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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
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• 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
• Graph (graph theory)
  ...eory]], a '''graph''' is a (usually [[finite]]) [[empty set | nonempty]] [[set]] of [[vertex|vertices]] that are joined by a number (possibly zero) of [[e ...then <math>E</math> may be defined using ordered pairs from the [[product set]] <math>V \times V</math>.
  8 KB (1,428 words) - 10:26, 27 August 2020
• Homomorphism
  If <math>A</math> and <math>B</math> are [[partially ordered set]]s, a homomorphism from <math>A</math> to <math>B</math> is a mapping <math
  2 KB (303 words) - 15:33, 11 February 2024
• 2007 AMC 12B Problems/Problem 17
  ...e number such that <math>ab^2=\log_{10}b</math>, what is the median of the set <math>\{0,1,a,b,1/b\}</math>? This puts <math>a</math> as the smallest in the set since it must be negative.
  1 KB (242 words) - 12:41, 19 June 2023
• 2006 Alabama ARML TST Problems
  ...glazed, and powdered donuts she wound up with. Find the number of possible ordered triples <math>(a, b, c)</math>. ...ng consecutively as he writes. When he stops, he realizes that there is no set of 5 composite integers among the ones he wrote such that each pair of thos
  5 KB (769 words) - 20:56, 24 March 2015
• Filter
  A '''filter''' on a [[set]] <math>X</math> is a structure of [[subset]]s of <math>X</math>. Let <math>\mathcal{F}</math> be a set of subsets of <math>X</math>. We say that <math>\mathcal{F}</math> is a fi
  2 KB (368 words) - 21:14, 13 October 2019
• Ultrafilter
  An '''ultrafilter''' is a [[set theory | set theoretic]] structure. An ultrafilter on a set <math>X</math> is a non-empty [[filter]] <math>\mathcal{F}</math> on <math>
  9 KB (1,685 words) - 20:28, 13 October 2019
• Free magma
  ...ure that is as general as possible&mdash;a magma generated from an initial set with no constraints or relations. The free magma generated from a [[set]] <math>X</math> is constructed as follows.
  4 KB (887 words) - 13:19, 6 July 2016
• Burnside's Lemma
  ...sult in [[group theory]] that is useful for counting the [[orbit]]s of a [[set]] on which a [[group]] [[group action|acts]]. ...<math>\alpha \in G</math>, let <math>\text{fix}(\alpha)</math> denote the set of [[fixed point]]s of <math>\alpha</math>. Then
  5 KB (757 words) - 18:11, 23 October 2023
• KGS math club
  |Devise a set of denominations, as few as possible, such that any integer value from 1 to |For which n can the complete graph K_n have its set of edges partitioned to form edge-disjoint Hamiltonian circuits or Hamitoni
  22 KB (3,358 words) - 15:17, 18 July 2017
• Field of fractions
  ...the '''quotient field'''), denoted by <math>\text{Frac}(R)</math>, as the set <math>\left\{\frac{a}{b} \mid a,b\in R, b\neq 0\right\}</math>. This is ana ...math>ad = bc</math>. Then we can define <math>\text{Frac}(R)</math> as the set of [[equivalence class|equivalence classes]] of <math>S</math> under <math>
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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
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• 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) - 22:09, 27 August 2023
• 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
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• 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
