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  • * Intermediate is recommended for students who can expect to pass the AMC 10/12. ...combinatorics, and number theory, along with sets of accompanying practice problems at the end of every section.
    23 KB (3,038 words) - 18:33, 15 February 2025
  • '''Fermat's Little Theorem''' is highly useful in [[number theory]] for simplifying the computation of exponents in [[modular arithmetic]] (w In contest problems, Fermat's Little Theorem is often used in conjunction with the [[Chinese Re
    15 KB (2,618 words) - 11:03, 19 February 2025
  • ==Group Theoretic Proof== ==Problems==
    4 KB (569 words) - 21:34, 30 December 2024
  • ...ntity given a name and usually denoted by a letter or symbol. Many contest problems test one's fluency with [[algebraic manipulation]]. ...tions of the normal operations seen arithmetic and high school algebra. [[Group]]s, [[ring]]s, [[field]]s, [[module]]s, and [[vector space]]s are common ob
    3 KB (360 words) - 08:31, 21 February 2025
  • ...e have plenty of synonyms for the word "set," like ''collection, ensemble, group'', etc., but those names really do not define the meaning of the word ''set ==Problems==
    11 KB (2,019 words) - 16:20, 7 July 2024
  • ...if and only if <math>n</math> is a [[perfect square]]. (Otherwise, we can group [[divisor]]s into pairs whose product is <math>n</math>.) Thus, <math>S(n) [[Category:Intermediate Number Theory Problems]]
    4 KB (647 words) - 01:29, 4 May 2021
  • ...are even except for <math>2^0</math>, so we begin with labeling an entire group "where one of the 1s is in the rightmost spot". In an equation, <math>2^n+1 ...th>4,2,1</math> in the first group and <math>-4,-2,-1</math> in the second group. For our number to be divisible by <math>9</math>, we have to get one of th
    8 KB (1,283 words) - 18:19, 8 May 2024
  • Clearly we need to find a group of numbers that multiply to 2004. We can list them all out since we know th [[Category:Intermediate Number Theory Problems]]
    2 KB (359 words) - 18:58, 24 December 2024
  • ...2+1)\times (38+1)</math> [[factor]]s by its [[prime factorization]]. If we group all of these factors (excluding <math>n</math>) into pairs that multiply to [[Category:Intermediate Number Theory Problems]]
    2 KB (407 words) - 07:14, 4 November 2022
  • Proof: This can be obtained directly from Turan's theorem in graph theory ...004</math>, and have each member of a group play every other member of the group. That yields <math>2 {1004 \choose 2} = 1007012</math> bouts.
    2 KB (256 words) - 04:32, 14 November 2024
  • A group of children held a grape-eating contest. When the contest was over, the win A group of <math>c</math> children held a grape-eating contest. When the contest wa
    4 KB (762 words) - 13:20, 29 December 2024
  • {{duplicate|[[2010 AMC 12A Problems|2010 AMC 12A #23]] and [[2010 AMC 10A Problems|2010 AMC 10A #24]]}} ...}25)</math>, and so is <math>6 \cdot 7 \cdot 8 \cdot 9</math>. How can we group terms to take advantage of this fact?
    10 KB (1,553 words) - 19:12, 14 October 2024
  • ...ple each person shakes hands with exactly two of the other people from the group. Let <math>N</math> be the number of ways this handshaking can occur. Consi ...erson shakes hands with two people, we can view all of these through graph theory as 'rings'. This will split it into four cases: Three rings of three, one
    7 KB (1,216 words) - 21:05, 29 November 2024
  • ...roblem. Therefore, I proved that you cannot use the Games theorem to solve problems. But using theorem 3.141592653589793238462643383 "2. Games can turn things ...oblems, which has around 73,000 bytes. EDIT: Gmaas has beaten 2008 most iT Problems! It is the longest AoPS Wiki article ever. Gmaas's article has 89,119 bytes
    69 KB (11,805 words) - 19:49, 18 December 2019
  • ...the others, we notice that <math>m=19</math> is the only solution in this group. Using the property that <math>ab\equiv(-a)(-b)\mod99</math>, it is clear t [[Category:Intermediate Number Theory Problems]]
    13 KB (2,039 words) - 08:52, 11 January 2025
  • ...text{th}</math> number is <math>a+n-1</math>. We know that the sum of the group of numbers is <math>100</math>, so [[Category:Intermediate Algebra Problems]]
    1 KB (185 words) - 13:02, 20 February 2020
  • ...rst term of the group with <math>n+1</math> terms and the last term of the group with <math>n</math> terms is <math>1.</math> This means that the differenc ...umber of terms in a group, and let <math>f(n)</math> be the last term in a group with <math>n</math> terms. We can write a system of equations to find a qu
    3 KB (443 words) - 12:00, 11 August 2018
  • ...s+(10^{321}-1)</math>. By the commutative and associative property, we can group it into <math>(10+10^2+10^3+\cdots+10^{321})-321</math>. We know the former [[Category:Intermediate Number Theory Problems]]
    3 KB (433 words) - 06:57, 9 February 2023
  • ...nine digits for the one repeat, which leaves eight distinct digits for the group of solo numbers. Thus, there are <math>105 \cdot 9 \cdot 8 \cdot 7 \cdot 6 ...math> ways to order the letters. The letter <math>B</math> represents the group of numbers that appear twice, which are already ordered in the above paragr
    5 KB (819 words) - 15:21, 20 December 2019
  • ...herefore, I proved that you cannot use the Almighty Gmaas theorem to solve problems. But using theorem 3.141592653589793238462643383 "2. Almighty Gmaas can tur ...ich has around 73,000 bytes. EDIT: Almighty Gmaas has beaten 2008 most iT Problems! It is the longest AoPS Wiki article ever. Almighty Gmaas's article has 89
    99 KB (14,096 words) - 22:49, 19 February 2025

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