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Generating function
...a [[sequence]] which is of interest. Therefore the power series (i.e. the generating function) is <math>c_0 + c_1 x + c_2 x^2 + \cdots </math> and the sequence Many generating functions can be derived using the [[Geometric sequence#Infinite|sum formul

4 KB (659 words) - 12:54, 7 March 2022
Generating functions
#REDIRECT [[Generating function]]

33 bytes (3 words) - 12:35, 6 July 2007
Talk:Generating function
...revweek=[[Asymptote (Geometry)]] [[AMC historical results]]|curweek=[[Generating function]] [[American Regions Mathematics League]]|nextweek=[[Concurre

228 bytes (26 words) - 19:57, 23 November 2007

Page text matches

User:Chess64
* [[Generating functions]]

1 KB (87 words) - 00:37, 21 July 2008
Combinatorics
...s competitions and can be approached by a variety of techniques, such as [[generating functions]] or the [[Principle of Inclusion-Exclusion|principle of inclusio

1 KB (208 words) - 02:12, 4 October 2020
Combination
* [[Generating function]]

4 KB (615 words) - 11:43, 21 May 2021
San Diego ARML
...k Tiefenbruck]] and is supported by the [[San Diego Math Circle]]. [[user:generating | Andy Niedermaier]] was a coach for 2007-2009, and [[user:MCrawford | Math

2 KB (378 words) - 16:34, 5 January 2010
Generating function
...a [[sequence]] which is of interest. Therefore the power series (i.e. the generating function) is <math>c_0 + c_1 x + c_2 x^2 + \cdots </math> and the sequence Many generating functions can be derived using the [[Geometric sequence#Infinite|sum formul

4 KB (659 words) - 12:54, 7 March 2022
Binomial Theorem
...function are termwise equal, the series at <math>x = a + b</math> is the [[Generating function#Convolutions|convolution]] of the series at <math>x = a</math> and

5 KB (935 words) - 13:11, 20 February 2024
Prime number
The [[Sieve of Eratosthenes]] is a relatively simplistic [[algorithm]] for generating a list of the first few prime numbers. It is a method in which the multiple The Sieve of Sundaram is a relatively simplistic [[algorithm]] for generating all odd prime numbers, less than <math>2n+2</math>. It is a method by which

6 KB (985 words) - 12:38, 25 February 2024
2005 USAMO Problems/Problem 4
..._i=C_{i-1}+4i</math>, and note that <math>C_0=1</math>. Now we can create generating functions. <math>F(x)=\sum_{i=0}^\infty C_ix^i</math>. Also, <math>G(x)=\

7 KB (1,276 words) - 20:51, 6 January 2024
2006 AMC 12A Problems/Problem 24
Alternatively, we can use a [[generating function]] to solve this problem. The goal is to find the generating function for the number of unique terms in the simplified expression (in te

8 KB (1,332 words) - 17:37, 17 September 2023
Combinatorics/Intermediate
* [[Generating functions]]

910 bytes (77 words) - 16:23, 18 May 2021
1985 AIME Problems/Problem 12
== Solution 6 (Generating Functions and Roots of Unity Filter / Casework) == .../math> states, <math>n</math> steps) is <math>(x+x^2+x^3)^n</math>, so the generating function of interest for this problem is <math>(x+x^2+x^3)^7</math>. Our go

17 KB (2,837 words) - 13:34, 4 April 2024
1992 AIME Problems/Problem 13
...th> can be equal with some value of <math>x</math>). MAA is pretty good at generating smooth combinations, so in this case, the AM-GM works; however, always try

4 KB (703 words) - 02:40, 29 December 2023
1996 AIME Problems/Problem 3
== Solution 2 (Generating Functions)==

3 KB (515 words) - 04:29, 27 November 2023
1998 AIME Problems/Problem 7
...for each of the terms, and obtain <math>(x+x^3+x^5\cdots)^4</math> as the generating function for the sum of the <math>4</math> numbers. We seek the <math>x^{98

5 KB (684 words) - 11:41, 13 August 2023
User:Sumech
So, generating sure our youths are genuinely using their period and perhaps not shifting a