• 1997 AHSME Problems/Problem 28
  How many ordered triples of integers <math>(a,b,c)</math> satisfy <math> |a+b|+c = 19 </math ...ur initial assumption that <math>c\ge 0</math>. Similarly, for the second set of four solutions, we have <math>|a + b| = 78</math>, which leads to <math>
  4 KB (679 words) - 17:21, 26 July 2021
• 2016 AMC 10A Problems/Problem 19
  We can set coordinates for the points. <math>D=(0,0), C=(6,0), B=(6,3),</math> and <m ...th>P</math>,<math>Q</math>, and <math>K</math> ordered from left to right. Set the values <math>BP=x</math>,<math>PK=x</math>,<math>BQ=4y</math> and <math
  8 KB (1,386 words) - 15:10, 8 October 2023
• 1989 AHSME Problems
  ...rt{x^{2}+1}-\frac{1}{x+\sqrt{x^{2}+1}} </math> is a rational number is the set of all: ..., and 4 shapes (circle, hexagon, square, triangle). How many blocks in the set are different from the 'plastic medium red circle' in exactly 2 ways? (The
  15 KB (2,247 words) - 13:44, 19 February 2020
• 2004 AMC 8 Problems/Problem 3
  ...or <math>18</math> people. If they shared, how many meals should they have ordered to have just enough food for the <math>12</math> of them? Set up the proportion <math>\frac{12\ \text{meals}}{18\ \text{people}}=\frac{x\
  700 bytes (106 words) - 00:54, 5 July 2013
• 2015 AIME I Problems
  ...math>(a_1,a_2,a_3)</math> with <math>1 \le a_1,a_2,a_3 \le 10</math>. Each ordered triple in <math>S</math> generates a sequence according to the rule <math>a Consider all 1000-element subsets of the set <math> \{ 1, 2, 3, ... , 2015 \} </math>. From each such subset choose the
  10 KB (1,615 words) - 21:48, 13 January 2024
• 2012 AIME I Problems
  ...es to the nine people so that exactly one person receives the type of meal ordered by that person. Let <math>B</math> be the set of all binary integers that can be written using exactly <math>5</math> zer
  10 KB (1,617 words) - 14:49, 2 June 2023
• 2012 AIME II Problems
  Find the number of ordered pairs of positive integer solutions <math>(m, n)</math> to the equation <ma ...than <math>2</math> from all multiples of <math>p</math>. For example, the set of <math>10</math>-safe numbers is <math>\{ 3, 4, 5, 6, 7, 13, 14, 15, 16,
  7 KB (1,228 words) - 12:16, 13 March 2020
• 2012 AIME II Problems/Problem 1
  Find the number of ordered pairs of positive integer solutions <math>(m, n)</math> to the equation <ma ...th>. So <math>20m=2000-12k</math>, where <math>k</math> is a member of the set <math>{0, 5, 10, 15...}</math>.
  3 KB (519 words) - 14:56, 2 June 2023
• 2012 USAMO Problems/Problem 1
  ...that the bounding condition becomes <math>a_n \le n \cdot a_1.</math> Now set <math>b_i \equiv \frac{a_i}{a_1},</math> and since a triangle with sideleng ...base case, we see inductively that in general <math>\{s_n\}</math> is the set of the first <math>n</math> Fibonacci numbers. To show this note that if <m
  5 KB (930 words) - 02:17, 7 March 2018
• 2015 IMO Problems/Problem 1
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> ...> Show that for all integers <math>n\geq 3</math>, there exists a balanced set consisting of <math>n</math> points. </li>
  4 KB (773 words) - 08:14, 19 July 2016
• 2012 AMC 12B Problems/Problem 16
  Let the ordered triple <math>(a,b,c)</math> denote that <math>a</math> songs are liked by A ...e number of ways to rearrange <math>AB, BJ, AJ</math>, and a song from the set <math>\{N, A, B, J, AB, BJ, AJ\}</math>.
