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• 2015 AIME I Problems
  ...the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is <math>\frac{m}{n}</math>, wher ...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
  10 KB (1,615 words) - 21:48, 13 January 2024
• 2012 AIME I Problems
  ...= z,</math> and the imaginary part of <math>z</math> is <math>\sin{\frac{m\pi}{n}}</math>, for relatively prime positive integers <math>m</math> and <mat ...he plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form <math>\tfrac{p}{q},</math> where <math>p<
  10 KB (1,617 words) - 14:49, 2 June 2023
• 2012 AIME II Problems/Problem 13
  ...ath> and <math>\sqrt{11}</math> from point <math>B</math>, so they are the two points where a circle centered at <math>A</math> with radius <math>\sqrt{11 There are two equilateral triangles with <math>\overline{AD_1}</math> as a side; let <mat
  13 KB (2,052 words) - 18:02, 5 February 2024
• 2017 AMC 8 Problems
  ...ves but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are tru ...with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?
  12 KB (1,771 words) - 09:41, 13 June 2024
• Mock AIME I 2012 Problems
  ...math>, <math>\cdots</math>, <math>9\}</math> is called ''golden'' if every two consecutive numbers in the permutation are relatively prime. How many ''gol ...ing (possibly the same string) chosen at random. He continues on by tying two randomly selected untied ends together until there are no more untied ends,
  7 KB (1,309 words) - 11:13, 8 April 2012
• 2015 IMO Problems/Problem 1
  if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math> regular polygons <math>P_n</math> with <math>n</math> sides: given any two vertices <math>A</math> and
  4 KB (773 words) - 08:14, 19 July 2016
• 1988 USAMO Problems/Problem 5
  ...{a_n}</math> are all <math>n</math>th roots of unity. If <math>\omega=e^{2\pi i/n}</math>, then the sum of <math>32</math>nd powers of these roots will b ...mials of degree <math>n</math> over the field <math>\mathbb{F}_2</math> of two elements, and on the left, we have the factorisation of this generating fun
  8 KB (1,348 words) - 09:44, 25 June 2022
• 2006 SMT/General Problems
  A geometric series is one where the ratio between each two consecutive terms is constant (ex. <math> 3, 6, 12, 24,\cdots </math>). The Given two <math> 2 </math>'s, "plus" can be changed to "times" without changing the r
  10 KB (1,477 words) - 16:02, 27 May 2012
• 2006 SMT/Team Problems
  ...e <math> M </math> minus the volume fo cone <math> N </math> is <math> 140\pi </math>, find the length of <math> \overline{BC} </math>. ...ath> S_n=\sum_{i=1}^{n-1}i\cdot S_{n-i} </math>. For example, if the first two elements are <math> 2 </math> and <math> 3 </math>, respectively, the third
  6 KB (910 words) - 17:32, 27 May 2012
• 2006 SMT/Advanced Topics Problems/Problem 1
  ...s the square root of himself, or the square root of <math> i </math>. What two answers should he give? ...math> i=\cos\left(\frac{\pi}{2}\right)+i\sin\left(\frac{\pi}{2}\right)=e^{\pi i/2} </math>.
  1 KB (170 words) - 20:12, 27 May 2012
• 1952 AHSME Problems
  Two high school classes took the same test. One class of <math> 20 </math> stud Two equal circles in the same plane cannot have the following number of common
  23 KB (3,556 words) - 15:35, 30 December 2023
• 1953 AHSME Problems
  \textbf{(B)}\ \frac{440}{\pi} \qquad \textbf{(C)}\ \frac{880}{\pi} \qquad
  21 KB (3,123 words) - 14:24, 20 February 2020
• 1955 AHSME Problems
  The ratio of the areas of two concentric circles is <math>1: 3</math>. If the radius of the smaller is <m Two numbers whose sum is <math>6</math> and the absolute value of whose differe
  22 KB (3,509 words) - 21:29, 31 December 2023
• 1998 USAMO Problems/Problem 6
  ..., we can easily alternate them, but if <math>n</math> is odd, we must have two adjacent sides be the same length, and that length must be the larger side
  5 KB (871 words) - 18:59, 10 May 2023
• 2012 AMC 10B Problems/Problem 16
  ...xtbf{(C)}\ 12\pi+\sqrt{3}\qquad\textbf{(D)}\ 10\pi+9\qquad\textbf{(E)}\ 13\pi</math> ...<math>\dfrac{1}{2}</math>, we get <math>2\sqrt{3}</math>. Since there are two 30-60-90 triangles in the equilateral triangle, we multiply the area of the
  4 KB (675 words) - 03:58, 23 January 2023
• 1986 USAMO Problems
  Two distinct circles <math>K_1</math> and <math>K_2</math> are drawn in the pla ...lar to a line through a point on or off the line), find a construction for two points <math>C</math> and <math>D</math> on <math>K_1</math> such that <mat
  3 KB (500 words) - 19:00, 23 October 2019
• 2008 AMC 8 Problems/Problem 21
  ...(6) = 96\pi</math>. The volume of the wedge is half this which is <math>48\pi \approx \boxed{\textbf{(C)}\ 151}</math>.
  1 KB (198 words) - 20:54, 29 October 2023
• 2008 AMC 8 Problems
  In <math>2005</math> Tycoon Tammy invested <math>100</math> dollars for two years. During the the first year investment showed a <math>20\%</math> gain. Over the two-year period, what was the change
  14 KB (2,035 words) - 15:23, 26 January 2024
• 2012 AMC 8 Problems
  A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inche ...ass. Her goal is to achieve an average grade of 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she r
  13 KB (1,835 words) - 08:51, 8 March 2024
• 2012 AMC 8 Problems/Problem 24
  ...D)}\hspace{.05in}\frac{\pi-1}{\pi}\qquad\textbf{(E)}\hspace{.05in}\frac{3}\pi </math> ...two areas, we find the answer is <math> \boxed{\textbf{(A)}\ \frac{4-\pi}{\pi}} </math>.
  2 KB (310 words) - 16:18, 10 November 2023

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