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• 2002 AIME II Problems/Problem 9
  Let <math>\mathcal{S}</math> be the [[set]] <math>\lbrace1,2,3,\ldots,10\rbrace</math> Let <math>n</math> be the numb Thus, there are <math>3^{10}-2\cdot2^{10}+1</math> ordered pairs of sets <math>(A,B)</math>. But since the question asks for the numbe
  3 KB (404 words) - 23:07, 4 May 2024
• Ring
  ...ra]], similar to a [[group]] or a [[field]]. A ring <math>R</math> is a [[set]] of elements closed under two [[operation]]s, usually called multiplicatio ...es lead to confusion when <math>R</math> is also an [[ordered set]].) The set of invertible elements of <math>R</math> constitute a group under multiplic
  6 KB (994 words) - 06:16, 8 April 2015
• 2005 Canadian MO Problems
  Let <math>S</math> be a set of <math>n\ge 3</math> points in the interior of a circle. Let's say that an ordered triple of positive integers <math>(a,b,c)</math> is <math>n</math>-''powerf
  2 KB (436 words) - 11:45, 26 December 2018
• System of equations
  A '''system of equations''' is a set of [[equation]]s which share the same [[variable]]s. Below is an example o Find the ordered pair <math>(x,y)</math> for which
  5 KB (784 words) - 23:27, 30 July 2020
• 2007 iTest Problems
  A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahe
  30 KB (4,794 words) - 23:00, 8 May 2024
• Least upper bound
  Given a [[subset]] <math>S</math> in some larger [[partially ordered set]] <math>R</math>, a '''least upper bound''' or '''supremum''', for <math>S< ...th>S</math> is said to be '''complete''' if any [[empty set | nonempty]] [[set|subset]] of <math>S</math> that is [[bounded]] above has a supremum.
  1,011 bytes (177 words) - 14:08, 5 March 2022
• 2004 AMC 10A Problems
  ...t of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the ...y collection of condiments. How many different kinds of hamburgers can be ordered?
  15 KB (2,092 words) - 20:32, 15 April 2024
• Ordered pair
  ...in which we have <math>\{2, 3\} = \{3, 2\}</math>. In general, we say two ordered pairs, <math>(x, y)</math> and <math>(a, b)</math> are the same if and only ...notion of an ordered pair can be naturally extended to that of an [[tuple|ordered tuple]].
  1 KB (179 words) - 20:40, 28 February 2020
• 2020 AMC 10A Problems
  ...ed quadruple of not necessarily distinct integers, each one of them in the set <math>\{0,1,2,3\}.</math> For how many such quadruples is it true that <mat
  13 KB (1,968 words) - 18:32, 29 February 2024
• Greatest lower bound
  Given a [[subset]] <math>S</math> in some larger [[partially ordered set]] <math>R</math>, a '''greatest lower bound''' or '''infimum''' for <math>S
  597 bytes (109 words) - 13:55, 5 March 2022
• 2002 USAMO Problems/Problem 3
  '''Lemma.''' For any set of ordered pairs of reals <math> \{ (x_i , y_i ) \}_{i=1}^{n} </math> with <math>x_i \
  4 KB (688 words) - 13:38, 4 July 2013
• 2007 AIME II Problems
  ...among the four letters in AIME or the four digits in <math>2007</math>. A set of plates in which each possible sequence appears exactly once contains N l Find the number of ordered triples <math>(a,b,c)</math> where <math>a</math>, <math>b</math>, and <mat
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2005 USAMO Problems
  ...h <math>n</math>, where <math>n</math> is a positive integer. For how many ordered 4-tuples <math>\left(k_1,k_2,k_3,k_4\right)</math> of nonnegative integers ...<math>f(n)</math> be the minimal <math>k</math> for which there exists a set <math>S</math> of <math>n</math> positive integers such that <math>s\left(
  4 KB (609 words) - 09:24, 14 May 2021
• 2002 USA TST Problems
  .../math> elements. Let <math> \displaystyle f </math> be a function from the set of two-element subsets of <math> \displaystyle S </math> to <math>\{0, \dot Find in explicit form all ordered pairs of positive integers <math> \displaystyle (m, n)</math> such that <ma
  3 KB (544 words) - 06:58, 3 August 2017
• Chicken McNugget Theorem
  Note that all purchasable integers are nonnegative, thus the set of non-purchasable integers is nonempty. ...ma</b>. Let <math>A_{N} \subset \mathbb{Z} \times \mathbb{Z}</math> be the set of solutions <math>(x,y)</math> to <math>xm+yn = N</math>. Then <math>A_{N}
  17 KB (2,748 words) - 19:22, 24 February 2024
• 2005 AMC 12A Problems/Problem 15
  ...e the shoelace formula. Our origin is the center of the circle. Denote the ordered pair for <math>D (x,y)</math>, and notice how <math>E</math> is a 180 degre We set point <math>A</math> as a mass of 2. This means that point <math>B</math> h
  14 KB (1,970 words) - 17:02, 18 August 2023
• 2006 AIME II Problems/Problem 7
  Find the number of [[ordered pair]]s of [[positive]] [[integer]]s <math> (a,b) </math> such that <math> ...th>. Therefore, there are <math>999 - (99 + 162) = \boxed{738}</math> such ordered pairs.
  7 KB (1,114 words) - 03:41, 12 September 2021
• Lexington Math Tournament
  ...will all involve some scenario relating to the overarching theme, and are ordered in approximately increasing difficulty. The themes rotate every year. ...m station and pick up the next set. However, once a team submits a problem set, they may not go back to it. Grading is immediate and scores are posted in
  3 KB (484 words) - 20:04, 12 March 2024
• Zorn's Lemma
  '''Zorn's Lemma''' is a [[set theory | set theoretic]] result which is equivalent to the [[Axiom of Choice]]. Let <math>A</math> be a [[partially ordered set]].
  9 KB (1,669 words) - 19:02, 1 August 2018
• 2008 AMC 12B Problems/Problem 23
  ...r <math>d</math> be the number of divisors <math>10^n</math> has. Then, we set up <math>\frac{d}{2}</math> pairs of divisors such that each pair <math>(a, ...an odd number of factors, as for perfect square <math>p^2</math>, we have ordered pair <math>(p,p)</math> that works. For even <math>n</math>, then, <math>10
  5 KB (814 words) - 18:02, 17 January 2023

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