  8 KB (1,462 words) - 11:33, 21 April 2024
• 2020 Mock Combo AMC 10
  Let <math>S</math> be the set of the first <math>2n</math> positive integers, and let <math>R</math> be a ...ath> for some <math>4 \leq k \leq n</math> such that there exists no other set of four consecutive integers in the permutation with sum greater than <math
  15 KB (2,452 words) - 03:03, 4 July 2020
• 2012 AMC 8 Problems
  ...nd sunrise were correct, but the sunset was wrong. When did the sun really set? Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice e
  13 KB (1,835 words) - 08:51, 8 March 2024
• 2013 AMC 12A Problems
  Let <math>S</math> be the set of positive integers <math>n</math> for which <math>\tfrac{1}{n}</math> has ...at either <math>0 < a - b \le 9</math> or <math>b - a > 9</math>. How many ordered triples <math>(x,y,z)</math> of elements of <math>S</math> have the propert
  14 KB (2,204 words) - 20:25, 22 November 2020
• 2013 AMC 12A Problems/Problem 20
  ...at either <math>0 < a - b \le 9</math> or <math>b - a > 9</math>. How many ordered triples <math>(x,y,z)</math> of elements of <math>S</math> have the propert Consider the 1st set of conditions for <math>x, y, z</math>. We get that there are
  5 KB (885 words) - 10:14, 29 October 2023
• 2013 AMC 10B Problems/Problem 25
  For <math>b_1,b_2<6</math>, there is a total of 7 ordered pairs that satisfy the condition. Namely, Since we know that <math>a_i<5</math>, for each of the ordered pairs <math>(b_1,b_2)</math>, there is respectively one and only one soluti
  10 KB (1,623 words) - 15:44, 31 August 2022
• 2013 AIME I Problems/Problem 15
  Let <math>N</math> be the number of ordered triples <math>(A,B,C)</math> of integers satisfying the conditions (d) each ordered triple <math>(A,B,C)</math> and each ordered triple <math>(b,a,c)</math> form arithmetic sequences. Find <math>N</math>.
  4 KB (661 words) - 23:14, 26 May 2023
• Mock AIME 6 2006-2007 Problems
  ...ts in the plane, no three of which lie on the same line. At most how many ordered triples of points <math>(A,B,C)</math> in <math>R</math> exist such that <m ...tit{n-unsound}</math> rationals. The sum of all the elements in the union set <math>S_2\cup S_3\cup\cdots\cup S_{14}</math> is <math>\frac mn</math>, whe
  7 KB (1,173 words) - 21:04, 7 December 2018
• Northeastern WOOTers Mock AIME I Problems
  Let the number of ordered tuples of positive odd integers <math> \left( x_1, x_2, \cdots, x_{42} \rig Define a ''T-Polyomino'' to be a set of 4 cells in a grid that form a T, as shown below. Dai wants to place T-Po
  8 KB (1,336 words) - 09:10, 30 May 2020
• Coordinate system
  ...lled the origin. Each point in the coordinate plane can be specified by an ordered pair of numbers, (x,y). The cordinate system is organized to 4 quadrants. I ...xis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the pe
  4 KB (591 words) - 01:23, 25 January 2016
• 1990 AHSME Problems/Problem 18
  ...99,100\}</math>, and then <math>b</math> is chosen at random from the same set. The probability that the integer <math>3^a+7^b</math> has units digit <mat ...h> each power's units digit is equally probable. There are <math>16</math> ordered pairs of units digits, and three of them <math>(1,7),(7,1),(9,9)</math> hav
  987 bytes (146 words) - 11:52, 4 February 2016
• 1969 AHSME Problems
  Let a binary operation <math>\star</math> on ordered pairs of integers be defined by <math>(a,b)\star (c,d)=(a-c,b+d)</math>. Th ...values satisfying the inequality <math>\frac{x^2-4}{x^2-1}>0</math> is the set of all <math>x</math> such that:
  16 KB (2,662 words) - 14:12, 20 February 2020
• 1987 AHSME Problems
  A student recorded the exact percentage frequency distribution for a set of measurements, as shown below. How many ordered triples <math>(a, b, c)</math> of non-zero real numbers have the property t
  16 KB (2,291 words) - 13:45, 19 February 2020
• 1976 AHSME Problems
  ...ximum number of points of intersection of pairs of lines from the complete set <math>\{L_1,L_2,\dots,L_{100}\}</math> is How many distinct ordered triples <math>(x,y,z)</math> satisfy the following equations?
