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• Principle of Inclusion-Exclusion
  ...element]]s in the [[union]] of a given group of [[set]]s, the size of each set, and the size of all possible [[intersection]]s among the sets. === Two Set Example ===
  9 KB (1,703 words) - 07:25, 24 March 2024
• Rearrangement Inequality
  ...d <math>B=\{b_1,b_2,\cdots,b_n\}</math> is a permutation of another finite set of real numbers, the quantity <math>a_1b_1+a_2b_2+\cdots+a_nb_n</math> is m Now for the general case. Again, without loss of generality, set <math>a_1 \leq a_2 \leq \cdots \leq a_n</math> and <math>b_1 \leq b_2 \leq
  5 KB (804 words) - 13:54, 26 January 2023
• Real number
  The set of real numbers, denoted by <math>\mathbb{R}</math>, is a subset of [[compl ==The set <math>\mathbb{R}~</math>==
  3 KB (496 words) - 23:22, 5 January 2022
• Constructive counting
  ...total possibilities of each step and assembles these to enumerate the full set. ...this problem, there are sometimes multiple independent ways to construct a set. In others, however, an alternative method is not apparent, as with the nex
  12 KB (1,896 words) - 23:55, 27 December 2023
• Function
  ...t of values to another set of values, assigning to each value in the first set exactly one value in the second. For instance, one function may map 1 to 1 Let <math>A</math>,<math>B</math> be [[set]]s and let <math>f</math> be a [[subset]] of <math>A\times B</math>, which
  10 KB (1,761 words) - 03:16, 12 May 2023
• Partition of an interval
  A '''partition''' <math>\mathcal{P}</math> is defined as the ordered <math>n</math>-[[tuple]] of real numbers <math>\mathcal{P}=(x_0,x_1,\ldots, ...agged partition''' <math>\mathcal{\dot{P}}</math> is defined as the set of ordered pairs <math>\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n</math>.
  1 KB (178 words) - 20:34, 6 March 2022
• Mathematical Kangaroo
  Past sets may be ordered from the US Math Kangaroo website. The Canadian and International (primari ...o does not have precalculus concepts (logs, complex numbers, trigonometry, set notation, or summation/product notation) whereas AMC 12 does.
  6 KB (936 words) - 15:38, 22 February 2024
• Cardinality
  ...1, \{2, 3\}, \{1, 2, 3\}\}</math> is 3, and the cardinality of the [[empty set]] is 0. The cardinality of a set <math>A</math> is denoted by <math>|A|</math>. In the above example, the c
  2 KB (263 words) - 00:54, 17 November 2019
• Polygon
  In their most general form, polygons are an ordered [[set]] of [[vertex|vertices]], <math>\{A_1, A_2, \ldots, A_n\}</math>, <math>n \
  2 KB (372 words) - 19:04, 30 May 2015
• Vector
  ...thin angle brackets or parentheses, <math>(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]].
  7 KB (1,265 words) - 13:22, 14 July 2021
• 2005 USAMO Problems/Problem 4
  ...<math>n</math>, where <math>n</math> is a positive integer. For how many ordered 4-tuples <math>(k_1, k_2, k_3, k_4)</math> of nonnegative integers can we ...math>C_i</math>, subtract <math>1</math> from each of the cuts to obtain a set of cuts that is counted in <math>C_{i-1}</math>. For example, if <math>\{2
  7 KB (1,276 words) - 20:51, 6 January 2024
• Well-Ordering theorem
  ...red set is a [[totally ordered set]] <math>(S,\prec)</math> for which each set <math>A\subseteq S</math> has a [[minimum]] element. [[Category:Set theory]]
  381 bytes (59 words) - 12:40, 2 June 2019
• Permutation
  ...re <math>r!</math> (the [[factorial]] of <math>r</math>) permutations of a set with <math>r</math> distinct objects. ...sider permutations of [[infinite]] sets. In this case, a permutation of a set <math>S</math> is simply a [[bijection]] between <math>S</math> and itself.
