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• Algebra/Intermediate
  * Techniques for Solving Equations ** [[System of equations]]
  2 KB (198 words) - 17:47, 3 November 2021
• Arithmetic sequence
  ...y two consecutive terms is constant. This constant is called the '''common difference''' of the sequence. ...d <math>4, 12, 36, 108, \ldots</math> are not arithmetic sequences, as the difference between consecutive terms varies.
  4 KB (736 words) - 02:00, 7 March 2024
• Modular arithmetic/Introduction
  ...the integers modulo 5. In modulo 5, two integers are congruent when their difference is a [[multiple]] of 5. Adding the two equations we get:
  15 KB (2,396 words) - 20:24, 21 February 2024
• Modular arithmetic/Intermediate
  ...'' <math>n</math>, or <math>a \equiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. ...roblems, including finding solutions to [[Diophantine equation|Diophantine equations]], testing whether certain large numbers are prime, and even some problems
  14 KB (2,317 words) - 19:01, 29 October 2021
• 2003 AMC 12B Problems
  ....35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank?
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2022 USAJMO Problems/Problem 1
  ...and (3), we have <math>m | d-gr^{n+2}+gr^{n+1}</math>. Reinterpreting both equations, To begin, we let the common difference of <math>\{a_n\}</math> be <math>d</math> and the common ratio of <math>\{g
  4 KB (792 words) - 00:29, 13 April 2024
• 2004 AIME I Problems/Problem 2
  ...onsecutive integers whose sum is <math>m.</math> The absolute value of the difference between the greatest element of <math>A</math> and the greatest element of ...th>A</math> has half the number of elements as set <math>B</math>, and the difference between the greatest terms of the two two sequences is <math>99</math> (for
  8 KB (1,437 words) - 21:53, 19 May 2023
• 1984 AIME Problems/Problem 15
  Rewrite the system of equations as <cmath>\frac{x^{2}}{t-1}+\frac{y^{2}}{t-3^{2}}+\frac{z^{2}}{t-5^{2}}+\fr ...ls on each side are equal at <math>t=4,16,36,64</math>, we can express the difference of the two polynomials by a quartic polynomial that has roots at <math>t=4,
  6 KB (1,050 words) - 18:07, 16 January 2024
• 1988 AIME Problems/Problem 13
  ...h> Since the coefficient of <math>x</math> must be zero, this gives us two equations, <math>F_{16}b + F_{17}a = 0</math> and <math>F_{15}b + F_{16}a + 1 = 0</ma ...<math>\frac{ax^3+bx^2+1}{x^2-x-1}</math>, we get the following systems of equations:
  10 KB (1,585 words) - 03:58, 1 May 2023
• 1988 AIME Problems/Problem 6
  ..., 2b, 3b, 4b</math>. Our method will be to use the given numbers to set up equations to solve for <math>a</math> and <math>b</math>, and then calculate <math>(* ...ue of <math>148 - 3a</math>. On the third column from the left, the common difference is <math>103 - 2b</math>, so that square also has a value of <math>2b + 3(1
  5 KB (878 words) - 23:06, 20 November 2023
• 1992 AIME Problems/Problem 8
  The sequence <math>\Delta(\Delta A)</math> is the second finite difference sequence, and the first <math>k-1</math> terms of this sequence can be comp Adding the above <math>k-1</math> equations we find that
  5 KB (778 words) - 21:36, 3 December 2022
• 1997 AIME Problems/Problem 14
  The [[Trigonometric identities|cosine difference identity]] simplifies that to ...ered at <math>(0,0)</math> with radius <math>\sqrt{2+\sqrt{3}}</math>. The equations of these circles are <math>(x-1)^2 = 1</math> and <math>x^2 + y^2 = 2 + \sq
  5 KB (874 words) - 22:30, 1 April 2022
• 2003 AIME I Problems/Problem 8
  Denote the first term as <math>a</math>, and the common difference between the first three terms as <math>d</math>. The four numbers thus are ...th>400+x^2=y^2</math>, where <math>y</math> is an integer. Factoring using difference of squares, we have
  5 KB (921 words) - 23:21, 22 January 2023
• 2003 AIME II Problems/Problem 8
  ...>f(3)=1848</math>. Plugging in the values for x gives us a system of three equations: ...er to find <math>f(8)</math> add <math>f(3)</math> enough times to get the difference between the <math>d_1d_2</math> and <math>ad_2+bd_1</math> terms, then add
  5 KB (793 words) - 15:18, 14 July 2023
• 2000 AIME II Problems/Problem 13
  Now, if we let <math>z = y + \frac{1}{y}</math>, we can get the equations By the difference of cubes formula, <math>2(1-y^3)=2(1-y)(1+y+y^2)</math>, so we have two cas
  6 KB (1,060 words) - 17:36, 26 April 2024
• 2000 AIME II Problems/Problem 11
  ...implies that <math>a^2 + b^2 = 1^2 + 7^2 = 50</math>. Combining these two equations yields ...y-coordinates of <math>C</math> and <math>D'</math> are, respectively, the difference between the x-coordinates and the y-coordinates of <math>A</math> and <math
  4 KB (750 words) - 22:55, 5 February 2024
• 2006 AMC 12A Problems/Problem 8
  ...lways read the problem VERY carefully before attempting; it could mean the difference of making the cutoff. ...irst term say <math>a</math>. Since the numbers are consecutive the common difference <math>d = 1</math>.
  3 KB (450 words) - 02:00, 13 January 2024
• 2000 AMC 12 Problems/Problem 8
  ...math>f(1) = 5</math>, and <math>f(2) = 13</math>, we get a system of three equations in three variables: Plugging in <math>c=1</math> into the last two equations gives
  7 KB (988 words) - 15:14, 10 April 2024
• 2007 iTest Problems
  ...years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages? Find the value of <math>c</math> such that the system of equations
  33 KB (5,177 words) - 21:05, 4 February 2023
• 2005 AMC 10A Problems
  ...th> of his bill and Joe tipped <math>20\%</math> of his bill. What was the difference, in dollars between their bills? The equations <math> 2x + 7 = 3 </math> and <math> bx - 10 = -2 </math> have the same sol
  14 KB (2,026 words) - 11:45, 12 July 2021

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