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• Algebra/Intermediate
  * Techniques for Solving Equations ** [[System of equations]]
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• 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 (1,051 words) - 04:52, 8 May 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
  30 KB (4,794 words) - 23:00, 8 May 2024
• 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
• LaTeX:Layout
  ...phing rules in LaTeX is very important when using display math. Notice the difference in the following: ...st equations, or even to past pages. Rather than having to manually number equations then change your text if the equation labels change, or having to manually
  30 KB (5,171 words) - 10:16, 4 April 2021
• 2007 AMC 12A Problems/Problem 17
  ...ic identities#Pythagorean Identities|Pythagorean identities]]: square both equations and add them up: This is just the cosine difference identity, which simplifies to <math>\cos (a - b) = \frac{1}{3} \Longrightar
  1,022 bytes (153 words) - 14:56, 7 August 2017
• 2004 AMC 12A Problems/Problem 14
  Let <math>d</math> be the common difference. Then <math>9</math>, <math>9+d+2=11+d</math>, <math>9+2d+20=29+2d</math> a ...h>, <math>11+d</math>, and <math>29+2d</math>. Thus, we get the following equations:
  4 KB (689 words) - 03:35, 16 January 2023
• 2003 AMC 12B Problems/Problem 24
  ...the line where <math>a<x<b</math> has slope <math>-1</math>, the positive difference in <math>y</math>-coordinates from <math>x=a</math> to <math>x=b</math> mus
  7 KB (1,183 words) - 11:47, 15 February 2016
• 2008 AMC 12B Problems/Problem 25
  ...frac{24}8=3</math>. Now we can solve for <math>r</math> by adding the two equations we just got to see that <math>2r=11</math>, or <math>r=\frac{11}2</math>.
  12 KB (2,015 words) - 20:54, 9 October 2022
• 2008 AIME II Problems
  ...aximum possible value of <math>\dfrac{a}{b}</math> for which the system of equations :(i) <math>n^2</math> can be expressed as the difference of two consecutive cubes;
  7 KB (1,167 words) - 21:33, 12 August 2020
• 2008 AMC 10A Problems/Problem 15
  Substituting equations <math>(1)</math> and <math>(2)</math> into <math>(5)</math> gives: We are asked the difference between Jan's and Ian's distances, or
  6 KB (1,033 words) - 15:19, 1 July 2021
• Dynamical Systems
  ...es of dynamical systems include the [[logistic equation]] and the [[Lorenz equations]].
  789 bytes (107 words) - 21:52, 18 October 2017
• 2008 iTest Problems
  The difference between two prime numbers is <math>11</math>. Find their sum. ...s. One of the problems Joshua and Alexis work on boils down to a system of equations:
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2009 AMC 12A Problems/Problem 23
  ...about the quadratic <math>ax^2+bx+c</math> (<math>a>0</math>) that (i) the difference of the two quadratic roots equals to <math>\sqrt{\Delta}/a</math>, and (ii)
  5 KB (862 words) - 02:04, 1 April 2024
• 2010 USAMO Problems/Problem 3
  Given <math>a_1</math>, from the equations <math>a_ia_{i+1} = 2i+1,\; 1\le i\le 2n-1</math>, The same equations <math>a_ia_{i+1} = 2i+1</math> can be used to compute the
  11 KB (1,889 words) - 13:45, 4 July 2013
• 2011 AMC 12B Problems/Problem 21
  Squaring the first and second equations, <math>\frac{x^2 + 2xy + y^2}{4}=100 a^2 + 20 ab + b^2</math> Subtracting the previous two equations, <math>\frac{x^2 + 2xy + y^2}{4} - xy = \frac{x^2 - 2xy + y^2}{4} = \left(\
  3 KB (507 words) - 19:48, 4 November 2023
• 2011 AMC 10B Problems/Problem 21
  The largest difference, <math>9,</math> must be between <math>w</math> and <math>z.</math> ...ven as a possibility in the problem. This means <math>1</math> must be the difference between <math>y</math> and <math>x.</math> We can express the possible conf
  8 KB (1,303 words) - 20:29, 5 September 2022
• 2002 AMC 10B Problems/Problem 19
  ...lize that the two equations are 100 terms apart, so by subtracting the two equations in a form like... ...we get the value of the common difference of every hundred terms one hundred times. So we have to divide the answer b
  3 KB (472 words) - 14:56, 17 August 2023
• Trigonometric Equations
  It is probably good to know how to solve trigonometric equations, which often involved brute force and the use of trigonometric identities. When solving trigonometric equations, it probably doesn't get easier than this. Using the unit circle or a graph
  8 KB (1,351 words) - 20:30, 10 July 2016