3 KB (538 words) - 04:18, 30 April 2018
2003 AIME II Problems/Problem 13
===Solution 6 (generating functions)=== The generating function for this is <math>(x+x^2)</math> since an ant on any vertex of the

15 KB (2,406 words) - 23:56, 23 November 2023
2006 USAMO Problems/Problem 5
...+1}</math>, <math>M\geq 0</math> be the length of the longest jump made in generating <math>J_{i_0,k_0}</math>. Such a jump can only be made from a number that i

7 KB (1,280 words) - 17:23, 26 March 2016
Vector space
== Generating Subset == ...ubset is said to be ''minimal'' if on removing any element it ceases to be generating.

3 KB (561 words) - 00:47, 21 March 2009
Combinatorics/Olympiad
* [[Generating functions]]

705 bytes (64 words) - 16:22, 18 May 2021
2005 Alabama ARML TST Problems/Problem 1
We use [[generating function]]s to represent the sum of the two dice rolls: <center><math>(x+x^

1 KB (210 words) - 01:30, 3 January 2023
2007 AIME I Problems/Problem 10
...n if two squares in the row are shaded, then the row is represented by the generating function <math>ab+ac+ad+bc+bd+cd</math>, which we can write as <math>P(a,b,

13 KB (2,328 words) - 00:12, 29 November 2023
AoPS Wiki:Words of the Week
* [[Generating function]] (diff: [http://www.artofproblemsolving.com/Wiki/index.php?title=

11 KB (1,427 words) - 16:25, 9 May 2021
2006 AIME II Problems/Problem 10
We can apply the concept of generating functions here. ...function for the next 5 games is <math>(1 + x)^{5}</math>. Thus, the total generating function for number of games he wins is

6 KB (983 words) - 13:42, 8 December 2021
Lucas' Theorem
* [[Generating function]]

1 KB (251 words) - 15:13, 11 August 2020
Talk:AMC historical results
...urweek=[[Asymptote (Geometry)]] [[AMC historical results]]|nextweek=[[Generating function]] [[American Regions Mathematics League]]}}

658 bytes (101 words) - 18:40, 20 February 2018
Talk:Asymptote (geometry)
...urweek=[[Asymptote (Geometry)]] [[AMC historical results]]|nextweek=[[Generating function]] [[American Regions Mathematics League]]}}

258 bytes (28 words) - 20:07, 3 December 2007
Talk:Generating function
...revweek=[[Asymptote (Geometry)]] [[AMC historical results]]|curweek=[[Generating function]] [[American Regions Mathematics League]]|nextweek=[[Concurre

228 bytes (26 words) - 19:57, 23 November 2007
Talk:American Regions Mathematics League
...revweek=[[Asymptote (Geometry)]] [[AMC historical results]]|curweek=[[Generating function]] [[American Regions Mathematics League]]|nextweek=[[Concurre

228 bytes (26 words) - 19:58, 23 November 2007
Talk:Concurrence
{{WotW|week=Nov 22-28|prevweek=[[Generating function]] [[American Regions Mathematics League]]|curweek=[[Concurren

654 bytes (102 words) - 16:34, 18 December 2013
Talk:Intel Science Talent Search
{{WotW|week=Nov 22-28|prevweek=[[Generating function]] [[American Regions Mathematics League]]|curweek=[[Concurren

214 bytes (25 words) - 22:55, 15 February 2008
2005 Alabama ARML TST Problems/Problem 12
The [[generating function]] for <math>a, b, c,</math> and <math>d</math> is <math>x+x^2+x^3+

1 KB (172 words) - 09:56, 18 June 2008
Partition (combinatorics)
...[Jacobi theta function]], in particular the [[Jacobi triple product]]. The generating function approach and the theta function approach can be used to study many == Generating Functions ==

10 KB (1,508 words) - 14:24, 17 September 2017
2008 AIME I Problems/Problem 9
==Solution 3 (Generating Functions)== We can model this as the generating function <cmath>\left(x^3+x^4+x^6\right)^{10}</cmath> where we want the coe

6 KB (909 words) - 15:39, 8 August 2022
Free magma
...math> to <math>M(X)</math>, and let <math>(u_{i},v_i)_{i\in I}</math> be a generating set of the equivalence relation <math>R(x,y)</math> defined as <math>f(x) =

4 KB (887 words) - 13:19, 6 July 2016
Derived group
'''Corollary 4.''' Let <math>X</math> be a generating subset of <math>G</math>. Then <math>D(G)</math> is the normal subgroup ge

4 KB (688 words) - 20:11, 28 May 2008
2008 Mock ARML 2 Problems/Problem 8
...the expansion of the coefficients of the product of two polynomials (or [[generating functions]]).

