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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
• Mock AIME I 2015 Problems
  ...be consecutive terms (in that order) in an arithmetic sequence with common difference <math>d</math>. Suppose <math>\cos b</math> and <math>\cos d</math> are ro ...h>, and <math>\gamma</math> are complex numbers that satisfy the system of equations <cmath>\begin{align*}\alpha+\beta+\gamma&=6,\\\alpha^3+\beta^3+\gamma^3&=87
  9 KB (1,463 words) - 14:48, 12 February 2017
• 1967 AHSME Problems/Problem 10
  ...oefficients of <math>y</math> and the constant terms, we get the system of equations: Thus, the difference is:
  1 KB (212 words) - 19:53, 28 September 2023
• 1981 AHSME Problems
  If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is ...be the largest positive real number which satisfies at least one of these equations. Then
  17 KB (2,633 words) - 15:44, 16 September 2023
• 2014 AMC 10A Problems/Problem 18
  ...of <math>A</math> and <math>B</math> is also <math>4</math>. Note that the difference in <math>y</math> value of <math>A</math> and <math>B</math> is <math>1</ma Now we have 3 equations with 3 variables:
  5 KB (791 words) - 03:18, 20 June 2022
• 1960 AHSME Problems
  Applied to a bill for <math>\textdollar{10,000}</math> the difference between a discount of <math>40</math>% and two successive discounts of <mat The pair of equations <math>3^{x+y}=81</math> and <math>81^{x-y}=3</math> has:
  21 KB (3,242 words) - 21:27, 30 December 2020
• 2014 AMC 12B Problems/Problem 16
  ...h>1</math> and <math>-1</math> for <math>x</math>, we obtain the following equations: Adding these two equations together, we get
  2 KB (331 words) - 21:57, 13 March 2023
• 2014 AIME I Problems/Problem 4
  Checking, we get that the common difference in Jon's speed and trains' speeds is <math>\frac{440}{9}</math> and the dif Adding both of the equations together, we get that
  5 KB (784 words) - 13:59, 30 November 2021
• 2014 AIME I Problems/Problem 6
  Begin by setting <math>x</math> to 0, then set both equations to <math>h^2=\frac{2013-j}{3}</math> and <math>h^2=\frac{2014-k}{2}</math>, ...ve integer <math>h</math> which has positive integer x-intercepts for both equations.
  7 KB (1,158 words) - 20:50, 8 December 2021
• 1973 AHSME Problems
  ...ers. For such representations of the even number 126, the largest possible difference between the two primes is Of the three equations
  18 KB (2,788 words) - 13:55, 20 February 2020
• 1970 AHSME Problems/Problem 15
  ...e <math>-1</math> and <math>2</math>. Similarly, the y-coordinates have a difference of <math>6</math>, so the trisections happen at <math>3</math> and <math>1< ...ints on it. Plugging in <math>(x, y) = (3, 4)</math> into all five of the equations works. The point <math>(2, 1)</math> doesn't work in any of the five lines
  1 KB (226 words) - 22:26, 13 July 2019
• 1982 AHSME Problems
  the difference between the first digit and the last digit is <math>2</math>? ...h>z</math>. Let <math>x</math> and <math>y</math> satisfy the simultaneous equations
  17 KB (2,500 words) - 19:05, 11 September 2023
• 1978 AHSME Problems
  ...pairs <math>(x,y)</math> of real numbers satisfying both of the following equations: What is the difference between the two smallest such integers?
  15 KB (2,432 words) - 01:06, 22 February 2024
• 1977 AHSME Problems
  ...rs turned up can be arranged to form an arithmetic progression with common difference one? For how many values of the coefficient a do the equations <cmath>\begin{align*}x^2+ax+1=0 \\ x^2-x-a=0\end{align*}</cmath> have a com
  15 KB (2,412 words) - 05:09, 27 November 2020
• 1975 AHSME Problems
  For which real values of m are the simultaneous equations <cmath>\text{I. The difference of the roots is odd.} \\
  16 KB (2,512 words) - 04:48, 27 November 2021
• 1957 AHSME Problems
  \textbf{(D)}\ \text{the difference of the digits}\qquad Given the system of equations
  26 KB (3,950 words) - 21:09, 31 August 2020
• 2015 AMC 10A Problems/Problem 16
  Note that we can add the two equations to yield the equation We can also subtract the two equations to yield the equation
  7 KB (1,197 words) - 11:49, 5 February 2024
• 2015 AIME I Problems/Problem 1
  ...in an alternating pattern between successive integers. Find the positive difference between integers <math>A</math> and <math>B</math>. ...o recognize the above, we may use Newton's Little Formula to semi-bash the equations.