  3 KB (422 words) - 11:01, 25 December 2020
• 2006 AIME I Problems
  Let set <math> \mathcal{A} </math> be a 90-element subset of <math> \{1,2,3,\ldots, Let <math> \mathcal{S} </math> be the set of real numbers that can be represented as repeating decimals of the form <
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2005 AMC 12A Problems
  How many ordered triples of [[integer]]s <math>(a,b,c)</math>, with <math>a \ge 2</math>, <m ...nct numbers <math>a</math> and <math>b</math> are chosen randomly from the set <math>\{ 2, 2^2, 2^3, \ldots, 2^{25} \}</math>. What is the probability tha
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2004 AMC 12A Problems
  For how many ordered pairs of positive integers <math>(x,y)</math> is <math>x+2y=100</math>? Let <math>S</math> be the set of points <math>(a,b)</math> in the coordinate plane, where each of <math>a
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2002 AMC 12A Problems
  ...th>\{1, 2, 3, 4, 5\}</math>, and Sergio randomly selects a number from the set <math>\{1, 2, \ldots, 10\}</math>. What is the probability that Sergio's nu ...nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2005 AMC 12B Problems
  ...<math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set Let <math>S</math> be the set of ordered triples <math>(x,y,z)</math> of real numbers for which
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12A Problems/Problem 25
  How many non-[[empty set | empty]] [[subset]]s <math>S</math> of <math>\{1,2,3,\ldots ,15\}</math> h ...to choose <math>k</math> elements from an ordered <math>n</math> element [[set]] without choosing two consecutive members?
  8 KB (1,405 words) - 11:52, 27 September 2022
• 2005 AMC 12B Problems/Problem 23
  Let <math>S</math> be the set of ordered triples <math>(x,y,z)</math> of real numbers for which There are real numbers <math>a</math> and <math>b</math> such that for all ordered triples <math>(x,y.z)</math> in <math>S</math> we have <math>x^{3}+y^{3}=a
  5 KB (786 words) - 16:49, 31 January 2023
• Set
  ...satisfy. These axioms are chosen to agree with our intuitive concept of a set, on one hand, and to allow various, sometimes quite sophisticated, mathemat ...t are called the [[element]]s of the set. A common misconception is that a set can have multiple indistinct elements, such as the following: <math>\{1,4,5
  11 KB (2,021 words) - 00:00, 17 July 2011
• 2004 AIME I Problems/Problem 12
  Let <math> S </math> be the set of [[ordered pair]]s <math> (x, y) </math> such that <math> 0 < x \le 1, 0<y\le 1, </mat
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• 2004 AIME I Problems
  ...<math> m </math> consecutive integers whose sum is <math> 2m, </math> and set <math> B </math> consists of <math> 2m </math> consecutive integers whose s ...are on adjacent sides of the square. The midpoints of the line segments in set <math> S </math> enclose a region whose area to the nearest hundredth is <m
  9 KB (1,434 words) - 13:34, 29 December 2021
• 1988 AIME Problems
  The function <math>f</math>, defined on the set of ordered pairs of positive integers, satisfies the following properties:
  6 KB (902 words) - 08:57, 19 June 2021
• 1993 AIME Problems
  How many ordered four-tuples of integers <math>(a,b,c,d)\,</math> with <math>0 < a < b < c < .../math>, are then drawn randomly and without replacement from the remaining set of <math>997</math> numbers. Let <math>p</math> be the probability that, af
  8 KB (1,275 words) - 06:55, 2 September 2021
• 1994 AIME Problems
  For certain ordered pairs <math>(a,b)\,</math> of real numbers, the system of equations ...lution is an ordered pair <math>(x,y)\,</math> of integers. How many such ordered pairs <math>(a,b)\,</math> are there?
  7 KB (1,141 words) - 07:37, 7 September 2018
• 1998 AIME Problems
  Find the number of [[ordered pair]]s <math>(x,y)</math> of positive integers that satisfy <math>x \le 2y Let <math>n</math> be the number of ordered quadruples <math>(x_1,x_2,x_3,x_4)</math> of positive odd [[integer]]s that
  7 KB (1,084 words) - 02:01, 28 November 2023
• 1999 AIME Problems
  There is a set of 1000 switches, each of which has four positions, called <math>A, B, C</m Let <math>\mathcal{T}</math> be the set of ordered triples <math>(x,y,z)</math> of nonnegative real numbers that lie in the pl
  7 KB (1,094 words) - 13:39, 16 August 2020
• 2000 AIME I Problems
  For how many ordered pairs <math>(x,y)</math> of integers is it true that <math>0 < x < y < 10^{ ...ays and at <math>14</math> miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area
  7 KB (1,204 words) - 03:40, 4 January 2023
• 2002 AIME I Problems
  ...e digits of Dick's age. Let <math>d</math> be Dick's present age. How many ordered pairs of positive integers <math>(d,n)</math> are possible? ...ct squares in the plane of the dodecagon have at least two vertices in the set <math>\{A_1,A_2,A_3,\ldots,A_{12}\}</math>?