1 KB (190 words) - 00:57, 31 May 2016
Approximation Techniques
...s an applied discipline, mathematicians have developed various methods for generating approximate solutions to intractable problems. The utility of such methods

2 KB (322 words) - 21:03, 11 February 2009
Alternating permutation
...tangent]] numbers. These latter names are the result of the remarkable [[generating function]] (or equivalently [[Taylor series]]) [[identity]]

2 KB (246 words) - 12:50, 6 August 2009
Characteristic polynomial
..._{n-1} + \cdots + a_kG_{n-k}</math> be a linear recurrence. Consider the [[generating function]] given by

19 KB (3,412 words) - 14:57, 21 September 2022
2003 AMC 12A Problems/Problem 22
== Solution 2 (Generating Functions) == ...r horizontally is equally likely, we can write all the possible paths as a generating function:

2 KB (321 words) - 08:40, 30 June 2023
2010 AIME I Problems/Problem 4
This can be solved quickly and easily with [[generating functions]]. The generating functions for these coins are <math>(1+x)</math>,<math>(1+x)</math>,and <ma

3 KB (470 words) - 22:15, 27 August 2023
2010 AIME I Problems/Problem 10
==Solution 5: Generating Functions== We will represent the problem using generating functions. Consider the generating function <cmath>f(x) = (1+x^{1000}+x^{2000}+\cdots+x^{99000})(1+x^{100}+x^{

7 KB (1,147 words) - 21:58, 23 January 2024
2010 AIME II Problems/Problem 11
...the first, second, third respective squares are <math>1</math>'s. Then the generating function representing the possible events that exclude a row of <math>1,1,1 Therefore, the generating function representing the possible grids where no row is filled with <math>

6 KB (1,057 words) - 01:58, 8 January 2023
2011 AIME II Problems/Problem 6
...h>a,x,y,k,b.</math> Now we use generating functions to finish. We find the generating function of the whole expression is <math>(x + x^2 + \cdots)^4 \cdot (x^2+x

9 KB (1,535 words) - 01:28, 16 January 2023
2012 AIME II Problems/Problem 3
Use [[generating functions]]. For each department, there is 1 way to pick 2 males, 4 ways to

2 KB (328 words) - 01:26, 28 January 2023
Mock AIME II 2012 Problems/Problem 5
Consider the generating function for a 12 sided die. When rolled n times, the generating function is <math>(x^1+x^2+\hdots+x^{12})^n</math>. This polynomial is clea

949 bytes (159 words) - 03:01, 5 April 2012
1988 USAMO Problems/Problem 5
.../math> of two elements, and on the left, we have the factorisation of this generating function that considers the breakdown of any given monic polynomial into mo

8 KB (1,348 words) - 09:44, 25 June 2022
1989 AHSME Problems/Problem 22
The process of choosing a block can be represented by a generating function. Each choice we make can match the 'plastic medium red circle' in

3 KB (507 words) - 20:48, 6 December 2021
2014 AIME I Problems/Problem 11
==Solution 5 (Generating Functions)==

11 KB (1,677 words) - 23:54, 4 February 2022
2007 iTest Problems/Problem 59
Recall the generating function of the Catalan numbers,

3 KB (528 words) - 10:56, 16 April 2024
North Suburban Mathematics League
| Generating Functions

4 KB (594 words) - 10:21, 31 August 2021
2015 AIME II Problems/Problem 12
==Solution 7 (Generating Functions)==

12 KB (2,047 words) - 22:02, 9 January 2024
2016 AIME II Problems/Problem 12
...asing the color number by <math>1,2,3\pmod4</math>, respectively. Thus the generating function that represents going through all six borders is <math>A(x)=(x+x^2