  2 KB (282 words) - 00:26, 9 January 2023
• 2015 AIME I Problems/Problem 10
  ...2</math>, and <math>f(2)=-12</math>. This provides the following system of equations. Using any four of these functions as a system of equations yields <math>d = |f(0)| = \boxed{072}</math>
  8 KB (1,474 words) - 10:00, 10 November 2023
• 2015 AIME II Problems/Problem 15
  ...nal distance between the centers of the circles is <math>4+1=5</math>. The difference in heights is <math>4-1=3</math>. So <math>BC=\sqrt{5^2-3^2}=4</math>. ...ale to <math>\triangle{AEC}</math>, their area ratio is 16. Divide the two equations for the two areas, we have
  31 KB (5,086 words) - 19:15, 20 December 2023
• Algebraic number theory
  ...and [[field|fields]]. In doing so, many questions concerning [[Diophantine equations]] are resolved, including the celebrated [[quadratic reciprocity]] theorem. The sum, difference, product, and quotient of any two algebraic numbers is itself an [[algebrai
  10 KB (1,646 words) - 15:04, 28 May 2020
• 2016 AIME II Problems/Problem 1
  ...r^{2}-25-(ar-9)=ar(r-1)-16</math>. We now have, letting, subtracting the 2 equations, <math>ar^{2}+-2ar+a=12</math>, so we have <math>3ar=432,</math> or <math>a For the common difference, <math>ak=5-(a-5)=ak^2-25-(ak-9)</math>. Simplifying, <math>k^2-2k+1=\frac{
  5 KB (788 words) - 02:50, 1 March 2024
• 2016 AMC 12A Problems/Problem 15
  ...two radii. <math>QQ' - PP' = 1</math> and <math>RR' - QQ' = 1</math>, the difference of the radii. Using pythagorean theorem, we find that <math>P'Q'</math> and ...a^2 + b^2 = 8 \\ ab = 2\sqrt{3} \end{cases}</cmath> Solving the system of equations, we get <math>a = \sqrt{6}</math> and <math>b = \sqrt{2}</math>. Alternativ
  8 KB (1,255 words) - 09:05, 5 September 2022
• 2016 AIME I Problems/Problem 11
  Since <math>P(x)</math> is a polynomial, the <math>k</math>th difference is constant, where <math>k=\deg(P(x))</math>. Thus we can list out the 0th, Since the 3rd difference of <math>P(x)</math> is constant, we can conclude that <math>\deg(P(x))=3</
  8 KB (1,415 words) - 14:00, 22 December 2021
• 2016 AIME I Problems/Problem 5
  ...s constant and is an integer, <math>d</math> must be a factor of the total difference, which is <math>374-319=55</math>. Also note that the number of pages Anh ...>. We then solve for <math>n</math> and <math>t</math> in their respective equations, getting <math>2n+10=68</math>. <math>n=29</math> We also get <math>2t+10
  5 KB (818 words) - 01:25, 10 January 2024
• 2016 AIME II Problems/Problem 9
  The common difference is <math>100-r - 1</math>, and so we can equate: <math>2(99-r)+100-r=1000-r .../math>, and <math>1, b, b^2, \ldots</math>. We can now write the given two equations as the following:
  6 KB (983 words) - 01:18, 2 February 2023
• 2017 AMC 10A Problems/Problem 24
  ...for each corresponding term (knowing that they must be equal), we have the equations: ...o <math>(1-a)g(x)</math>. Equating the coefficients, we get <math>3</math> equations. We will tackle the situation one equation at a time, starting the <math>x<
  10 KB (1,708 words) - 23:16, 7 October 2023
• 2017 AMC 10A Problems/Problem 14
  ...e cost of his soda, while the cost of his soda was <math>5\%</math> of the difference between <math>A</math> and the cost of his movie ticket. To the nearest who We can create two equations:
  3 KB (438 words) - 15:54, 4 July 2023
• 2017 AMC 12A Problems/Problem 23
  ...for each corresponding term (knowing that they must be equal), we have the equations: ...o <math>(1-a)g(x)</math>. Equating the coefficients, we get <math>3</math> equations. We will tackle the situation one equation at a time, starting the <math>x<
  10 KB (1,861 words) - 10:47, 17 October 2021
• 2017 AIME II Problems/Problem 6
