This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 14842

2023 BmMT, Ind. Round

[b]p1.[/b] If $x$ is $20\%$ of $23$ and $y$ is $23\%$ of $20$, compute $xy$ . [b]p2.[/b] Pablo wants to eat a banana, a mango, and a tangerine, one at a time. How many ways can he choose the order to eat the three fruits? [b]p3.[/b] Let $a$, $b$, and $c$ be $3$ positive integers. If $a + \frac{b}{c} = \frac{11}{6}$ , what is the minimum value of $a + b + c$? [b]p4.[/b] A rectangle has an area of $12$. If all of its sidelengths are increased by $2$, its area becomes $32$. What is the perimeter of the original rectangle? [b]p5.[/b] Rohit is trying to build a $3$-dimensional model by using several cubes of the same size. The model’s front view and top view are shown below. Suppose that every cube on the upper layer is directly above a cube on the lower layer and the rotations are considered distinct. Compute the total number of different ways to form this model. [img]https://cdn.artofproblemsolving.com/attachments/b/b/40615b956f3d18313717259b12fcd6efb74cf8.png[/img] [b]p6.[/b] Priscilla has three octagonal prisms and two cubes, none of which are touching each other. If she chooses a face from these five objects in an independent and uniformly random manner, what is the probability the chosen face belongs to a cube? (One octagonal prism and cube are shown below.) [img]https://cdn.artofproblemsolving.com/attachments/0/0/b4f56a381c400cae715e70acde2cdb315ee0e0.png[/img] [b]p7.[/b] Let triangle $\vartriangle ABC$ and triangle $\vartriangle DEF$ be two congruent isosceles right triangles where line segments $\overline{AC}$ and $\overline{DF}$ are their respective hypotenuses. Connecting a line segment $\overline{CF}$ gives us a square $ACFD$ but with missing line segments $\overline{AC}$, $\overline{AD}$, and $\overline{DF}$. Instead, $A$ and $D$ are connected by an arc defined by the semicircle with endpoints $A$ and $D$. If $CF = 1$, what is the perimeter of the whole shape $ABCFED$ ? [img]https://cdn.artofproblemsolving.com/attachments/2/5/098d4f58fee1b3041df23ba16557ed93ee9f5b.png[/img] [b]p8.[/b] There are two moles that live underground, and there are five circular holes that the moles can hop out of. The five holes are positioned as shown in the diagram below, where $A$, $B$, $C$, $D$, and $E$ are the centers of the circles, $AE = 30$ cm, and congruent triangles $\vartriangle ABC$, $\vartriangle CBD$, and $\vartriangle CDE$ are equilateral. The two moles randomly choose exactly two of the five holes, hop out of the two chosen holes, and hop back in. What is the probability that the holes that the two moles hop out of have centers that are exactly $15$ cm apart? [img]https://cdn.artofproblemsolving.com/attachments/c/e/b46ba87b954a1904043020d7a211477caf321d.png[/img] [b]p9.[/b] Carson is planning a trip for $n$ people. Let $x$ be the number of cars that will be used and $y$ be the number of people per car. What is the smallest value of $n$ such that there are exactly $3$ possibilities for $x$ and $y$ so that $y$ is an integer, $x < y$, and exactly one person is left without a car? [b]p10.[/b] Iris is eating an ice cream cone, which consists of a hemisphere of ice cream with radius $r > 0$ on top of a cone with height $12$ and also radius $r$. Iris is a slow eater, so after eating one-third of the ice cream, she notices that the rest of the ice cream has melted and completely filled the cone. Assuming the ice cream did not change volume after it melted, what is the value of $r$? [b]p11.