  17 KB (2,835 words) - 14:36, 8 September 2021
• 2014 UMO Problems
  Completely describe the set of all right triangles with positive integer-valued legs such that when fou ...with ordered lists of integers in the following way. He starts out with an ordered list
  6 KB (982 words) - 15:11, 22 October 2018
• 2016 AMC 12A Problems
  ...} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordina How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(
  15 KB (2,348 words) - 17:20, 19 January 2024
• 2015 AMC 12A Problems
  ...y four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. ...<math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For eve
  13 KB (2,117 words) - 12:33, 24 August 2023
• 2015 AMC 12B Problems
  <math>\textbf{(A)}\; \text{the set of positive integers} \\ \textbf{(B)}\; \text{the set of composite positive integers} \\
  13 KB (2,064 words) - 13:39, 1 October 2022
• 2015 AIME I Problems/Problem 9
  ...math>(a_1,a_2,a_3)</math> with <math>1 \le a_1,a_2,a_3 \le 10</math>. Each ordered triple in <math>S</math> generates a sequence according to the rule <math>a ...h>|y-x|</math> or <math>|z-y|</math> is less than or equal to 1, then that ordered triple meets the criteria.
  6 KB (1,003 words) - 20:35, 28 July 2023
• 2016 AMC 12B Problems/Problem 16
  ...math>(\text{exg}~ (10,69) ~\text{or} ~(15,46))</math> we will have several ordered pairs <math>(c,d)</math> where <math>c<d</math> ...brief testing, we find that the the other 7 factor pairs produce 7 viable ordered pairs. This means we have found <math>\boxed{\textbf{(E)}\ 7}</math> ways t
  8 KB (1,396 words) - 07:17, 31 March 2023
• 2016 AMC 12B Problems
  A set of teams held a round-robin tournament in which every team played every oth There are exactly <math>77,000</math> ordered quadruplets <math>(a, b, c, d)</math> such that <math>\gcd(a, b, c, d) = 77
  14 KB (2,037 words) - 19:09, 29 July 2023
• 2016 AIME II Problems/Problem 12
  ...complement, we wish to count the number of colorings in which at least one set of adjacent sections are the same color. There are six possible sets of adj ...>, given the bug's starting number <math>k</math>, is simply the number of ordered quintuplets of positive integers <math>(a_1, a_2, a_3, a_4, a_5)</math> tha
  20 KB (3,328 words) - 21:13, 5 April 2024
• 2006 iTest Problems
  ...ns are identical) and where a group of students is treated as an unordered set of people. Compute the remainder obtained when <math>N</math> is divided by For each positive integer <math>n</math> let <math>S_n</math> denote the set of positive integers <math>k</math> such that <math>n^k-1</math> is divisib
  31 KB (4,811 words) - 00:02, 4 November 2023
• 2017 AMC 12A Problems/Problem 25
  ...ctly 8 angles from the set <math>\{z^2,z^6\}</math> and exactly 4 from the set <math>\{z^1,z^3,z^5,z^7\}</math>. If we found a good combination where we h ...h>2a + 6b</math> is even for all <math>a + b =8, </math> and the number of ordered quadruples <math>(a_1, a_2, a_3, a_4)</math> such that <math>a_i \in \{1, 3
  18 KB (2,878 words) - 01:47, 16 December 2023
• 2017 AMC 12A Problems
  Let <math>S</math> be the set of points <math>(x,y)</math> in the coordinate plane such that two of the t How many ordered pairs <math>(a,b)</math> such that <math>a</math> is a positive real number
  15 KB (2,418 words) - 16:58, 7 November 2022
• 2018 AMC 10A Problems
  ...ance in miles to the nearest town. Which of the following intervals is the set of all possible values of <math>d</math>? How many ordered pairs of real numbers <math>(x,y)</math> satisfy the following system of eq
  14 KB (2,171 words) - 21:10, 4 November 2023
• 2017 IMO Problems/Problem 6
  ...sor of <math>x</math> and <math>y</math> is <math>1</math>. Given a finite set <math>S</math> of primitive points, prove that there exist a positive integ
  582 bytes (105 words) - 03:09, 19 November 2023
• 2018 AMC 12A Problems
  ...ance in miles to the nearest town. Which of the following intervals is the set of all possible values of <math>d</math>? ...math> such that <math>m+10<n+1</math>, both the mean and the median of the set <math>\{m, m+4, m+10, n+1, n+2, 2n\}</math> are equal to <math>n</math>. Wh
  15 KB (2,380 words) - 18:52, 7 April 2022
• Basic Programming With Python
  In the print statement, we set <math>x</math> as 2, and the function returns <math>2 + 1 = 3,</math> so th ...tion() is said to be '''called''' in the final line of the code. Again, we set <math>x</math>, our parameter, as "print_function."
