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• 2010 AMC 8 Problems/Problem 25
  ...climb one stair is <math>1</math>. There are <math>2</math> ways to climb two stairs: <math>1</math>,<math>1</math> or <math>2</math>. For 3 stairs, ther ...e from #2, or used a single step from #3. The ways to get to each of these steps are <math>1+2+4=7</math> ways to get to step 4. The pattern can then be ext
  2 KB (243 words) - 11:12, 15 July 2024
• 2013 AMC 10A Problems/Problem 18
  ...DE</math> is <math>\frac{15}{8}</math>, and from there split the base into two parts, <math>x</math>, and <math>4-x</math>, such that <math>x</math> is th
  4 KB (592 words) - 22:19, 2 November 2023
• 2013 AMC 12B Problems
  ...t is <math>15</math> steps by <math>20</math> steps. Each of Mr. Green’s steps is <math>2</math> feet long. Mr. Green expects a half a pound of potatoes p ...kens and <math>75</math> blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and anoth
  15 KB (2,458 words) - 23:05, 4 July 2024
• 2013 AMC 12B Problems/Problem 2
  ...it is <math>15</math> steps by <math>20</math> steps. Each of Mr. Green's steps is <math>2</math> feet long. Mr. Green expects a half a pound of potatoes p
  1 KB (160 words) - 17:02, 27 January 2015
• 2013 AMC 12B Problems/Problem 11
  Two bees start at the same spot and fly at the same rate in the following direc Let A and B begin at <math>(0,0,0)</math>. In <math>6</math> steps, <math>A</math> will have done his route twice, ending up at <math>(2,2,2)<
  2 KB (287 words) - 15:57, 10 January 2021
• 2013 AMC 10B Problems
  ...it is <math>15</math> steps by <math>20</math> steps. Each of Mr. Green's steps is <math>2</math> feet long. Mr. Green expects a half a pound of potatoes ...ir three-point shots, which resulted in 54 points. They attempted 50% more two-point shots than three-point shots. How many three-point shots did they att
  12 KB (1,926 words) - 21:54, 6 October 2022
• 2013 USAJMO Problems/Problem 4
  ...- 2^i)</math> and <math>f(b- 2^{a-1})</math> are odd; since there are just two odd numbers, the ending sum for any <math>b</math> is even. Finally, consi ...ath>2^n</math> has <math>0</math> for <math>f</math> because there will be two numbers in DS that give a <math>f</math> of one: <math>2^{n-1}</math> and <
  6 KB (1,069 words) - 14:05, 28 November 2022
• Symmetry
  ...h>\angle C = \angle B;</math> the other cases hold by symmetry because the steps are the same. It is known that the composition of two axial symmetries with non-parallel axes is a rotation centered at
  19 KB (3,292 words) - 13:04, 13 May 2024
• 1952 AHSME Problems/Problem 49
  ...> and <math>\triangle ABC</math>, we can find the scale factor between the two triangle by simply dividing <math>N_{2}O</math>(not nitrous oxide) by <math ...ves you the ratio of <math>3:3:1</math> so I did not need to use that many steps to calculate the coordinate of <math>N_2</math>, directly attaining it to b
  7 KB (1,136 words) - 10:01, 23 December 2023
• 1965 AHSME Problems
  ...le ABC</math> in such a way that one of its vertices is <math>A</math> and two of its sides lie along <math>AB</math> and <math>AC</math>. \textbf{(D) }\ \text{more than two, but finite}\qquad
  19 KB (2,897 words) - 10:43, 20 July 2024
• 2014 UMO Problems
  are certain lists <math>(m, n)</math> of length two that he might end up writing an infinite number of times. Find the four legal directions indicated. A path is a series of steps, starting at <math>A</math> and ending at <math>B</math>, such
  6 KB (982 words) - 15:11, 22 October 2018
• 2014 UMO Problems/Problem 6
  the four legal directions indicated. A path is a series of steps, starting at <math>A</math> and ending at <math>B</math>, such MP("Legal Steps",(11.4,4.2));
  3 KB (628 words) - 19:22, 20 May 2021
• 2015 AMC 10A Problems/Problem 3
  ...<math>2[2x+(x-1)+(x-2)+...+1]</math> toothpicks. So, for <math>x=5</math> steps, we have <math>2[2(5)+4+3+2+1]=40</math> toothpicks. So our answer is <math ...obs with <math>3</math> picks each and <math>3</math> more blobs that have two picks each. We add <math>2\cdot3+3\cdot2=12</math> more for a total increas
  2 KB (393 words) - 03:20, 1 June 2024
• 2015 AMC 10B Problems
  Isaac has written down one integer two times and another integer three times. The sum of the five numbers is <math ...week but never on two consecutive days. On Monday she plays basketball and two days later golf. She swims and plays tennis, but she never plays tennis the
  14 KB (2,247 words) - 11:38, 26 March 2024
• 2015 AMC 12B Problems
  Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, ...s an isosceles triangle attached to each side. Each of these triangles has two sides of length 8. The isosceles triangles are folded to make a pyramid wit
  13 KB (2,064 words) - 13:39, 1 October 2022
• 2015 AMC 12B Problems/Problem 21
  ...he staircase. Let <math>s</math> denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of <math>s</math>? ...le Dash takes 12, a difference of 18. If <math>s=70</math>, Cozy takes 35 steps while Dash takes 14, a difference of 21. When <math>s</math> increases fro
  6 KB (1,007 words) - 13:54, 12 October 2022
• 2015 AMC 10B Problems/Problem 21
  ...e staircase. Let <math>s</math> denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of <math>s</math>? Let <math>n</math> be the number of steps. We have
  6 KB (983 words) - 01:51, 16 June 2022
• 2015 AIME I Problems/Problem 8
  ...hatever <math>n</math> is, it has to have 3 digits, because if it had only two, the maximum of <math>s(n)</math> is 18. ...now when to carry the digits. Note that if you don't understand any of the steps I take, just try adding any 3-digit number to 864 regularly (using the old-
  10 KB (1,797 words) - 14:06, 21 January 2024
• 2015 AIME I Problems/Problem 5
  ...>2</math> of the remaining <math>8</math> socks at random and on Wednesday two of the remaining <math>6</math> socks at random. The probability that Wedn ...casework. If the third sock's color matches the color of one of the first two socks (which occurs with probability <math>\dfrac{2}{6} = \dfrac{1}{3}</mat
  5 KB (760 words) - 23:33, 23 January 2024
• 2015 AIME I Problems/Problem 10
  Since <math>f(x)</math> is a third degree polynomial, it can have at most two bends in it where it goes from up to down, or from down to up. By drawing a coordinate axis, and two lines representing <math>12</math> and <math>-12</math>, it is easy to see
  8 KB (1,474 words) - 11:02, 15 August 2024

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