  ...<math>139</math>. The two values of <math>n</math> that satisfy one of the equations are <math>168</math> and <math>27</math>. Summing these together gives us t ...n is larger than <math>n</math> itself. Let <math>x</math> be the positive difference between that result and <math>n</math>, so that <math>\sqrt{n^2+85n+2017}=n
  7 KB (1,096 words) - 21:03, 12 March 2021
• 1981 AHSME Problems/Problem 4
  If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is: ...ath>y.</math> We can use the information given in the problem to write two equations:
  754 bytes (126 words) - 15:48, 28 January 2021
• 2018 AMC 12B Problems/Problem 17
  We then make sure we consider fractions with higher positive difference between the denominator and numerator. And we also do not forget that the n ...the mediant <math>\frac{9}{16}</math> is between the two fractions, with a difference of <math>\boxed{\textbf{(A) } 7}.</math> Suppose that the answer was not <m
  11 KB (1,937 words) - 00:18, 23 October 2023
• 2018 AMC 10B Problems/Problem 20
  From <math>(5),</math> we have the following system of <math>336</math> equations: We add these equations up to get <cmath>f(2018)-f(2)=6\cdot336=2016,</cmath> from which <math>f(20
  9 KB (1,490 words) - 02:11, 11 September 2023
• 1961 AHSME Problems/Problem 26
  ...first term of the [[arithmetic sequence]] be <math>a</math> and the common difference be <math>d</math>. Substituting in values results in this [[system of equations]].
  1 KB (210 words) - 11:38, 22 May 2018
• 1963 AHSME Problems/Problem 16
  Let <math>d</math> be the common difference of the [[arithmetic sequence]], so <math>a = b-d</math> and <math>c = b+d</ Cross-multiply in both equations to get a [[system of equations]].
  1 KB (242 words) - 13:05, 5 June 2018
• 2018 AIME I Problems/Problem 2
  We have these equations: Taking the first two equations we see that <math>29a+14c=13b</math>. Combining the two gives <math>a=4, b=
  3 KB (510 words) - 18:36, 21 February 2024
• 2018 AIME II Problems/Problem 12
  ...)^2\end{align}</cmath>Eliminating <math>\cos\theta</math> in the above two equations and solving for <math>\cos\phi</math> we get<cmath>\cos\phi = \frac{3}{5}\q Hence <math>AP=CP</math> (note that <math>BP=DP</math> makes no difference here).
  18 KB (2,912 words) - 13:12, 24 January 2024
• 2018 AIME II Problems/Problem 13
  ...ight)^3</math>. We can call this value <math>x</math>, to keep our further equations looking clean. .... Since they sum to 1, that means the odds probability will be half of the difference above one-half. Subbing in our earlier result from the intermediate step, t
  11 KB (1,860 words) - 13:12, 24 January 2024
• 2004 Indonesia MO Problems/Problem 5
  Given a system of equations: The difference between the two above equations is
  1 KB (171 words) - 18:14, 28 July 2018
• 2008 iTest Problems/Problem 68
  ...t terms of a grouping <math>(1,5,12,22 \cdots)</math> have the same second difference, so the series of numbers can be modeled by a [[quadratic]] function. ...last term in a group with <math>n</math> terms. We can write a system of equations to find a quadratic function.
  3 KB (443 words) - 13:00, 11 August 2018
• 2019 AMC 12A Problems/Problem 25
  ...from <math>(2)</math> gives <math>pq(q-1)=-360</math>. Dividing these two equations gives <math>q+1=-1</math>, so <math>q=-2</math>. Substituting back, we get ...e that the problem seems quite complicated, but since it is an AMC 12, the difference between the largest angle of <math>\triangle A_nB_nC_n</math> and <math>60^
  5 KB (933 words) - 22:23, 2 January 2024
• 2019 AMC 10A Problems/Problem 7
  ...be <math>s = \sqrt{2} + \sqrt{20}</math>, so the area reduces nicely to a difference of squares, making it <math>\boxed{\textbf{(C) }6}</math>.
