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Swimming poolFri, 20 Jul 2018 06:30:19 +0000
http://www.amc8.net/threads/swimming-pool.636/
http://www.amc8.net/threads/swimming-pool.636/invalid@example.com (Cherry)CherryTanner wants to fill his swimming pool using two hoses, each of which sprays water at a constant rate. Hose A fills the pool in \(a\) hours when used by itself, where \(a\) is a positive integer. Hose B fills the pool in \(b\) hours when used by itself, where \(b\) is a positive integer. When used together, Hose A and Hose B fill the pool in 6 hours. How many different possible values are there for \(a\)?

\( \textbf{(A)}\ 5\qquad...\)

Swimming pool]]>0CubeWed, 18 Jul 2018 09:45:11 +0000
http://www.amc8.net/threads/cube.635/
http://www.amc8.net/threads/cube.635/invalid@example.com (Cherry)CherryOne face of a cube contains a circle, as shown. This cube rolls without sliding on a four by four checkerboard. The cube always begins a path on the bottom left square in the position shown and completes the path on the top right square. During each move, an edge of the cube remains in contact with the board. Each move of the cube is either to the right or up. For each path, a face of the cube contacts seven different squares on the checkerboard, including the bottom left and top right...

Cube]]>0TicketsWed, 18 Jul 2018 09:43:12 +0000
http://www.amc8.net/threads/tickets.634/
http://www.amc8.net/threads/tickets.634/invalid@example.com (Cherry)CherryA box contains a total of 400 tickets that come in five colours: blue, green, red, yellow and orange. The ratio of blue to green to red tickets is 1 : 2 : 4. The ratio of green to yellow to orange tickets is 1 : 3 : 6. What is the smallest number of tickets that must be drawn to ensure that at least 50 tickets of one colour have been selected?

\( \textbf{(A)}\ 50\qquad \textbf{(B)}\ 246\qquad \textbf{(C)}\ 148\qquad \textbf{(D)}\ 196\qquad \textbf{(E)}\ 115\qquad\)]]>0positive integersWed, 18 Jul 2018 07:03:39 +0000
http://www.amc8.net/threads/positive-integers.633/
http://www.amc8.net/threads/positive-integers.633/invalid@example.com (Cherry)CherryTwo different 2-digit positive integers are called a reversal pair if the position of the digits in the first integer is switched in the second integer. For example, 52 and 25 are a reversal pair. The integer 2015 has the property that it is equal to the product of three different prime numbers, two of which are a reversal pair. Including 2015, how many positive integers less than 10 000 have this same property?

positive integers]]>0Positive IntegersMon, 16 Jul 2018 06:24:26 +0000
http://www.amc8.net/threads/positive-integers.632/
http://www.amc8.net/threads/positive-integers.632/invalid@example.com (bridge)bridgeAshley writes out the first 2017 positive integers. She then underlines any of the 2017 integers that is a multiple of 2, and then underlines any of the 2017 integers that is a multiple of 3, and then underlines any of the 2017 integers that is a multiple of 5. Finally, Ashley finds the sum of all the integers which have not been underlined. What is this sum?

\( \textbf{(A)}\ 542708\\\textbf{(B)}\ 543213\\\textbf{(C)}\ 542203\\\textbf{(D)}\ 543326\\\textbf{(E)}\ 543618\)]]>0Different schedulesWed, 11 Jul 2018 08:25:09 +0000
http://www.amc8.net/threads/different-schedules.631/
http://www.amc8.net/threads/different-schedules.631/invalid@example.com (bridge)bridgeSix soccer teams are competing in a tournament in Waterloo. Every team is to play three games, each against a different team. (Note that not every pair of teams plays a game together.) Judene is in charge of pairing up the teams to create a schedule of games that will be played. Ignoring the order and times of the games, how many different schedules are possible?

\(\text{(A)} 90 \qquad\text{(B)} 100 \qquad\text{(C)} 80 \qquad\text{(D)} 60 \qquad\text{(E)} 70\)]]>0N to GTue, 10 Jul 2018 06:07:33 +0000
http://www.amc8.net/threads/n-to-g.630/
http://www.amc8.net/threads/n-to-g.630/invalid@example.com (Cherry)CherryNadia walks along a straight path that goes directly from her house (N) to her Grandmother's house (G). Some of this path is on flat ground, and some is downhill or uphill. Nadia walks on flat ground at 5 km/h, walks uphill at 4 km/h, and walks downhill at 6 km/h. It takes Nadia 1 hour and 36 minutes to walk from N to G and 1 hour and 39 minutes to walk from G to N. If 2.5 km of the path between N and G is on flat ground, the total distance from N to G is closest to

\( \textbf{(A)}\ 8.0...\)

N to G]]>02731Fri, 06 Jul 2018 08:34:33 +0000
http://www.amc8.net/threads/2731.629/
http://www.amc8.net/threads/2731.629/invalid@example.com (Cherry)CherryThe rst four rows of a table with columns V , W, X, Y , and Z are shown. For each row, whenever integer n appears in column V , column W contains the integer 2n + 1, column X contains 3n + 1, column Y contains 5n+1, and column Z contains 7n+1. For every row after the first, the number in column V is the smallest positive integer that does not yet appear in any previous row. The integer 2731 appears in column W. The complete list of columns in which 2731 appears is
...

