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• 2008 iTest Problems
  ...hat are the product of two consecutive integers. Let <math>B</math> be the set of positive integers that are the product of three consecutive integers. Fi ...lean up the storage shed. After clearing away some trash, Joshua and Wendy set aside give boxes that belong to the two of them that they decide to take up
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2009 AIME I Problems/Problem 11
  Consider the set of all triangles <math>OPQ</math> where <math>O</math> is the origin and <m *There are 48 ordered pairs <math>(x_2,x_1)</math> such that their positive difference is 2.
  8 KB (1,319 words) - 15:01, 16 August 2020
• Ascending chain condition
  Let <math>S</math> be a [[partially ordered set]]. We say that <math>S</math> satisfies the '''ascending chain condition'' ...th>S</math> satisfies the '''descending chain condition''' ('''DCC'''). A set <math>S</math> with an ordering <math>\leqslant</math> satisfies ACC if and
  2 KB (314 words) - 18:00, 15 December 2018
• 1990 AJHSME Problems
  When three different numbers from the set <math>\{ -3, -2, -1, 4, 5 \} </math> are multiplied, the largest possible p ...s of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards?
  15 KB (2,059 words) - 15:03, 6 October 2021
• 2010 AMC 12A Problems/Problem 21
  ...g <math>2p+2q+2r = 10</math>. We just need to take the lowest value in the set, square root it, and subtract the resulting value from 5 to get <math>\boxe ...^2+r^2+4pq+4pr+4qr = 29</math> to obtain <cmath>p^2+q^2+r^2=21</cmath> The ordered triple {16,4,1} sums to 21, and the answer choices are all positive integer
  5 KB (969 words) - 21:33, 22 June 2022
• 2010 AMC 12B Problems
  ...s less than <math>20</math> are there exactly two distinct elements in the set <math>\{i^x, (1+i)^y, z\}</math>, where <math>i=\sqrt{-1}</math>? ...c</math> are randomly and independently selected with replacement from the set <math>\{1, 2, 3,\dots, 2010\}</math>. What is the probability that <math>ab
  12 KB (1,845 words) - 13:00, 19 February 2020
• 2010 AIME I Problems
  ...he number of minimally intersecting ordered triples of sets for which each set is a subset of <math>\{1,2,3,4,5,6,7\}</math>. Find the remainder when <mat '''Note''': <math>|S|</math> represents the number of elements in the set <math>S</math>.
  8 KB (1,243 words) - 21:58, 10 August 2020
• 2010 AIME I Problems/Problem 7
  ...he number of minimally intersecting ordered triples of sets for which each set is a subset of <math>\{1,2,3,4,5,6,7\}</math>. Find the remainder when <mat '''Note''': <math>|S|</math> represents the number of elements in the set <math>S</math>.
  2 KB (255 words) - 17:03, 9 August 2018
• Mock AIME 2 2010 Answer Key
  ...th>S=\{ 1,2,3,4,5\}</math> (where the product of the elements of the empty set is taken to be 1). If we pair each subset <math>P</math> with its complemen .../math> themselves, and let <math>C=\{1,2,3,4,5\} - (A\cup B)</math> be the set of the remaining values. Notice that all possible values of <math>f(f(x))</
  36 KB (6,214 words) - 20:22, 13 July 2023
• 2010 AIME II Problems/Problem 8
  Let <math>N</math> be the number of [[ordered pair]]s of nonempty sets <math>\mathcal{A}</math> and <math>\mathcal{B}</ma Let us [[partition]] the set <math>\{1,2,\cdots,12\}</math> into <math>n</math> numbers in <math>A</math
  4 KB (699 words) - 20:57, 20 July 2023
• 2010 AMC 12B Problems/Problem 15
  ...s less than <math>20</math> are there exactly two distinct elements in the set <math>\{i^x, (1+i)^y, z\}</math>, where <math>i=\sqrt{-1}</math>? ...ath>i^{x}=(1+i)^{y}\neq z</math>, we have <math>5\cdot 1\cdot 19=95</math> ordered triples.
  2 KB (360 words) - 17:29, 26 May 2023
• 1993 AHSME Problems
  How many ordered pairs <math>(m,n)</math> of positive integers are solutions to Let <math>S</math> be the set of points on the rays forming the sides of a <math>120^{\circ}</math> angle
  20 KB (2,814 words) - 08:15, 27 June 2021
• 2021 AIME II Problems/Problem 6
  .../math> denote the number of elements in <math>S</math>. Find the number of ordered pairs <math>(A,B)</math> such that <math>A</math> and <math>B</math> are (n ...<math>X</math>, <math>Y</math>, <math>Z</math>, <math>W</math> is an empty set.
  8 KB (1,364 words) - 01:02, 29 January 2024
• 2011 AIME II Problems/Problem 14
  ...e <math>(i, j, k)</math>. The <math>n</math>th position is defined by this ordered triple where <math>i</math> is <math>n \mod 2</math>, <math>j</math> is <ma The ordered triple (or position) in which 1 can be placed has 2 options for i, 3 for j,
  10 KB (1,581 words) - 16:55, 27 May 2024
• 2011 AIME II Problems/Problem 13
  ...degrees is rather friendly in terms of ordered-pair representation! We can set <math>A=(0, 12)</math>, <math>B=(12,12)</math>, <math>C=(12, 0)</math>, <ma
  13 KB (2,055 words) - 05:25, 9 September 2022
• List
  ...ts''' in [[Python]] are used to store multiple objects in a single ordered set. Unlike [[tuple]]s, lists are mutable, so entries can be added, removed, o ...ents of a list can be changed with a simple reassignment. For example, to set the i'th element of myList to x, you would use
  3 KB (470 words) - 12:27, 9 September 2021
• Sequence (Python)
  In [[Python]], '''sequence''' is the generic term for an ordered set. There are several types of sequences in Python, the following three are t
  3 KB (443 words) - 16:53, 19 February 2024
• Tuple
  ...ples''' in [[Python]] are used to store multiple items in a single ordered set. However, unlike [[list]]s, tuples are ''immutable'' - they can't be chang
  1 KB (196 words) - 17:18, 19 April 2011
• 2011 USAMO Problems/Problem 6
  Let <math>A</math> be a set with <math>|A| = 225</math>, meaning that <math>A</math> has 225 elements. ...9 elements in common. Since <math>\textstyle \binom{10}2 = 45,</math> each set is in 45 triples and thus will have 45 elements. We can now throw in 60 mor
  7 KB (1,209 words) - 12:50, 25 August 2023
• 1997 AJHSME Problems/Problem 14
  .... What is the difference between the largest and smallest integers in the set? When these numbers are ordered in ascending order, 5, the median, falls right in the middle, which is the
  949 bytes (150 words) - 22:29, 17 December 2020

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