[/b] As Natasha begins eating brunch between $11:30$ AM and $12$ PM, she notes that the smaller angle between the minute and hour hand of the clock is $27$ degrees. What is the number of degrees in the smaller angle between the minute and hour hand when Natasha finishes eating brunch $20$ minutes later? [b]p12.[/b] On a regular hexagon $ABCDEF$, Luke the frog starts at point $A$, there is food on points $C$ and $E$ and there are crocodiles on points $B$ and $D$. When Luke is on a point, he hops to any of the five other vertices with equal probability. What is the probability that Luke will visit both of the points with food before visiting any of the crocodiles? [b]p13.[/b] $2023$ regular unit hexagons are arranged in a tessellating lattice, as follows. The first hexagon $ABCDEF$ (with vertices in clockwise order) has leftmost vertex $A$ at the origin, and hexagons $H_2$ and $H_3$ share edges $\overline{CD}$ and $\overline{DE}$ with hexagon $H_1$, respectively. Hexagon $H_4$ shares edges with both hexagons $H_2$ and $H_3$, and hexagons $H_5$ and $H_6$ are constructed similarly to hexagons H_2 and $H_3$. Hexagons $H_7$ to $H_{2022}$ are constructed following the pattern of hexagons $H_4$, $H_5$, $H_6$. Finally, hexagon H_{2023} is constructed, sharing an edge with both hexagons H2021 and H2022. Compute the perimeter of the resulting figure. [img]https://cdn.artofproblemsolving.com/attachments/1/d/eaf0d04676bac3e3c197b4686dcddd08fce9ac.png[/img] [b]p14.[/b] Aditya’s favorite number is a positive two-digit integer. Aditya sums the integers from $5$ to his favorite number, inclusive. Then, he sums the next $12$ consecutive integers starting after his favorite number. If the two sums are consecutive integers and the second sum is greater than the first sum, what is Aditya’s favorite number? [b]p15.[/b] The $100^{th}$ anniversary of BMT will fall in the year $2112$, which is a palindromic year. Compute the sum of all years from $0000$ to $9999$, inclusive, that are palindromic when written out as four-digit numbers (including leading zeros). Examples include $2002$, $1991$, and $0110$. [b]p16.[/b] Points $A$, $B$, $C$, $D$, and $E$ lie on line $r$, in that order, such that $DE = 2DC$ and $AB = 2BC$. Let $M$ be the midpoint of segment $\overline{AC}$. Finally, let point $P$ lie on $r$ such that $PE = x$. If $AB = 8x$, $ME = 9x$, and $AP = 112$, compute the sum of the two possible values of $CD$. [b]p17.[/b] A parabola $y = x^2$ in the xy-plane is rotated $180^o$ about a point $(a, b)$. The resulting parabola has roots at $x = 40$ and $x = 48$. Compute $a + b$. [b]p18.[/b] Susan has a standard die with values $1$ to $6$. She plays a game where every time she rolls the die, she permanently increases the value on the top face by $1$. What is the probability that, after she rolls her die 3 times, there is a face on it with a value of at least $7$? [b]p19.[/b] Let $N$ be a $6$-digit number satisfying the property that the average value of the digits of $N^4$ is $5$. Compute the sum of the digits of $N^4$. [b]p20.[/b] Let $O_1$, $O_2$, $...$, $O_8$ be circles of radius $1$ such that $O_1$ is externally tangent to $O_8$ and $O_2$ but no other circles, $O_2$ is externally tangent to $O_1$ and $O_3$ but no other circles, and so on. Let $C$ be a circle that is externally tangent to each of $O_1$, $O_2$, $...$, $O_8$. Compute the radius of $C$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2010 China Team Selection Test, 1

Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings: (1) $\sum_{v\in V} f(v)=|E|$; (2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$. Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.

2011 Indonesia TST, 2

Let $n$ be a integer and $n \ge 3$, and $T_1T_2...T_n$ is a regular n-gon. Distinct $3$ points $T_i , T_j , T_k$ are chosen randomly. Determine the probability of triangle $T_iT_jT_k$ being an acute triangle.

2010 Romania National Olympiad, 1

Let $S$ be a subset with $673$ elements of the set $\{1,2,\ldots ,2010\}$. Prove that one can find two distinct elements of $S$, say $a$ and $b$, such that $6$ divides $a+b$.

2023 Vietnam National Olympiad, 6

There are $n \geq 2$ classes organized $m \geq 1$ extracurricular groups for students. Every class has students participating in at least one extracurricular group. Every extracurricular group has exactly $a$ classes that the students in this group participate in. For any two extracurricular groups, there are no more than $b$ classes with students participating in both groups simultaneously. a) Find $m$ when $n = 8, a = 4 , b = 1$ . b) Prove that $n \geq 20$ when $m = 6 , a = 10 , b = 4$. c) Find the minimum value of $n$ when $m = 20 , a = 4 , b = 1$.

2023 Dutch BxMO TST, 3

We play a game of musical chairs with $n$ chairs numbered $1$ to $n$. You attach $n$ leaves, numbered $1$ to $n$, to the chairs in such a way that the number on a leaf does not match the number on the chair it is attached to. One player sits on each chair. Every time you clap, each player looks at the number on the leaf attached to his current seat and moves to sit on the seat with that number. Prove that, for any $m$ that is not a prime power with$ 1 < m \leq n$, it is possible to attach the leaves to the seats in such a way that after $m$ claps everyone has returned to the chair they started on for the first time.

1997 Cono Sur Olympiad, 1

We have $98$ cards, in each one we will write one of the numbers: $1, 2, 3, 4,...., 97, 98$. We can order the $98$ cards, in a sequence such that two consecutive numbers $X$ and $Y$ and the number $X - Y$ is greater than $48$, determine how and how many ways we can make this sequence!!

2021 Baltic Way, 9

We are given $2021$ points on a plane, no three of which are collinear. Among any $5$ of these points, at least $4$ lie on the same circle. Is it necessarily true that at least $2020$ of the points lie on the same circle?

2012 Tournament of Towns, 3

For a class of $20$ students several field trips were arranged. In each trip at least four students participated. Prove that there was a field trip such that each student who participated in it took part in at least $1/17$-th of all field trips.

2012 IFYM, Sozopol, 8

On a chess tournament two teams $A$ and $B$ are playing between each other and each consists of $n$ participants. It was noticed that however they arranged them in pairs, there was at least one pair that already played a match. Prove that there can be chosen $a$ chess players from $A$ and $b$ chess players from $B$ so that $a+b>n$ and each from the first chosen group has played a match earlier with each from the second group.

1998 Tournament Of Towns, 1

A $ 20\times20\times20$ block is cut up into 8000 non-overlapping unit cubes and a number is assigned to each. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $ 1\times20\times20$ slices parallel to the faces of the block contain this unit cube. Find the sume of all numbers of the cubes outside these slices.

2019 CMIMC, 2

How many ways are there to color the vertices of a cube red, blue, or green such that no edge connects two vertices of the same color? Rotations and reflections are considered distinct colorings.

2013 Poland - Second Round, 3

We have tiles (which are build from squares of side length 1) of following shapes: [asy] unitsize(0.5 cm); draw((1,0)--(2,0)); draw((1,1)--(2,1)); draw((1,0)--(1,1)); draw((2,0)--(2,1)); draw((0,1)--(1,1)); draw((0,2)--(1,2)); draw((0,1)--(0,2)); draw((1,1)--(1,2)); draw((0, 0)--(1, 0)); draw((0, 0)--(0, 1)); draw((5,0)--(6,0)); draw((5,1)--(6,1)); draw((5,0)--(5,1)); draw((6,0)--(6,1)); draw((4,1)--(5,1)); draw((5,2)--(6,2)); draw((5,1)--(5,2)); draw((6,1)--(6,2)); draw((4, 0)--(5, 0)); draw((4, 0)--(4, 1)); draw((6,2)--(7,2)); draw((7,1)--(7,2)); draw((6,1)--(7,1)); draw((11,0)--(12,0)); draw((11,1)--(12,1)); draw((11,0)--(11,1)); draw((12,0)--(12,1)); draw((10,1)--(11,1)); draw((10,2)--(11,2)); draw((10,1)--(10,2)); draw((11,1)--(11,2)); draw((10, 0)--(11, 0)); draw((10, 0)--(10, 1)); draw((9, 2)--(9, 1)); draw((9,1)--(10, 1)); draw((9,2)--(10,2)); [/asy] For each odd integer $n \ge 7$, determine minimal number of these tiles needed to arrange square with side of length $n$. (Attention: Tiles can be rotated, but they can't overlap.)