  33 KB (5,277 words) - 22:14, 3 June 2023
• 2017 IMO Problems
  Let <math>\mathbb{R}</math> be the set of real numbers , determine all functions ...sor of <math>x</math> and <math>y</math> is <math>1</math>. Given a finite set <math>S</math> of primitive points, prove that there exist a positive integ
  4 KB (720 words) - 12:12, 5 August 2021
• 2019 AMC 10C Problems
  Find the number of ordered pairs <math>(a, b, c, d)</math> that satisfy these conditions. A tripod has three legs of length 3 feet, 4 feet, and 4 feet. It is set up so that the angle between any two legs is <math>90^{\circ}</math>. The h
  12 KB (1,917 words) - 12:14, 29 November 2021
• 2020 AMC 10B Problems
  For how many ordered pairs of integers <math>(x,y)</math> satisfy the equation <cmath>x^{2020} + ...h its interior. For real <math>r\geq0</math>, let <math>S(r)</math> be the set of points in <math>3</math>-dimensional space that lie within a distance <m
  16 KB (2,497 words) - 21:14, 12 November 2023
• 2019 AIME II Problems/Problem 4
  ...from <math>1^2</math> to <math>36^2</math>, and how many ways they can be ordered ...nce of <math>4</math> numbers whose product is a perfect square. To form a set, we can simply select zero to two groups of size <math>2</math> or <math>3<
  19 KB (2,942 words) - 21:22, 21 January 2024
• What is the definition of Pure Mathematics?
  A '''system of equations''' is a set of [[equation]]s which share the same [[variable]]s. An example of a syste Find the ordered pair <math>(x,y)</math> for which
  35 KB (5,882 words) - 18:08, 28 June 2021
• 2020 AIME I Problems/Problem 11
  ...th>f(x)=x^2+ax+b</math> and <math>g(x)=x^2+cx+d.</math> Find the number of ordered triples <math>(a,b,c)</math> of integers with absolute values not exceeding ...> can be anything. However, <math>c</math> can also be anything, as we can set the root of <math>g</math> (not equal to <math>f(2) = f(4)</math>) to any i
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• Mock AMC 10B Problems
  For a particular positive integer <math>n</math>, the number of ordered sextuples of positive integers <math>(a, b, c, d, e, f)</math> that satisfy ...nd stops as soon as it touches the ground again. What is the volume of the set of points swept out by the larger log as it rolls over the smaller one?
  13 KB (2,059 words) - 02:59, 21 January 2021
• 2001 Pan African MO Problems/Problem 4
  ...<math>(0,0),(0,a),(a,0),(a,a)</math>. It seems as if the solution to the set involves only <math>0</math> and <math>a</math>. To prove this, we need to ...i+1})^2 = (n+1)a^2</math>. The inductive step holds, so solutions are all ordered pairs <math>(x_1, x_2, \cdots x_n)</math> where <math>x_i = 0</math> or <ma
  3 KB (629 words) - 00:26, 18 December 2019
• 2020 AMC 10A Problems/Problem 18
  ...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 ...actly one term to be odd, one term to be even. Because of symmetry, we can set <math>ad</math> to be odd and <math>bc</math> to be even, then multiply by
  6 KB (1,044 words) - 13:50, 4 April 2024
• 2020 AMC 10B Problems/Problem 25
  ...e properties that factors are greater than <math>1</math>, and differently ordered products are counted separately. ...ly totaling the number of ways there are to insert a <math>3</math> into a set of numbers that multiply to <math>32</math>.
