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• Combination
  ...objects are chosen is irrelevant. We are generally concerned with finding the number of combinations of size <math>r</math> from an original set of size The common forms of denoting the number of combinations of <math>{r}</math> objects from a set of <math>{n}<
  4 KB (615 words) - 11:43, 21 May 2021
• Induction
  ...at the result holds for <math>n=k+1</math>. In this way, we can show that the result holds for all positive integers; we will have shown that it works fo ...t, so does <math>n=k+2</math>. If you wish, you can similarly induct over the powers of 2.
  5 KB (768 words) - 20:45, 1 September 2022
• Binomial Theorem
  The '''Binomial Theorem''' states that for [[real]] or [[complex]] <math>a</mat ...cted are the same as the entries in the <math>n</math>th row of [[Pascal's Triangle]].
  5 KB (935 words) - 13:11, 20 February 2024
• Combinatorial identity
  == Pascal's Identity == Pascal's Identity states that
  12 KB (1,993 words) - 23:49, 19 April 2024
• Pascal's triangle
  '''Pascal's triangle''' is a triangle which contains the values from the [[binomial expansion]]; its various properties play a large role in [[combi ...se k}}=2^n</math>, the sum of the values on row <math>n</math> of Pascal's Triangle is <math>2^n</math>.
  5 KB (838 words) - 17:20, 3 January 2023
• 1992 AIME Problems
  Find the sum of all positive rational numbers that are less than 10 and that have de ...>.503</math>. What's the largest number of matches she could've won before the weekend began?
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1992 AIME Problems/Problem 4
  ...ch entry is the sum of the two entries above it. The first few rows of the triangle are shown below. ...ow of [[Pascal's Triangle]] do three consecutive entries occur that are in the ratio <math>3 :4 :5</math>?
  3 KB (476 words) - 14:13, 20 April 2024
• 2003 AIME I Problems/Problem 13
  ...math> representation has more <math>1</math>'s than <math>0</math>'s. Find the [[remainder]] when <math> N </math> is divided by <math>1000</math>. ...+1</math> digits in base <math>2</math> notation, with <math>k+1</math> of the digits being <math>1</math>'s.
  4 KB (651 words) - 19:42, 7 October 2023
• 2002 AIME II Problems/Problem 12
  ...be the ratio of shots made to shots attempted after <math>n</math> shots. The probability that <math>a_{10} = .4</math> and <math>a_n\le.4</math> for all ...umber of such paths using a Pascal's Triangle type method below, computing the number of paths to each point that only move right and up.
  7 KB (1,127 words) - 13:34, 19 June 2022
• Geometry/Olympiad
  * [[Orthic triangle]] **[[Pascal's Theorem]]
  2 KB (242 words) - 10:16, 18 June 2023
• Combinatorics/Introduction
  ...comes more advanced, the scope of combinatorics grows, leading students to the [[Combinatorics/Intermediate | intermediate]] level. ** [[Pascal's triangle]]
  2 KB (159 words) - 21:39, 6 October 2023
• Power's of 2 in pascal's triangle
  == Pascal's Triangle == ...Triangle is a triangular array of numbers where each number is the sum of the two numbers above it. It Looks something like this:
  2 KB (341 words) - 16:57, 16 June 2019
• 2007 AIME II Problems/Problem 10
  ...[relatively prime]]. Find <math>m+n+r.</math> (The set <math>S-A</math> is the set of all elements of <math>S</math> which are not in <math>A.</math>) **<math>A</math> must have either 0, 6, or 1, 5 elements. The total probability is <math>\frac{2}{64} + \frac{2}{64} = \frac{4}{64}</math
  8 KB (1,367 words) - 11:48, 23 October 2022
• 2007 AIME II Problems/Problem 13
  ...f <math>0</math>'s and <math>1</math>'s in the bottom row is the number in the top square a [[multiple]] of <math>3</math>? Label each of the bottom squares as <math>x_0, x_1 \ldots x_9, x_{10}</math>.
  3 KB (600 words) - 11:10, 22 January 2023
• Pascal Triangle Related Problems
  ==The Triangle== Here are lines zero through eight of Pascal's triangle:
  4 KB (513 words) - 20:18, 3 January 2023
• Pascal's Theorem
  </asy>|right|A diagram of the theorem}} ...]] is inscribed in a [[conic section]], then the points of intersection of the pairs of its opposite sides are collinear:
  4 KB (712 words) - 21:57, 25 April 2020
• Triangular number
  ...riangular numbers''' are the numbers <math>T_n</math> which are the sum of the first <math>n</math> [[natural number]]s from <math>1</math> to <math>n</ma The <math>n^{th}</math> triangular number is the sum of all natural numbers from one to n.