EMCC Accuracy Rounds, 2022

[b]p1.[/b] At a certain point in time, $20\%$ of seniors, $30\%$ of juniors, and $50\%$ of sophomores at a school had a cold. If the number of sick students was the same for each grade, the fraction of sick students across all three grades can be written as $\frac{a}{b}$ , where a and b are relatively prime positive integers. Find $a + b$. [b]p2.[/b] The average score on Mr. Feng’s recent test is a $63$ out of $100$. After two students drop out of the class, the average score of the remaining students on that test is now a $72$. What is the maximum number of students that could initially have been in Mr. Feng’s class? (All of the scores on the test are integers between $0$ and $100$, inclusive.) [b]p3.[/b] Madeline is climbing Celeste Mountain. She starts at $(0, 0)$ on the coordinate plane and wants to reach the summit at $(7, 4)$. Every hour, she moves either $1$ unit up or $1$ unit to the right. A strawberry is located at each of $(1, 1)$ and $(4, 3)$. How many paths can Madeline take so that she encounters exactly one strawberry? [b]p4.[/b] Let $E$ be a point on side $AD$ of rectangle $ABCD$. Given that $AB = 3$, $AE = 4$, and $\angle BEC = \angle CED$, the length of segment $CE$ can be written as $\sqrt{a}$ for some positive integer $a$. Find $a$. [b]p5.[/b] Lucy has some spare change. If she were to convert it into quarters and pennies, the minimum number of coins she would need is $66$. If she were to convert it into dimes and pennies, the minimum number of coins she would need is $147$. How much money, in cents, does Lucy have? [b]p6.[/b] For how many positive integers $x$ does there exist a triangle with altitudes of length $20$, $22$, and $x$? [b]p7.[/b] Compute the number of positive integers $x$ for which $\frac{x^{20}}{x+22}$ is an integer. [b]p8.[/b] Vincent the Bug is crawling along an octagonal prism. He starts on a fixed vertex $A$, visits all other vertices exactly once by traveling along the edges, and returns to $A$. Find the number of paths Vincent could have taken. [b]p9.[/b] Point $U$ is chosen inside square $ALEX$ so that $\angle AUL = 90^o$. Given that $UL = 56$ and $UE = 65$, what is the sum of all possible values for the area of square $ALEX$? [b]p10.[/b] Miranda has prepared $8$ outfits, no two of which are the same quality. She asks her intern Andrea to order these outfits for the new runway show. Andrea first randomly orders the outfits in a list. She then starts removing outfits according to the following method: she chooses a random outfit which is both immediately preceded and immediately succeeded by a better outfit and then removes it. Andrea repeats this process until there are no outfits that can be removed. Given that the expected number of outfits in the final routine can be written as $\frac{a}{b}$ for some relatively prime positive integers $a$ and $b$, find $a + b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2021 Bangladesh Mathematical Olympiad, Problem 8

Shakur and Tiham are playing a game. Initially, Shakur picks a positive integer not greater than $1000$. Then Tiham picks a positive integer strictly smaller than that.Then they keep on doing this taking turns to pick progressively smaller and smaller positive integers until some one picks $1$. After that, all the numbers that have been picked so far are added up. The person picking the number $1$ wins if and only if this sum is a perfect square. Otherwise, the other player wins. What is the sum of all possible values of $n$ such that if Shakur starts with the number $n$, he has a winning strategy?

2014 Contests, 2

We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A [i]bicoloured triangulation[/i] is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which share an edge have different colours. We call a positive integer $n \ge 4$ [i]triangulable[/i] if every regular $n$-gon has a bicoloured triangulation such that for each vertex $A$ of the $n$-gon the number of black triangles of which $A$ is a vertex is greater than the number of white triangles of which $A$ is a vertex. Find all triangulable numbers.

2021-IMOC, C5

A drunken person walks randomly on a tree. Each time, he chooses uniformly at random a neighbouring node and walks there. Show that wherever his starting point and goal are, the expected number of steps the person takes to reach the goal is always an integer. [i]houkai[/i]

2023 USA TSTST, 7

The Bank of Pittsburgh issues coins that have a heads side and a tails side. Vera has a row of 2023 such coins alternately tails-up and heads-up, with the leftmost coin tails-up. In a [i]move[/i], Vera may flip over one of the coins in the row, subject to the following rules: [list=disc] [*] On the first move, Vera may flip over any of the $2023$ coins. [*] On all subsequent moves, Vera may only flip over a coin adjacent to the coin she flipped on the previous move. (We do not consider a coin to be adjacent to itself.) [/list] Determine the smallest possible number of moves Vera can make to reach a state in which every coin is heads-up. [i]Luke Robitaille[/i]

1991 IMTS, 3

On a 8 x 8 board we place $n$ dominoes, each covering two adjacent squares, so that no more dominoes can be placed on the remaining squares. What is the smallest value of $n$ for which the above statement is true?

LMT Accuracy Rounds, 2023 S6

Aidan, Boyan, Calvin, Derek, Ephram, and Fanalex are all chamber musicians at a concert. In each performance, any combination of musicians of them can perform for all the others to watch. What is the minimum number of performances necessary to ensure that each musician watches every other musician play?

