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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) - 03:05, 19 May 2024
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—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>

2 KB (439 words) - 14:09, 4 March 2022

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