  2 KB (275 words) - 08:39, 7 July 2021
• Pascal's Identity
  '''Pascal's Identity''' is a useful theorem of [[combinatorics]] dealing with [[combi ...dentity is also known as Pascal's Rule, Pascal's Formula, and occasionally Pascal's Theorem.
  3 KB (496 words) - 15:43, 11 April 2024
• 2008 Mock ARML 2 Problems
  ...>. Given that <math>m\angle{ABC} + m\angle{BCD} = 270^{\circ}</math>, find the area of <math>ABCD</math>. ...rs, each of which is less than <math>30</math>, is <math>2397</math>, find the sum of all positive integers with exactly three proper divisors, each of wh
  4 KB (582 words) - 21:57, 8 May 2019
• 1995 AHSME Problems/Problem 27
  ...sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown. ...h>f(n)</math> denote the sum of the numbers in row <math>n</math>. What is the remainder when <math>f(100)</math> is divided by 100?
  5 KB (682 words) - 09:45, 18 February 2022
• 2008 iTest Problems
  ...s. Michael is their oldest child, and Wendy their oldest daughter. Tony is the youngest child. Twins Joshua and Alexis are <math>12</math> years old. When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo.
  71 KB (11,749 words) - 01:31, 2 November 2023
• Sierpinski triangle
  ...angle, named after [[Waclaw Sierpinki]], is formed by parity of [[Pascal's triangle]], coloring a cell black if odd and white if even.
  257 bytes (37 words) - 17:08, 9 March 2015
• 2009 USAMO Problems/Problem 5
  ...ath> again at points <math>P</math> and <math>Q</math>, respectively. Let the line through <math>G</math> parallel to <math>\overline{AB}</math> intersec \end{align*}</cmath> depending on the configuration.
  6 KB (973 words) - 19:24, 18 October 2018
• Binomial
  A '''binominal''' is a polynominal with two terms, the sum of two monominals. *The binomial <math>a^2-b^2</math> can be [[factoring|factored]] as a product of
  605 bytes (98 words) - 16:39, 30 June 2021
• 2008 iTest Problems/Problem 77
  ...tion, Wendy looks for something to do. She starts working on a project for the math team. ...e are sixteen students, including Wendy, who are about to be sophomores on the math team. Elected as a math team officer,
  5 KB (788 words) - 05:15, 28 January 2019
• 2011 AMC 10B Problems/Problem 23
  What is the hundreds digit of <math>2011^{2011}?</math> To compute this, we use a clever application of the [[binomial theorem]].
  9 KB (1,287 words) - 20:37, 20 August 2023
• 1993 AHSME Problems/Problem 22
  ...ich it rests. Find the smallest possible number which could be assigned to the top block. ...gate down the block pyramid, its sort of like a three dimensional Pascal's triangle.
  7 KB (780 words) - 08:19, 27 June 2021
• 1971 AHSME Problems
  The number of digits in the number <math>N=2^{12}\times 5^8</math> is ...bricks, then the number of days it will take <math>c</math> men working at the same rate to lay <math>b</math> bricks, is
  18 KB (2,703 words) - 20:50, 11 September 2023
• 2014 IMO Problems/Problem 4
  ...math>BM</math> and <math>CN</math> intersect on the circumcircle of <math>\triangle{ABC}</math>. ...d <math>CN</math>, call it point <math>D</math>, is on the circumcircle of triangle <math>ABC</math>. In other words, we are trying to prove <math>\angle {BDC}
  13 KB (1,801 words) - 07:29, 1 May 2024
• 1986 AHSME Problems/Problem 23
  ...rime factorized as <math>2^5\cdot5^5\cdot7^5</math>. Finally, we can count the number of factors of this number.
  843 bytes (116 words) - 18:52, 1 April 2018
• 2016 AMC 12A Problems/Problem 19
  ...hen he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process ...to <math>8</math> heads, we are guaranteed to hit <math>4</math> heads, so the sum here is <math>\binom{8}{2}+\binom{8}{1}+\binom{8}{0}=28+8+1=37</math>.
  5 KB (881 words) - 13:34, 7 October 2023
• 2006 iTest Problems
  Find the number of positive integral divisors of 2006. Find the harmonic mean of 10 and 20.
  31 KB (4,811 words) - 00:02, 4 November 2023
• 2016 USAMO Problems/Problem 3
  Let <math>\triangle ABC</math> be an acute triangle, and let <math>I_B, I_C,</math> and <math>O</math> denote its <math>B</math This problem can be proved in the following two steps.
  6 KB (998 words) - 21:36, 17 October 2022
• 2016 AMC 8 Problems/Problem 15
  What is the largest power of <math>2</math> that is a divisor of <math>13^4 - 11^4</mat ...he right-hand side because it doesn't contain any factor of 2, we see that the greatest power of <math>2</math> that is a divisor <math>13^4 - 11^4</math>
  3 KB (481 words) - 03:02, 2 March 2024
• 2017 USAJMO Problems/Problem 2
  Consider the equation ...e infinitely many pairs <math>(x,y)</math> of positive integers satisfying the equation.