  16 KB (2,560 words) - 20:26, 11 November 2023
• 2020 AMC 10B Problems/Problem 9
  How many ordered pairs of integers <math>(x, y)</math> satisfy the equation <cmath>x^{2020}+ ...<math>(0,2)</math> gives a total of <math>\boxed{\textbf{(D) }4}</math> ordered pairs.
  8 KB (1,221 words) - 16:29, 5 February 2024
• 2020 AIME I Problems
  Let <math>S</math> be the set of positive integers <math>N</math> with the property that the last four di ...e numbers are chosen independently and at random with replacement from the set <math>S</math> and labeled <math>a_1,a_2,</math> and <math>a_3</math> in th
  7 KB (1,257 words) - 17:51, 5 February 2022
• Mock AIME 2 Pre 2005 Problems/Problem 14
  ...y the number of distinct subsets <math>S</math> such of our aforementioned set <math>E_1, E_2, ..., E_n</math> such that the sum of the members in <math>S The only basic ordered pair <math>(E_1, E_2)</math> that offers a nonzero number of legal placemen
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• AMC 12C 2020
  ...th>(0, 0), (6, 0)</math>, and <math>(0, 5)</math>. Let <math>S</math> be a set of rigid transformations consistsing of rotataions <math>90, 180</math>, an Let there be multiple ordered pairs <math>(n, k)</math> where <math>n</math> and <math>k</math> are posit
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• Ordinal
  ...math> and <math>b</math> is the ordinal that describes the order type of a set with order type a concatenated with one of order type b. Warning! Ordinal a Every ordinal characterizes the order type of the ordered ordinals less than it. For example, <math>0,1,2,\dotsb,\omega</math> has or
  5 KB (811 words) - 14:16, 7 June 2020
• 2020 AIME II Problems/Problem 14
  ...ordered triple can be identified with a multiset of three elements of the set of <math>2003</math> integers <math>\{17,18,19,\ldots,2019\}</math>, which
  7 KB (1,186 words) - 15:31, 5 January 2024
• 2020 FMC 10A Problems
  Let <math>S</math> denote the set of all positive integers <math>n</math> that satisfy <math>0 \leq n \leq 10 ...hat he is on <math>(x, y)</math>, he will randomly choose one point in the set <math>\{ (x-1, y-1), (x, y-1), (x+1, y-1) \}</math> to travel to. The proba
  14 KB (2,267 words) - 12:49, 9 June 2020
• 2020 FMC 12A Problems
  Let <math>S</math> denote the set of all positive integers <math>n</math> that satisfy <math>0 \leq n \leq 10 ...integers (which are ordered accordingly) not including <math>1</math> are ordered based on the placement of the <math>1</math> in the permutation. For instan
  15 KB (2,388 words) - 13:24, 9 June 2020
• 2020 USAMO Problems
  Denote by <math>A</math> the set of all integers <math>a</math> such that <math>1 \le a < p</math>, and both ...value of <math>N</math> over all possible choices of the <math>100</math> ordered pairs.
