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• OpenCourseWare
  '''OpenCourseWare''' is a unique resource offered by [[MIT]]. There are course decriptions, sample syllabi,
  289 bytes (40 words) - 12:20, 3 July 2006
• Empty set
  ...erally denoted <math>\emptyset</math> or <math>\varnothing</math>) is the (unique) [[set]] containing no elements. It is therefore a [[subset]] of every set.
  489 bytes (84 words) - 21:33, 27 February 2020
• 2005 AIME I Problems
  ...> A(0,12), B(10,9), C(8,0), </math> and <math> D(-4,7). </math> There is a unique square <math> S </math> such that each of the four points is on a different
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 14
  ...h> A(0,12), B(10,9), C(8,0),</math> and <math> D(-4,7). </math> There is a unique [[square]] <math> S </math> such that each of the four points is on a diffe
  3 KB (561 words) - 14:11, 18 February 2018
• 2004 AIME II Problems/Problem 4
  ...digit, there are <math>9</math> possibilities. When <math>n</math> has two unique digits there are two cases. Case 1: two digits are the same with each other
  3 KB (508 words) - 01:16, 19 January 2024
• 1983 AIME Problems
  For <math>\{1, 2, 3, \ldots, n\}</math> and each of its nonempty subsets a unique '''alternating sum''' is defined as follows. Arrange the numbers in the sub
  7 KB (1,104 words) - 03:13, 27 May 2024
• 1987 AIME Problems
  What is the largest positive integer <math>n</math> for which there is a unique integer <math>k</math> such that <math>\frac{8}{15} < \frac{n}{n + k} < \fr
  6 KB (869 words) - 15:34, 22 August 2023
• 1988 AIME Problems
  ..., <math>w_4 = 1 + 27i</math>, and <math>w_5 = -14 + 43i</math>, there is a unique mean line with y-intercept <math>3</math>. Find the slope of this mean line
  6 KB (902 words) - 08:57, 19 June 2021
• 1989 AIME Problems
  ...h between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let <math>D^{}_{}</math> be the difference betw is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_
  7 KB (1,045 words) - 20:47, 14 December 2023
• 1991 AIME Problems
  ...\ldots,a_n^{}</math> are positive real numbers whose sum is 17. There is a unique positive integer <math>n^{}_{}</math> for which <math>S_n^{}</math> is also
  7 KB (1,106 words) - 22:05, 7 June 2021
• 1997 AIME Problems
  ...f(f(x))=x</math> for all values except <math>\frac{-d}{c}</math>. Find the unique number that is not in the range of <math>f</math>.
  7 KB (1,098 words) - 17:08, 25 June 2020
• 2000 AIME II Problems
  Every positive integer <math>k</math> has a unique factorial base expansion <math>(f_1,f_2,f_3,\ldots,f_m)</math>, meaning tha
  6 KB (947 words) - 21:11, 19 February 2019
• 1983 AIME Problems/Problem 13
  ...or <math>\{1, 2, 3, \ldots, n\}</math> and each of its non-empty subsets a unique '''alternating sum''' is defined as follows. Arrange the numbers in the sub
  5 KB (894 words) - 22:02, 5 April 2024
• 1983 AIME Problems/Problem 15
  ...assume that the answer is unique, but would need to prove that this is the unique solution. This can be proven as follows.
  20 KB (3,497 words) - 15:37, 27 May 2024
• 1984 AIME Problems/Problem 11
  ...remove <math>4</math> trees that aren't birch. What you are left with is a unique arrangement of <math>5</math> birch trees and <math>3</math> other trees th ...ve <math>4</math> trees. Adding a tree between each pair of people gives a unique arrangement of <math>5</math> nonadjacent birch trees.
  7 KB (1,115 words) - 00:52, 7 September 2023
• 1987 AIME Problems/Problem 8
  What is the largest positive integer <math>n</math> for which there is a unique integer <math>k</math> such that <math>\frac{8}{15} < \frac{n}{n + k} < \fr ...\frac{7n}{8}</math>. Thus, <math>48n < 56k < 49n</math>. <math>k</math> is unique if it is within a maximum [[range]] of <math>112</math>, so <math>n = 112</
  2 KB (393 words) - 16:59, 16 December 2020
• 1988 AIME Problems/Problem 11
  ...<math>w_4 = 1 + 27i</math>, and <math>w_5 = - 14 + 43i</math>, there is a unique mean line with <math>y</math>-intercept 3. Find the [[slope]] of this mean
  2 KB (422 words) - 00:22, 6 September 2020
• 1989 AIME Problems/Problem 14
  is true for a unique choice of non-negative integer <math>m</math> and digits <math>a_0,a_1,\ldo
  2 KB (408 words) - 17:28, 16 September 2023
• 1989 AIME Problems/Problem 11
  ...h between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique [[mode]] (most frequent value). Let <math>D</math> be the difference betwee
  5 KB (851 words) - 18:01, 28 December 2022
• 1991 AIME Problems/Problem 15
  ...\ldots,a_n^{}</math> are positive real numbers whose sum is 17. There is a unique positive integer <math>n^{}_{}</math> for which <math>S_n^{}</math> is also
  4 KB (658 words) - 16:58, 10 November 2023

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