20 KB (3,328 words) - 21:13, 5 April 2024
1986 USAMO Problems/Problem 5
...functions to approach this problem -- specifically, we will show that the generating functions of <math>S(n)</math> and <math>T(n)</math> are equal. Let us start by finding the generating function of <math>S(n).</math> This function counts the total number of 1's

5 KB (975 words) - 14:32, 30 August 2018
1996 USAMO Problems/Problem 6
...hat have no solutions in <math>S</math> a similar proof holds, but instead generating the terms <math>S_{m+1} = 10N - M</math> and <math>S_{m+2} = -5N + M</math>

3 KB (584 words) - 07:56, 16 April 2018
2017 AMC 12A Problems/Problem 25
...lem using elementary counting methods. This solution proceeds by a cleaner generating function. ...nom{4}{a_1,a_3,a_5,a_7}</math>. We want to add these all up. We proceed by generating functions.

18 KB (2,878 words) - 01:47, 16 December 2023
2017 USAJMO Problems/Problem 2
To solve part (b), we look back at the original method of generating solutions. Define <math>a_n</math> and <math>b_n</math> to be the pair repr

10 KB (1,657 words) - 19:08, 24 April 2023
2018 AMC 10A Problems/Problem 11
We can use generating functions, where <math>(x+x^2+...+x^6)</math> is the function for each die.

3 KB (498 words) - 20:57, 7 October 2023
2018 AIME I Problems/Problem 12
==Solution 8 (Generating Functions)== ...t or out of the set. Therefore, given <math>n\in U</math>, the probability generating function is

26 KB (4,044 words) - 13:58, 24 January 2024
2018 AIME I Problems/Problem 10
==Solution 6(Generating Functions???)== Therefore the generating function for the paths beginning at <math>1</math> or <math>A</math> can be

11 KB (1,934 words) - 12:18, 29 March 2024
2019 AIME II Problems/Problem 4
==Solution 7(Generating functions)== Now, using multi-variable generating functions, we get:

19 KB (2,942 words) - 21:22, 21 January 2024
User:Tinwor
...for this site is far better than others. A winner can certainly obtain the generating amount in the profile. BCA, BRI, BNI and even Mandiri banks support via Tua

3 KB (510 words) - 01:47, 17 February 2020
User:Shivakannan
...s of Combinatorics & Number Theory. He has done research on partitions and generating functions. Occasionally he has tried to spot new assumptions and theorems i

302 bytes (44 words) - 12:43, 27 March 2020
Pólya Enumeration Theorem
...as the [https://artofproblemsolving.com/wiki/index.php/Generating_function generating function] ...x <math>Z(t_1,t_2,...,t_N)</math> where <math>|\mathcal{X}|=N</math>. The generating function for the number of ways to paint <math>\mathcal{X}</math> in colors

2 KB (308 words) - 19:51, 24 November 2021
2021 AMC 12A Problems/Problem 15
==Solution 3 (Generating Functions)== ...>. By expanding the binomials and distributing, <math>f(x,y)</math> is the generating function for different groups of basses and tenors. That is, <cmath>f(x,y)=

7 KB (1,152 words) - 14:13, 29 February 2024
User:Rynverpol
...ing the goodies. The web site progression of long island assists the folks generating their website quickly. Folks and the long island SEO company favour mobile-

3 KB (520 words) - 06:26, 8 January 2021
User:Midelran
In the current generating, a large number of guys are gaining multiple issues as part of their indust

4 KB (571 words) - 10:49, 21 August 2021
User:Seynnem
While the factor comes to life at generating revenue, casino may be the foremost method links towards the thought proces

3 KB (577 words) - 02:25, 4 June 2021
2022 AIME II Problems/Problem 13
==Solution 4 (Generating Function Bash)== Notice <math>\text{lcm}(105,70,42,30) = 210</math>, so we can rewrite the generating function as

