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• 2007 AMC 12A Problems/Problem 19
  ...th>), we can use an easy application of the [[Shoelace Theorem]] to figure out the values of X. Since we already know the two distances (positive y and ne
  4 KB (565 words) - 17:01, 2 April 2023
• 2007 AMC 12A Problems/Problem 25
  Call a set of integers ''spacy'' if it contains no more than one out of any three consecutive integers. How many [[subset]]s of <math>\{1,2,3,\l Since each of the elements of the subsets must be spaced at least two apart, a divider counting argument can be used.
  9 KB (1,461 words) - 23:07, 27 January 2024
• KGS math club
  |A, B and C have a cycle race from E to F. All three set out at 10am at different speeds from E. A is 5km/h faster than B and 10km/h fas |31 points are equally spaced in a circle. In how many distinct ways can you pick 6 of them such that eve
  22 KB (3,358 words) - 15:17, 18 July 2017
• 2008 iTest Problems
  ...is, he measures the distance between the centers of the trees, which comes out to be around <math>15</math> feet. He then measures <math>2</math> feet to ...days after the first day Tony places the spider on the wall) will Tony run out of room to place the spider three feet higher?
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2005 AMC 8 Problems
  ...r blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. How many socks must the Martian remove ...lay a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on poin
  13 KB (1,821 words) - 22:18, 5 December 2023
• 2010 AMC 12B Problems/Problem 21
  ...,3,5,7</math>, we can factor the product <math>(x-1)(x-3)(x-5)(x-7)</math> out of <math>R(x)</math> to obtain a new polynomial <math>Q(x)</math> such that The evenly-spaced data suggests using [[discrete derivative|discrete derivatives]] to tackle
  7 KB (1,166 words) - 00:04, 26 February 2024
• KGS math club/solution 11 5
  ...this final meeting at the end of the day). These meetings will be equally spaced over the day (by symmetry -- for example by running time backwards). These ...se <math>S=s_1s_2\cdots s_n</math> at each stage instead of having to work out the lowest common multiple at every stage, the drawback is that the numbers
  4 KB (712 words) - 10:51, 11 May 2011
• 1996 AHSME Problems
  A father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges <math>4.95</math> evenly spaced around a circle. All quadruples are equally likely to be chosen.
  15 KB (2,343 words) - 13:39, 19 February 2020
• 1989 AHSME Problems
  ...a five-minute interval. Assume vehicles in the westbound lane are equally spaced. Which of the following is closest to the number of westbound vehicles pres //commented out labeling because it doesn't look right.
  15 KB (2,247 words) - 13:44, 19 February 2020
• 2012 AMC 10A Problems/Problem 25
  ...repetition, let us say that <math>(x_{1},\dotsc,x_{k})</math> is <i>spaced-out</i> if <math>|x_{i} - x_{j}| > 1</math> for all <math>i \ne j</math>. ...th> has volume <math>n^{k}</math>. Let <math>S</math> be the set of spaced-out <math>k</math>-tuples <math>(x_{1},\dotsc,x_{k})</math>. The desired probab
  13 KB (2,133 words) - 01:22, 6 February 2024
• 2013 USAMO Problems/Problem 2
  For a positive integer <math>n\geq 3</math> plot <math>n</math> equally spaced points around a circle. Label one of them <math>A</math>, and place a mark ...nts of each tuple refer to the first and second pass respectively. Listing out the state transitions (while following the rule that we can't use the same
  14 KB (2,076 words) - 20:29, 10 July 2023
• 1963 AHSME Problems
  Thereafter, <math>7</math> out of every <math>8</math> counted were red. If, in all, <math>90</math> % or ...points <math>R</math> and <math>S</math>, <math>76</math> miles apart, set out at the same time to walk towards each other.