1999 All-Russian Olympiad, 4

A frog is placed on each cell of a $n \times n$ square inside an infinite chessboard (so initially there are a total of $n \times n$ frogs). Each move consists of a frog $A$ jumping over a frog $B$ adjacent to it with $A$ landing in the next cell and $B$ disappearing (adjacent means two cells sharing a side). Prove that at least $ \left[\frac{n^2}{3}\right]$ moves are needed to reach a configuration where no more moves are possible.

2011 Princeton University Math Competition, A1 / B2

Consider the sum $\overline{a b} + \overline{ c d e}$, where each of the letters is a distinct digit between $1$ and $5$. How many values are possible for this sum?

2016 Indonesia Juniors, day 2

p1. Given $f(x)=\frac{1+x}{1-x}$ , for $x \ne 1$ . Defined $p @ q = \frac{p+q}{1+pq}$ for all positive rational numbers $p$ and $q$. Note the sequence with $a_1,a_2,a_3,...$ with $a_1=2 @3$, $a_{n}=a_{n-1}@ (n+2)$ for $n \ge 2$. Determine $f(a_{233})$ and $a_{233}$ p2. It is known that $ a$ and $ b$ are positive integers with $a > b > 2$. Is $\frac{2^a+1}{2^b-1}$ an integer? Write down your reasons. p3. Given a cube $ABCD.EFGH$ with side length $ 1$ dm. There is a square $PQRS$ on the diagonal plane $ABGH$ with points $P$ on $HG$ and $Q$ on $AH$ as shown in the figure below. Point $T$ is the center point of the square $PQRS$. The line $HT$ is extended so that it intersects the diagonal line $BG$ at $N$. Point $M$ is the projection of $N$ on $BC$. Determine the volume of the truncated prism $DCM.HGN$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/22c26f2c7c66293ad7065a3c8ce3ac2ffd938b.png[/img] 4. Nine pairs of husband and wife want to take pictures in a three-line position with the background of the Palembang Ampera Bridge. There are $4$ people in the front row, $6$ people in the middle row, and $ 8$ people in the back row. They agreed that every married couple must be in the same row, and every two people next to each other must be a married couple or of the same sex. Specify the number of different possible arrangements of positions. p5. p5. A hotel provides four types of rooms with capacity, rate, and number of rooms as presented in the following table. [b] type of room, capacity of persons/ room, day / rate (Rp.), / number of rooms [/b][img]https://cdn.artofproblemsolving.com/attachments/3/c/e9e1ed86887e692f9d66349a82eaaffc730b46.jpg[/img] A group of four families wanted to stay overnight at the hotel. Each family consists of husband and wife and their unmarried children. The number of family members by gender is presented in the following table. [b]family / man / woman/ total[/b] [img]https://cdn.artofproblemsolving.com/attachments/4/6/5961b130c13723dc9fa4e34b43be30c31ee635.jpg[/img] The group leader enforces the following provisions. I. Each husband and wife must share a room and may not share a room with other married couples. II. Men and women may not share the same room unless they are from the same family. III. At least one room is occupied by all family representatives (“representative room”) IV. Each family occupies at most $3$ types of rooms. V. No rooms are occupied by more than one family except representative rooms. You are asked to arrange a room for the group so that the total cost of lodging is as low as possible. Provide two possible alternative room arrangements for each family and determine the total cost.

1990 Czech and Slovak Olympiad III A, 6

Let $k\ge 1$ be an integer and $\mathsf S$ be a family of 2-element subsets of the index set $\{1,\ldots,2k\}$ with the following property: if $\mathsf M_1,\ldots,\mathsf M_{2k}$ are arbitrary sets such that \[\mathsf M_i\cap\mathsf M_j\neq\emptyset\quad\Leftrightarrow\quad\{i,j\}\in\mathsf S,\] then the union $\mathsf M_1\cup\ldots\cup\mathsf M_{2k}$ contains at least $k^2$ elements. Show that there is a suitable family $\mathsf S$ for any integer $k\ge1.$

2017 South East Mathematical Olympiad, 8

Tags: set , combinatorics
Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set $$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$ Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, x_3, y_1, y_2, y_3$ such that $x_1 < x_2 < x_3, y_1 < y_2 < y_3$ and $$(x_1, y_2), (x_2, y_1), (x_2, y_2), (x_2, y_3), (x_3, y_2) \in A.$$ Determine the largest possible number of elements in $A$.