  7 KB (1,079 words) - 22:24, 10 November 2023
• 2019 AMC 10B Problems/Problem 4
  ...to deduce <cmath>y = 2</cmath> and plug this into one of the previous line equations. We get <cmath>x+4 = 3 \Rightarrow x=-1</cmath> Thus the common point is <m ...math>, and <math>c</math> form an arithmetic progression, so if the common difference is <math>d</math>, we can say <math>a,b,c = a, a+d, a+2d.</math> Now we hav
  4 KB (597 words) - 10:24, 24 June 2023
• 2019 AMC 10B Problems/Problem 11
  ...2, and the amount of green marbles in jar 2, respectively. We now have the equations, ...t. Substituting and dividing, we find <math>x = 5</math>. Thus to find the difference of the blue marbles we must do
  3 KB (575 words) - 21:15, 18 October 2023
• 2019 IMO Problems/Problem 1
  ...sfies the above for any '''integral''' constant c, and that this family of equations is unique. ...(-x-k)}{k},</cmath> which means that <math>f</math> is linear. (Functional equations don't work like that unfortunately)
  4 KB (680 words) - 01:42, 4 May 2024
• 2020 AMC 10B Problems/Problem 23
  ...tion <math>R</math> and rotation <math>L</math> cancel each other out, the difference between the numbers of them define the final position. The probability of t Equations <math>(2)</math>, <math>(3)</math>, <math>(4)</math> are equivalent. Here I
  10 KB (1,653 words) - 23:33, 3 August 2023
• 2020 AMC 10B Problems/Problem 14
  Equating the equations, we have ...triangle and the shaded area. The area of one of these mini-sectors is the difference between a <math>60^{\circ}</math> sector of the semicircle and the equilate
  17 KB (2,392 words) - 12:36, 24 December 2023
• 1982 AHSME Problems/Problem 24
  So, we have a system of equations: ...).</math> Substituting this into #1, we can now solve for x, and we have a difference of squares, or <math>100-x^2/4=78.</math> This yields <math>x^2/4=22,</math
  3 KB (449 words) - 16:58, 11 October 2020
• 2024 AIME II Problems
  ...ath>z</math> be positive real numbers that satisfy the following system of equations: ...tangent to <math>S</math> along a circle with radius <math>r_o</math>. The difference <math>r_i-r_o</math> can be written as <math>\frac{m}{n}</math>, where <mat
  8 KB (1,236 words) - 23:11, 12 March 2024
• 1955 AHSME Problems/Problem 19
  Two numbers whose sum is <math>6</math> and the absolute value of whose difference is <math>8</math> are roots of the equation: The first two hints can be expressed as the following system of equations:
  1 KB (207 words) - 11:11, 12 July 2021
• 2023 AIME I Problems/Problem 11
  ...h (1,2), because 1 is still unusable as it is consecutive with 2. The only difference is we now have only 4-10 to work with. Using the same pattern as before, we ==Solution 3 (Double Recursive Equations)==
  15 KB (2,414 words) - 06:57, 26 November 2023
• 2023 AIME II Problems
  ...th>x,y,</math> and <math>z</math> be real numbers satisfying the system of equations ...<math>2 \times 6</math> grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by <math>3.</math> One
  8 KB (1,370 words) - 21:34, 28 January 2024
• 2021 Fall AMC 10A Problems/Problem 17
  ...the heights at pillar <math>C</math> and pillar <math>D</math> is half the difference between the heights at <math>B</math> and <math>E,</math> so Solving these equations, we get <math>h_E = \boxed{\textbf{(D) } 17}</math>.
  10 KB (1,705 words) - 00:13, 17 July 2023
• 1972 AHSME Problems/Problem 16
  Let <math>a</math> be first and <math>b</math> be second. We can then get equations based on our knowledge: <math>b-a = 9-b</math> and <math>b/a = a/3</math>. for some common difference <math>d</math> and common ratio <math>r</math>. We can use these to obtain
  2 KB (370 words) - 13:44, 4 April 2024
• 2020 AMC 8 Problems/Problem 25
  ...he side lengths have a sum of <math>3322</math> or <math>2020</math> and a difference of <math>S_2</math>, the answer must be <math>\dfrac{3322 - 2020}{2} = \dfr ...h>r+s</math>. Then, <math>r+s+s+r+s=3322</math>. Now, we have 2 systems of equations.