  10 KB (1,657 words) - 19:08, 24 April 2023
• Bill's Triangle
  ...top two. It was found by Bill9000. (If anyone finds someone who found this triangle before, please tell Bill9000.) ==How to Make Bill's Triangle==
  2 KB (307 words) - 20:25, 12 December 2017
• 2018 IMO Problems
  ...lines <math>DE</math> and <math>FG</math> are either parallel or they are the same line. ...numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from <math>1</math> to <math>10
  3 KB (544 words) - 08:34, 27 May 2023
• 2018 IMO Problems/Problem 3
  ...numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from <math>1</math> to <math>10 Does there exist an anti-Pascal triangle with <math>2018</math> rows which contains every integer from <math>1</math
  4 KB (626 words) - 01:45, 19 November 2023
• 2008 iTest Problems/Problem 60
  Consider the Harmonic Table ...emainder when the sum of the reciprocals of the <math>2007</math> terms on the <math>2007^\text{th}</math> row gets divided by <math>2008</math>.
  3 KB (527 words) - 01:29, 2 November 2023
• 2008 iTest Problems/Problem 78
  Feeling excited over her successful explorations into Pascal's Triangle, Wendy formulates a second problem to use during a future Jupiter Falls Hig ''How many of the first 2010 rows of Pascal's Triangle (Rows 0 through 2009) have exactly 256 odd entries?''
  2 KB (234 words) - 15:09, 1 August 2021
• A choose b
  Here is the formula for a choose b: <math>\binom{a}{b}=\frac{a!}{b!(a-b)!}</math>. This a choose b counts the number of ways you can pick b things from a set of a things. For example <m
  3 KB (407 words) - 20:03, 25 July 2021
• Pascal's Triangle and its patterns and properties
  = Pascal's Identity = Pascal's Identity states that
  9 KB (1,531 words) - 15:22, 17 June 2019
• Pascal's Triangle and Pythagorean Theorem
  = Pascal's Identity = Pascal's Identity states that
  15 KB (2,425 words) - 09:25, 13 February 2020
• What is the definition of Pure Mathematics?
  ...icle is not finished. Everyone is welcomed to edit, BUT ONLY IN GOOD WAYS! The AoPS Secret Governemtn has a backup of this page. = What is the definition of Pure Mathematics? =
  35 KB (5,882 words) - 18:08, 28 June 2021
• 2021 JMPSC Accuracy Problems/Problem 14
  What is the leftmost digit of the product <cmath>\underbrace{161616 \cdots 16}_{100 \text{ digits }} \times \ We conclude that the leftmost digit must be <math>\boxed{4}</math>.
  2 KB (209 words) - 17:25, 11 July 2021
• 2006 iTest Problems/Problem 10
  ...number of elements in the first <math>64</math> rows of Pascal's Triangle that are divisible by <math>4</math>. ...d\mathrm{(H)}\,1134\quad\mathrm{(I)}\,1256\quad\mathrm{(J)}\,\text{none of the above}</math>
  4 KB (612 words) - 23:33, 3 November 2023
• 2020 USOJMO Problems/Problem 4
  ...ersection of <math>AE</math> and <math>(ABCD)</math> and <math>H</math> be the intersection of <math>DF</math> and <math>(ABCD)</math>. By Pascal's on <math>GDCBAH</math>, we see that the intersection of <math>GH</math> and <math>BC</math>, <math>E</math>, and <m
  6 KB (1,131 words) - 19:15, 6 October 2023
• 2023 USAMO Problems/Problem 6
  ...wo distinct points <math>D</math> and <math>F</math>. If <math>E</math> is the intersection of lines <math>DF</math> and <math>BC</math>, prove that <math ...verline{II_A}</math> and <math>\overline{BC}</math> is <math>P</math>, and the intersection of lines <math>\overline{I_BI_C}</math> and <math>\overline{BC
  14 KB (2,600 words) - 23:37, 10 March 2024
• 2023 AMC 12A Problems/Problem 20
  ...e previous row. What is the units digits of the sum of the 2023 numbers in the 2023rd row? First, let <math>R(n)</math> be the sum of the <math>n</math>th row. Now, with some observation and math instinct, we can
  9 KB (1,414 words) - 09:11, 22 February 2024
• Mathboy282
  The older edits (2019-2022): [[2020 AIME II Problems/Problem 13]] Solution 8 (The same circle)
  10 KB (1,118 words) - 05:33, 13 January 2024
• Sharygin Olympiads, the best
  ...h school students. It consists of two rounds – correspondence and final. The correspondence round lasts 3 months. ...d and the serial number of the problem. Solutions are often different from the original ones.
  29 KB (4,997 words) - 18:06, 16 May 2024