  4 KB (614 words) - 13:47, 22 November 2023
• 2021 AIME I Problems/Problem 3
  ...th>, we can first choose two different numbers <math>a > b</math> from the set <math>\{0,1,2,\ldots,10\}</math> in <math>\binom{11}{2}=55</math> ways. Thi As indicated by the X-marks, the ordered pairs <math>(a,b)=(10,0),(10,1),(10,2),(10,3),(10,4)</math> generate <math>
  7 KB (1,174 words) - 08:21, 13 May 2023
• 2022 AIME I Problems/Problem 6
  ...uence<cmath>3, 4, 5, a, b, 30, 40, 50</cmath>is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression
  8 KB (1,205 words) - 22:55, 26 March 2023
• 2022 AIME I Problems/Problem 12
  For any finite set <math>X</math>, let <math>| X |</math> denote the number of elements in <ma where the sum is taken over all ordered pairs <math>(A, B)</math> such that <math>A</math> and <math>B</math> are s
  9 KB (1,471 words) - 16:41, 1 February 2024
• 2013 IMO Problems
  Let <math>\mathbb Q_{>0}</math> be the set of all positive rational numbers. Let <math>f:\mathbb Q_{>0}\to\mathbb R</m ...>M</math> be the number of beautiful labelings, and let N be the number of ordered pairs <math>(x, y)</math> of positive integers such that <math>x + y \le n<
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• 2021 AIME II Problems
  ...</math> and two sides with lengths <math>4</math> and <math>10</math>. The set of all <math>s</math> for which <math>\tau(s)</math> is nonempty, but all t .../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
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• 2020 CIME II Problems
  Over all ordered triples of positive integers <math>(a,b,c)</math> for which <math>a+b+c^2=a ...exists an integer <math>n\ge2020</math> such that when the elements of the set <math>\{1,2,\ldots,n\}</math> are sorted lexicographically from least to gr
  8 KB (1,298 words) - 18:32, 7 January 2021
• 2021 AMC 12A Problems/Problem 15
  ...pairs <math>(t,b')</math> have one-to-one correspondence, we consider the ordered pairs <math>(t,b')</math> instead. The requirements become <math>t\equiv8-b Consider the set of all <math>2^{8+6}=2^{14}</math> possible choirs that can be formed. For
  7 KB (1,152 words) - 14:13, 29 February 2024
• 2021 AMC 12A Problems/Problem 23
  ...</math> by <math>2</math> square centered at <math>(3x, 3y)</math> for all ordered pairs of integers <math>(x, y).</math> ...th>(0, 0)</math>. (minus the teleportations) Since counting the complement set is easier, we'll count the number of <math>4</math>-step paths such that Fr
  17 KB (2,801 words) - 07:29, 4 November 2022
• 2013 Mock AIME I Problems
  Find the number of ordered positive integer triplets <math>(a,b,c)</math> such that <math>a</math> eve ...ith imaginary part greater than <math>0</math>. Let <math>T</math> be the set of all <math>9</math>th primitive roots of unity with imaginary part greate
  7 KB (1,149 words) - 17:16, 15 December 2020
• 2021 April MIMC 10
  ...y)</math> which both <math>x</math> and <math>y</math> are elements in the set <math>\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\}</math>, <math>x</math> and < What can be a description of the set of solutions for this: <math>x^{2}+y^{2}=|2x+|2y||</math>?
  15 KB (2,366 words) - 17:45, 19 September 2021
• 2021 AIME II Problems/Problem 9
  ...> such that <math>m</math> and <math>n</math> are positive integers in the set <math>\{1, 2, ..., 30\}</math> and the greatest common divisor of <math>2^m To count the ordered pairs <math>(m,n),</math> we perform casework on the number of factors of <
  7 KB (1,212 words) - 15:54, 15 April 2024
• 2022 AIME I Problems
  ...i\sqrt{3}}{2},</math> where <math>i = \sqrt{-1}.</math> Find the number of ordered pairs <math>(r,s)</math> of positive integers not exceeding <math>100</math ...e sequence <cmath>3,4,5,a,b,30,40,50</cmath> is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression
  9 KB (1,520 words) - 19:06, 2 January 2023
• 2021 Fall MIMC 10
  ...y)</math> which both <math>x</math> and <math>y</math> are elements in the set <math>\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\}</math>, <math>x</math> and < What can be a description of the set of solutions for this: <math>x^{2}+y^{2}=|2x+|2y||</math>?