  8 KB (1,374 words) - 21:09, 27 July 2023
• 1985 AIME Problems/Problem 12
  For simplicity purposes, we set <math>c=\frac14,</math> which gives <cmath>Q(k)=-\frac13Q(k-1).</cmath> Thus, our desired number of paths is equivalent to the number of ordered septuples of positive integers <math>(b_1, b_2, \ldots, b_7)</math> such th
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1985 AIME Problems/Problem 10
  ...a</math> in the range <math>0<k<1000</math>, or <math>49\cdot12=588</math> ordered pairs <math>(a,b)</math>. If <math>a=0</math>, <math>b\neq0</math>, which includes <math>11</math> ordered pairs.
  12 KB (1,859 words) - 18:16, 28 March 2022
• 1988 AIME Problems/Problem 8
  The function <math>f</math>, defined on the set of ordered pairs of positive integers, satisfies the following properties:
  4 KB (538 words) - 13:24, 12 October 2021
• 1993 AIME Problems/Problem 7
  .../math>, are then drawn randomly and without replacement from the remaining set of <math>997</math> numbers. Let <math>p</math> be the probability that, af There is a total of <math>P(1000,6)</math> possible ordered <math>6</math>-tuples <math>(a_1,a_2,a_3,b_1,b_2,b_3).</math>
  5 KB (772 words) - 09:04, 7 January 2022
• 1997 AIME Problems/Problem 10
  ...deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the followin ...\binom{27}{2} = 27*13 = 351</math> possibilities. Note, however, that each set is generated by <math>{3\choose 2} = 3</math> pairs, so we've overcounted b
  3 KB (585 words) - 19:37, 25 April 2022
• 1998 AIME Problems/Problem 15
  ...for any <math>i</math> and <math>j</math>. Let <math>D_{40}</math> be the set of all dominos whose coordinates are no larger than 40. Find the length of We can draw a comparison between the domino a set of 40 points (labeled 1 through 40) in which every point is connected with
  9 KB (1,671 words) - 22:10, 15 March 2024
• 1998 AIME Problems/Problem 13
  If <math>\{a_1,a_2,a_3,\ldots,a_n\}</math> is a [[set]] of [[real numbers]], indexed so that <math>a_1 < a_2 < a_3 < \cdots < a_n ...all possible subsets of <math>\{1,2,\ldots,8\}</math>. Since the sets are ordered, a <math>9</math> must go at the end; hence we can just append a <math>9</m
  2 KB (384 words) - 19:02, 20 October 2023
• 1998 AIME Problems/Problem 7
  Let <math>n</math> be the number of ordered quadruples <math>(x_1,x_2,x_3,x_4)</math> of positive odd [[integer]]s that ...however note that the quadruples all need to be odd. This motivates us to set <math>x_i= 2y_i +1</math>, as for all integers <math>y_i</math>, <math>2y_i
  5 KB (684 words) - 11:41, 13 August 2023
• 1999 AIME Problems/Problem 8
  Let <math>\mathcal{T}</math> be the set of ordered triples <math>(x,y,z)</math> of nonnegative [[real number]]s that lie in th
  3 KB (445 words) - 19:40, 4 July 2013
• 2003 AIME I Problems/Problem 9
  Call the number <math>\overline{abcd}</math>. Then <math>a+b=c+d</math>. Set <math>a+b=x</math>. ...\leq k \leq 18</math>, we notice that there are <math>(18 - k) + 1</math> ordered pairs with a sum of <math>k</math>.
  4 KB (696 words) - 11:55, 10 September 2023
• 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

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