn by any player must be between $1$ and $k$, inclusive.
The winner is the one who takes the last marble. What is the sum of all $n$ for which $B$ has a winning strategy?
[b]p2.[/b] How many ways can your rearrange the letters of "Alejandro" such that it contains exactly one pair of adjacent vowels?
[b]p3.[/b] Assuming real values for $p, q, r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $q + 6i$, and the product of the other two roots is $3 - 4i$. Find the smallest value of $q$.
[b]p4.[/b] Lisa has a $3$D box that is $48$ units long, $140$ units high, and $126$ units wide. She shines a laser beam into the box through one of the corners, at a $45^o$ angle with respect to all of the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a $45^o$ angle. Compute the distance the laser beam travels until it hits one of the eight corners of the box.
[u]Part 2[/u]
[b]p5.[/b] How many ways can you divide a heptagon into five non-overlapping triangles such that the vertices of the triangles are vertices of the heptagon?
[b]p6.[/b] Let $a$ be the greatest root of $y = x^3 + 7x^2 - 14x - 48$. Let $b$ be the number of ways to pick a group of $a$ people out of a collection of $a^2$ people. Find $\frac{b}{2}$ .
[b]p7.[/b] Consider the equation
$$1 -\frac{1}{d}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c},$$
with $a, b, c$, and $d$ being positive integers. What is the largest value for $d$?
[b]p8.[/b] The number of non-negative integers $x_1, x_2,..., x_{12}$ such that $$x_1 + x_2 + ... + x_{12} \le 17$$
can be expressed in the form ${a \choose b}$ , where $2b \le a$. Find $a + b$.
[u]Part 3[/u]
[b]p9.[/b] In the diagram below, $AB$ is tangent to circle $O$. Given that $AC = 15$, $AB = 27/2$, and $BD = 243/34$, compute the area of $\vartriangle ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/b/f/b403e5e188916ac4fb1b0ba74adb7f1e50e86a.png[/img]
[b]p10.[/b] If
$$\left[2^{\log x}\right]^{[x^{\log 2}]^{[2^{\log x}]...}}= 2, $$
where $\log x$ is the base-$10$ logarithm of $x$, then it follows that $x =\sqrt{n}$. Compute $n^2$.
[b]p11.[/b]
[b]p12.[/b] Find $n$ in the equation $$133^5 + 110^5 + 84^5 + 27^5 = n^5, $$ where $n$ is an integer less than $170$.
[u]Part 4[/u]
[b]p13.[/b] Let $x$ be the answer to number $14$, and $z$ be the answer to number $16$. Define $f(n)$ as the number of distinct two-digit integers that can be formed from digits in $n$. For example, $f(15) = 4$ because the integers $11$, $15$, $51$, $55$ can be formed from digits of $15$. Let $w$ be such that $f(3xz - w) = w$. Find $w$.
[b]p14.[/b] Let $w$ be the answer to number $13$ and $z$ be the answer to number $16$. Let $x$ be such that the coefficient of $a^xb^x$ in $(a + b)^{2x}$ is $5z^2 + 2w - 1$. Find $x$.
[b]p15.[/b] Let $w$ be the answer to number $13$, $x$ be the answer to number $14$, and $z$ be the answer to number $16$. Let $A$, $B$, $C$, $D$ be points on a circle, in that order, such that $\overline{AD}$ is a diameter of the circle. Let $E$ be the intersection of $\overleftrightarrow{AB}$ and $\overleftrightarrow{DC}$, let $F$ be the intersection of $\overleftrightarrow{AC}$ and $\overleftrightarrow{BD}$, and let $G$ be the intersection of $\overleftrightarrow{EF}$ and $\overleftrightarrow{AD}$. Now, let $AE = 3x$, $ED = w^2 - w + 1$, and $AD = 2z$. If $FG = y$, find $y$.
[b]p16.[/b] Let $w$ be the answer to number $13$, and $x$ be the answer to number $16$. Let $z$ be the number of integers $n$ in the set $S = \{w,w + 1, ... ,16x - 1, 16x\}$ such that $n^2 + n^3$ is a perfect square. Find $z$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Malmer Pebane, Fames Jung, and Weven Dare are perfect logicians that always tell the truth. Malmer decides to pose a puzzle to his friends: he tells them that the day of his birthday is at most the number of the month of his birthday. Then Malmer announces that he will whisper the day of his birthday to Fames and the month of his birthday to Weven, and he does exactly that.
After Malmer whispers to both of them, Fames thinks a bit, then says “Weven cannot know what Malmer’s birthday is.”
After that, Weven thinks a bit, then says “Fames also cannot know what Malmer’s birthday is.”
This exchange repeats, with Fames and Weven speaking alternately and each saying the other can’t know Malmer’s birthday. However, at one point, Weven instead announces “Fames and I can now know what Malmer’s birthday is. Interestingly, that was the longest conversation like that we could have possibly had before both figuring out Malmer’s birthday.”
Find Malmer’s birthday.
Alberto and Bianca play a game on a square board. Alberto begins. On their turn, players place a $1 \times 2$ or $2 \times 1$ domino on two empty squares on the board. The player who cannot put a domino loses. Determine who has a winning strategy (and prove it) if the board is:
i) $3 \times 3$
ii) $3 \times 4$
A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or
[*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter.
[i]Proposed by Aron Thomas[/i]