  20 KB (3,122 words) - 14:17, 20 February 2020
• 1998 CEMC Gauss (Grade 7) Problems
  Ten points are spaced equally around a circle. How many different chords can be formed by joining ...lengths, in cm, of 1, 1, 2, 2, 3, 3, 4, 4, ... as shown. Dana’s pen runs out of ink after the total of all the lengths he has drawn is 3000 cm. What is
  10 KB (1,590 words) - 16:43, 29 January 2021
• 2015 AMC 12B Problems
  The Tigers beat the Sharks 2 out of the 3 times they played. They then played <math>N</math> more times, and Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person ge
  13 KB (2,064 words) - 13:39, 1 October 2022
• 2015 AMC 12B Problems/Problem 22
  Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person ge Case 3: The movement of seats cancel out each other.
  9 KB (1,623 words) - 18:48, 30 October 2022
• 2016 AMC 10B Problems
  ...>20</math> fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly <math>4</mat ...h>ZW</math>, <math>YX</math>, and <math>ED</math> are parallel and equally spaced. What is the ratio of the area of hexagon <math>WCXYFZ</math> to the area o
  12 KB (1,868 words) - 17:50, 30 December 2023
• 2016 AMC 10B Problems/Problem 11
  ...>20</math> fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly <math>4</mat
  2 KB (383 words) - 22:34, 15 April 2022
• 2016 AMC 12B Problems/Problem 9
  ...>20</math> fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly <math>4</mat
  2 KB (288 words) - 14:30, 19 January 2021
• 2016 AMC 12B Problems
  ...er <math>(4)</math>, marks out <math>6</math>, skips <math>8</math>, marks out <math>10</math>, and so on to the end. Josh continues in this manner until ...>20</math> fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly <math>4</mat
  14 KB (2,037 words) - 19:09, 29 July 2023
• 2017 AMC 12B Problems/Problem 12
  ...oots lying in the first quadrant, because the imaginary parts would cancel out with those from the fourth quadrant. We have two <math>30-60-90</math> tria
  3 KB (449 words) - 01:54, 11 February 2019
• 2008 iTest Problems/Problem 2
  ...is, he measures the distance between the centers of the trees, which comes out to be around <math>15</math> feet. He then measures <math>2</math> feet to
  2 KB (366 words) - 00:15, 22 June 2018
• 2019 AMC 8 Problems
  There are <math>81</math> grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edge ...>0</math> points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is th
  17 KB (2,585 words) - 15:27, 4 May 2024
• 2021 AMC 10B Problems
  Three equally spaced parallel lines intersect a circle, creating three chords of lengths <math>3 ...yrone, and Kim play a card game. Each person is given <math>2</math> cards out of a set of <math>10</math> cards numbered <math>1,2,3, \dots,10.</math> Th
  17 KB (2,418 words) - 12:52, 5 November 2023
• 2021 Fall AMC 10A Problems
  ...me distance in <math>12</math> equal leaps. The telephone poles are evenly spaced, and the <math>41</math>st pole along this road is exactly one mile (<math> ...</math> rolls all the way around the outside of the same square and sweeps out a region of area <math>2A</math>. The value of <math>s</math> can be writte
  14 KB (2,162 words) - 21:33, 2 November 2023
• 2022 AMC 8 Problems
  Three positive integers are equally spaced on a number line. The middle number is <math>15,</math> and the largest num ...0</math> attempts in the first half of a game, and <math>10</math> baskets out of <math>10</math> attempts in the second half. Candace took <math>12</math
  20 KB (2,805 words) - 21:26, 22 February 2024
• 2023 AIME I Problems/Problem 1
  Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diam ...t the person opposite to him is a woman is <math>\frac{9}{13}</math> since out of the <math>13</math> people who can sit opposite to him, <math>9</math> c
  11 KB (1,803 words) - 17:17, 12 January 2024
• 2024 AIME I Problems/Problem 3
  ...r otherwise. In other words, there will be <math>2</math> losing positions out of every <math>5</math> consecutive values of n. As <math>n</math> ranges f ...trial and error we see that <math>2, 5, 7,</math> and <math>10</math> are spaced alternating in between 2 and 3 apart. This can also be proven with modular
  10 KB (1,628 words) - 18:50, 7 February 2024