2731]]>0Largest value of NThu, 05 Jul 2018 03:27:28 +0000
http://www.amc8.net/threads/largest-value-of-n.628/
http://www.amc8.net/threads/largest-value-of-n.628/invalid@example.com (Cherry)CherryIn Mrs. Warner's class, there are 30 students. Strangely, 15 of the students have a height of 1.60 m and 15 of the students have a height of 1.22 m. Mrs. Warner lines up n students so that the average height of any four consecutive students is greater than 1.50 m and the average height of any seven consecutive students is less than 1.50 m. What is the largest possible value of n?

Largest value of N]]>0463Wed, 04 Jul 2018 08:39:50 +0000
http://www.amc8.net/threads/463.627/
http://www.amc8.net/threads/463.627/invalid@example.com (Cherry)CherryIn a sequence of 10 terms, the rst term is 1, the second term is x, and each term after the second is the sum of the previous two terms. For example, if x = 11, the sequence would be 1, 11, 12, 23, 35, 58, 93, 151, 244, 295. For some values of x, the number 463 appears in the sequence. If x is a positive integer, what is the sum of all the values of x for which 463 appears in the sequence?

463]]>0SKY classTue, 03 Jul 2018 03:32:23 +0000
http://www.amc8.net/threads/sky-class.626/
http://www.amc8.net/threads/sky-class.626/invalid@example.com (Cherry)CherryAt the beginning of the winter, there were at least 66 students registered in a ski class. After the class started, eleven boys transferred into this class and thirteen girls transferred out. The ratio of boys to girls in the class was then 1 : 1. Which of the following is not a possible ratio of boys to girls before the transfers?

SKY class]]>0Greg, Charlize, and AzarahTue, 03 Jul 2018 03:30:24 +0000
http://www.amc8.net/threads/greg-charlize-and-azarah.625/
http://www.amc8.net/threads/greg-charlize-and-azarah.625/invalid@example.com (Cherry)CherryGreg, Charlize, and Azarah run at dierent but constant speeds. Each pair ran a race on a track that measured 100 m from start to nish. In the rst race, when Azarah crossed the nish line, Charlize was 20 m behind. In the second race, when Charlize crossed the nish line, Greg was 10 m behind. In the third race, when Azarah crossed the nish line, how many metres was Greg behind?

Greg, Charlize, and Azarah]]>0arithmetic sequenceTue, 03 Jul 2018 03:27:15 +0000
http://www.amc8.net/threads/arithmetic-sequence.624/
http://www.amc8.net/threads/arithmetic-sequence.624/invalid@example.com (Cherry)CherryAn arithmetic sequence is a sequence in which each term after the rst is obtained by adding a constant to the previous term. For example, 2,4,6,8 and 1,4,7,10 are arithmetic sequences.
In the grid shown, the numbers in each row must form an arithmetic sequence and the numbers in each column must form an arithmetic sequence. The value of x is

arithmetic sequence]]>0alternating sumMon, 02 Jul 2018 06:31:21 +0000
http://www.amc8.net/threads/alternating-sum.623/
http://www.amc8.net/threads/alternating-sum.623/invalid@example.com (Cherry)CherryThe alternating sum of the digits of 63 195 is 6 - 3 + 1 - 9 + 5 = 0. In general, the alternating sum of the digits of a positive integer is found by taking its leftmost digit, subtracting the next digit to the right, adding the next digit to the right, then subtracting, and so on. A positive integer is divisible by 11 exactly when the alternating sum of its digits is divisible by 11. For example, 63 195 is divisible by 11 since the alternating sum of its digits is equal to 0, and 0 is...

alternating sum]]>0(1056; 1007)Tue, 26 Jun 2018 08:23:28 +0000
http://www.amc8.net/threads/1056-1007.622/
http://www.amc8.net/threads/1056-1007.622/invalid@example.com (Cherry)CherryZeus starts at the origin (0; 0) and can make repeated moves of one unit either up, down, left or right, but cannot make a move in the same direction twice in a row. For
example, he cannot move from (0; 0) to (1; 0) to (2; 0). What is the smallest number of moves that he can make to get to the point (1056; 1007)?

\(\textbf{(A)} \text{ 2112} \\ \textbf{(B)} \text{ 2161} \\ \textbf{(C)} \text{ 2063} \\ \textbf{(D)} \text{ 2111} \\ \textbf{(E)} \text{ 2113}\)]]>0Possible valuesMon, 25 Jun 2018 09:09:32 +0000
http://www.amc8.net/threads/possible-values.621/
http://www.amc8.net/threads/possible-values.621/invalid@example.com (Cherry)CherryResidents were surveyed in order to determine which flowers to plant in the new Public Garden. A total of N people participated in the survey. Exactly \(\dfrac{9}{14}\) of those surveyed said that the colour of the flower was important. Exactly \(\dfrac{7}{12}\) of those surveyed said that the smell of the flower was important. In total, 753 people said that both the colour and smell were important. How many possible values are there for N?

\(

\textbf{(A) }22 \qquad...\)

Possible values]]>0Slackrope WalkerThu, 21 Jun 2018 09:50:35 +0000
http://www.amc8.net/threads/slackrope-walker.620/
http://www.amc8.net/threads/slackrope-walker.620/invalid@example.com (bridge)bridgeA “slackrope walker” is much like a tightrope walker except that the rope on which he performs is not pulled tight. Paul, a slackrope walker, has a rope tied to two 15 m high poles which are 14 m apart. When he is standing on the rope 5 m away from one of the poles, he is 3 m above the ground. How long is the rope?