7 KB (1,102 words) - 13:41, 25 April 2024
User:Sidenard
In some cases people feel that generating revenue is amazingly really difficult and they can require operate difficul

3 KB (551 words) - 03:15, 12 July 2021
2021 Fall AMC 12B Problems/Problem 17
==Solution 4 (Generating Function)== Use a generating function, define <math>c_{n}\cdot x^{n}</math> be <math>c_{n}</math> ways f

8 KB (1,309 words) - 00:31, 6 January 2023
Talk:2021 Fall AMC 12B Problems/Problem 17
Use generating function, define <math>c_{n}\cdot x^{n}</math> be <math>c_{n}</math> ways f

961 bytes (169 words) - 18:52, 28 November 2021
2022 AMC 8 Problems/Problem 25
==Solution 6 (Generating Function)== ...<math>1</math> to <math>2</math> would be a change of <math>1.</math> This generating function is equal to <math>(x+x^2+x^3)^4.</math> It is clear that we want t

9 KB (1,513 words) - 22:38, 18 February 2024
User:Unteneld
...ww.eddysflooringamerica.com/ vslots88] can access in excess of $1 million, generating it amongst the simplest methods to win considerable on the net.

3 KB (612 words) - 01:22, 12 January 2024
User:Antonio2
...lso involved entertainment to gamers. We are actually constantly servicing generating even more thrilling and practical video game functions in addition to a var

8 KB (1,378 words) - 05:04, 19 October 2022
User:Lyeismos
...ed.Unlike arrays, strings can be split into many parts. This is useful for generating different combinations of strings. The'slice' function returns the elements

5 KB (760 words) - 05:10, 31 July 2023
User:Crazyvideogamez
Next, let us represent the number of solutions to each equation as a generating function. The equation Then, to find the total number of solutions, we sum each of the generating functions corresponding to each equation and derive the coefficient of <mat

7 KB (1,077 words) - 14:44, 8 February 2024
User:Roderough
...in other interesting ways of quantity correct Christmas, like for example generating a selfmade snowman, stringing popcorn, or possibly resulting in a Christmas

4 KB (711 words) - 07:18, 10 August 2023
User:Daughet
...sperity, it's a really good option to target a niche as you concentrate at generating expert inside in which room or space. Choosing a good Youtube take a look a

4 KB (607 words) - 06:08, 21 September 2023
2023 AMC 12A Problems/Problem 24
...t we can generate the number of valid sequences of <math>A</math> by first generating all sequences of <math>b</math> such that <math>b_i\leq{}b_{i+1}</math> for

9 KB (1,370 words) - 09:08, 19 February 2024
2023 AMC 10B Problems/Problem 11
== Solution 6 (generating functions) == ...th> bills, <math>\$5</math> bills, and <math>\$10</math> bills. We can use generating functions to find the coefficient of <math>x^{80}</math>:

6 KB (945 words) - 15:44, 5 February 2024
User:Afly
===2019 AIME II #4 Solution 7 (Generating functions)=== Now, using multi-variable generating functions, we get:

32 KB (4,110 words) - 20:58, 10 April 2024
2023 AMC 12B Problems/Problem 19
We use the generating functions approach to solve this problem. ==Solution 10 (Generating Functions)==

15 KB (2,516 words) - 20:31, 15 January 2024
User:Cyantist
...lution_10_.28Generating_Functions.29 2023 AMC 12B Problem 19 Solution 10] (Generating Functions)

942 bytes (105 words) - 17:11, 4 February 2024
User:Palsam
...used for data centres, or "Data Centre Power application" is made up of a generating plant combined with an uninterruptible power supply (UPS). The UPS contains In addition to generating power generators can also be utilized to supply backup power for ranching o

6 KB (914 words) - 04:06, 10 January 2024
User:Tanmosris
...enough gold during WoW classic season, it is recommended to take on income-generating jobs like Herbalism and Mining. These professions offer a consistent source

4 KB (605 words) - 06:50, 29 January 2024
User:ErinBoyd24
.... With Quantum AI, you can take control of your financial future and start generating passive income with ease. Don't miss out on the chance to benefit from this

594 bytes (87 words) - 16:42, 5 March 2024

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