• 1984 AHSME Problems/Problem 12
  <math> 2, 4, 8, 14, 22, .... </math>. We notice that the difference between succesive terms of the sequence are <math> 2, 4, 6, 8, .... </math> ...'''(2)''' and '''(2)''' from '''(3)''' yields the two-variable [[system of equations]]
  2 KB (325 words) - 18:10, 30 November 2013
• 2003 AMC 10B Problems/Problem 24
  The difference between consecutive terms is <math>(x-y)-(x+y)=-2y.</math> Therefore we can ...-\frac35.</math> Substituting the value for <math>y</math> into any of the equations, we get <math>x=-\frac98.</math> Finally,
  4 KB (779 words) - 16:16, 12 March 2024
• 1950 AHSME Problems
  The values of <math>y</math> which will satisfy the equations <cmath>\begin{array}{rcl} 2x^{2}+6x+5y+1&=&0\\ 2x+y+3&=&0 \end{array}</cmat ...qquad\\ \textbf{(D)}\ y^{2}+y-12=0\qquad \textbf{(E)}\ \text{None of these equations} </math>
  22 KB (3,306 words) - 19:50, 3 May 2023
• 2012 AMC 10A Problems/Problem 24
  Add the two equations. ...c)^2 = 9 \Rightarrow a-c = 3</math>, since <math>a-c</math> is the biggest difference. It is impossible to determine by inspection whether <math>a-b = 1</math> o
  2 KB (398 words) - 14:32, 5 December 2022
• 2012 AMC 10A Problems/Problem 17
  Using difference of cubes in the numerator and cancelling out one <math>(a-b)</math> in the An alternate method of solving the system of equations involves solving the second equation for <math>a</math>, by plugging it int
  6 KB (1,024 words) - 01:35, 1 October 2023
• Mock AIME II 2012 Problems/Problem 15
  ...>a_0</math>, and we are going to be taking two more differences to get the equations equal to <math>0</math>. These two more differences subtract the <math>b_1 ...th>5b_2+b_1+6=0</math> and <math>7b_2+b_1+12=0</math>. Subtract these two equations to give us <math>2b_2+6=0</math> or <math>b_2=-3</math>. Now, substitute t
  4 KB (660 words) - 15:55, 8 March 2015
• 1952 AHSME Problems
  The difference of the roots of <math> x^2-7x-9=0 </math> is: ...the sum of the first five terms, the ratio of the first term to the common difference is:
  23 KB (3,556 words) - 15:35, 30 December 2023
• 1954 AHSME Problems
  The solution of the equations ...of the larger circle}\\ \textbf{(D)}\ CB\sqrt{3}\\ \textbf{(E)}\ \text{the difference of the two radii} </math>
  23 KB (3,535 words) - 16:29, 24 April 2020
• 1955 AHSME Problems
  ...math>1: 3</math>. If the radius of the smaller is <math>r</math>, then the difference between the Two numbers whose sum is <math>6</math> and the absolute value of whose difference is <math>8</math> are roots of the equation:
  22 KB (3,509 words) - 21:29, 31 December 2023
• 2012 AMC 8 Problems/Problem 9
  ...number of four-legged mammals be <math>y</math>. We can now use systems of equations to solve this problem. Write two equations:
  2 KB (371 words) - 18:58, 15 April 2023
• 2013 AMC 10B Problems/Problem 19
  ...ve that <math>a=b+d</math> and that <math>c=b-d</math>. Plugging those two equations into <math>b^2=4ac</math>, we have <math>b^2=4(b^2-d^2)=4b^2-4d^2</math> wh Setting the two equations equal, we have <math>4ac=\frac{a^2+2ac+c^2}{4}</math>.
  5 KB (969 words) - 19:14, 15 August 2023
• 2013 AMC 12B Problems/Problem 17
  What is the difference between the maximum and minimum possible values of <math>c</math>? ...e then find the roots of <math>c</math> that satisfy equality and find the difference of the roots. This gives the answer, <math>\boxed{\textbf{(D)} \ \frac{16}{
  7 KB (1,225 words) - 14:59, 8 August 2021
• 2013 AIME I Problems/Problem 15
  ...ubling equation <math>(2)</math> would give <math>2B - 2a + 2y</math>. The difference between them would be <math>3y</math>. Since <math>p|\{(1), (2), (3)\}</mat The difference between <math>(4)</math> and <math>(5)</math> is <math>x+1</math>, which sh
  4 KB (661 words) - 23:14, 26 May 2023
• 2013 AIME I Problems/Problem 14
  This is a system of equations; rearrange and rewrite to get <cmath>P(1 + 2 \sin \theta) + 2Q \cos \theta ...t add <math>0.5i\sin\phi</math> to the numerator to make the denominator a difference (or rather a sum) of squares. The denominator does not matter. Only the num
  10 KB (1,641 words) - 20:03, 3 January 2024
• 2003 AMC 12B Problems/Problem 7
  ....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? ...he number of nickels, dimes, and quarters, respectively, we can set up two equations:
  1 KB (163 words) - 00:30, 5 January 2014

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