  14 KB (2,226 words) - 23:39, 12 September 2021
• 2021 April MIMC 10 Problems/Problem 14
  ...y)</math> which both <math>x</math> and <math>y</math> are elements in the set <math>\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\}</math>, <math>x</math> and < We can begin by converting all the elements in the set to Modular of <math>5</math>. Then, we realize that all possible elements t
  1 KB (186 words) - 02:05, 15 June 2021
• 2023 IOQM Problems
  ...n \leq 1000</math>. Let <math>M_n</math> be the number of integers in the set Find the number of elements in the set
  7 KB (1,225 words) - 03:05, 4 May 2024
• 1994 IMO Problems
  ...ts</math>, <math> a_m</math> be <math> m</math> different numbers from the set <math> \{1, 2,\ldots, n\}</math> such that for any two indices <math> i</ma ...ger <math>k</math>, let <math>f(k)</math> be the number of elements in the set
  3 KB (545 words) - 14:01, 19 April 2024
• 2023 AMC 12A Problems
  Consider the set of complex numbers <math>z</math> satisfying <math>|1+z+z^{2}|=4</math>. Th How many ordered pairs of positive real numbers <math>(a,b)</math> satisfy the equation
  15 KB (2,168 words) - 05:11, 4 February 2024
• 2023 INMO Problems/Problem 1
  Let S be a finite set of positive integers. Assume that there are precisely 2023 ordered pairs (x, y) in S × S so that the product xy is a perfect square. Prove th
  431 bytes (70 words) - 11:48, 1 November 2023
• 2023 IOQM/Problem 2
  Find the number of [[elements]] in the [[set]] Finding the no. of [[elements]] in the [[set]] means finding no. of [[ordered pairs]] of (<math>a</math>, <math>b</math>)
  2 KB (299 words) - 02:48, 4 May 2024
• 2023 IOQM/Problem 20
  For any finite non empty set X of integers, let max(X) denote the largest element of X and |X| denote the number of elements in X . If N is the number of ordered
  368 bytes (72 words) - 01:02, 27 October 2023
• 2023 AMC 12B Problems/Problem 23
  Note that there are only <math>3</math> primes in the set <math>\{1,2,3,4,5,6\}</math>: <math>2,3,</math> and <math>5</math>. Thus if ...in this scenario because there is only one factor of <math>5</math> in the set. Because of this, having <math>j</math> fives in our prime factorization of
  13 KB (2,194 words) - 19:10, 18 December 2023
• Mock AIME 6 2006-2007 Problems/Problem 4
  ...ts in the plane, no three of which lie on the same line. At most how many ordered triples of points <math>(A,B,C)</math> in <math>R</math> exist such that <m
  878 bytes (151 words) - 02:42, 3 January 2024
• 2024 AIME II Problems/Problem 14
  ...partition of <math>\left\{ 1, 2, \cdots , n \right\}</math> (we allow one set to be empty). ...<math>{\rm gcd} \left( z, z-1 \right) = 1</math>, there must exist such an ordered partition, such that <math>P_A | z</math> and <math>P_{\bar A} | z-1</math>
  4 KB (682 words) - 17:07, 2 May 2024
• 2024 AIME II Problems/Problem 9
  ...rn. Once the first blank row is chosen, the rest of the blank rows must be ordered similarly. For example, with 2 black chips on the left, there will be 3 bla ...way to place the chips, and every way to place the chips corresponds to a set of rows and columns occupied by the white pieces.
  6 KB (995 words) - 12:52, 25 March 2024
• 2024 USAMO Problems
  ...nct number of red beads. Determine, with proof, all possible values of the ordered pair <math>(m, n)</math>. Note: For a finite set <math>S,|S|</math> denotes the number of elements in <math>S</math>.
  4 KB (625 words) - 21:00, 23 March 2024