  4 KB (606 words) - 13:22, 1 January 2024
• Quartic Equation
  These types of equations are extremely hard to solve; however, there are very clever methods for sol ==Solving Quartic Equations==
  13 KB (2,376 words) - 16:56, 19 February 2024
• 2021 AIME II Problems/Problem 15
  ...root is equal to the perfect square, <math>m^2</math>. Thus, after using a difference of squares, we have ..., which yields <math>n=\boxed{258}</math>, which, if plugged in to for our equations of <math>f(n)</math> and <math>g(n)</math>, will yield the desired ratio, a
  14 KB (2,569 words) - 09:28, 28 March 2024
• Jadhav Theorem
  ...icable for any 3 terms of an Arithmetic Progression with a constant common difference between them. This theorem is derived by Jyotiraditya Jadhav. ...the first and the third term (ac) will always be the square of the common difference (d).
  2 KB (346 words) - 05:52, 1 April 2021
• 2022 AMC 10B Problems
  ...ence between a pair of primes is equal to <math>2</math>, and the positive difference between the cubes of the two primes is <math>31106</math>. What is the sum ...the first <math>n</math> terms of an arithmetic sequence that has a common difference of <math>2</math>. The quotient <math>\frac{S_{3n}}{S_n}</math> does not de
  15 KB (2,224 words) - 13:10, 20 February 2024
• 2021 Fall AMC 12A Problems/Problem 24
  Let <math>k</math> be the common difference of the arithmetic progression of the side-lengths. It follows that <math>b, ...f these sides form an arithmetic sequence, we have the following system of equations:
  9 KB (1,500 words) - 01:18, 29 August 2022
• 2023 AIME I Problems/Problem 5
  ...{2}</math>, and therefore <math>b + d = a\sqrt{2}</math>. By squaring both equations, we obtain Thus, the equations <math>PA \cdot PC = 56</math> and <math>PB \cdot PD = 90</math> can be writ
  19 KB (3,107 words) - 23:31, 17 January 2024
• Chebyshev polynomials of the first kind
  ...) = \cos ny \cos y + \sin ny \sin y. \end{align*}</cmath> The sum of these equations is <cmath> \cos ((n+1)y) + \cos ((n-1)y) = 2 \cos ny \cos y;</cmath> rearra
  10 KB (1,919 words) - 15:24, 26 June 2023
• Chebyshev polynomials of the second kind
  ...- y) = \sin((n+1)y)\cos y - \cos((n+1)y)\sin y.</cmath> The sum of these equations is <cmath>\sin((n+2)y) + \sin ny = 2\sin((n+1)y)\cos y;</cmath> rearranging
  2 KB (392 words) - 22:12, 11 March 2022
• 2022 AMC 10A Problems/Problem 3
  ...h>40</math> less than the second number. What is the absolute value of the difference between the first and second numbers? ...ath> and the second number is <math>47.</math> Their absolute value of the difference is <math>|42-47|=\boxed{\textbf{(E) } 5}.</math>
  2 KB (333 words) - 11:35, 24 March 2024
• 2023 AMC 8 Problems/Problem 25
  ...h>241-20=221</math>, and the maximum–<math>250-13=237</math>. There is a difference of <math>13</math> between them, so only <math>17</math> and <math>18</math ...ge to satisfy this inequality. On the other hand, we can now find that the difference will be <math>17</math>, which satisfies this inequality.
  4 KB (580 words) - 14:19, 6 April 2024
• 2023 AMC 10A Problems/Problem 23
  ...is another pair of two integers that multiply to <math>n</math> but have a difference of 23, one integer must be greater than <math>a</math>, and one must be sma ...h>(2y+2x+43)</math> and <math>(2y-2x+3)</math> must be integer, we get two equations.
  8 KB (1,344 words) - 01:16, 3 March 2024
• 2023 AMC 12A Problems/Problem 6
  ...of <math>\overline{AB}</math> is <math>(6, 2)</math>. What is the positive difference between the <math>x</math>-coordinates of <math>A</math> and <math>B</math> ..._{2}(6+m)=2+n</math> and <math>\log_{2}(6-m)=2-n</math>. Now add these two equations to obtain <math>\log_{2}(6+m)+\log_{2}(6-m)=4</math>. By logarithm rules, w
  3 KB (530 words) - 08:37, 22 February 2024
• 2024 AIME I Problems/Problem 15
  ...this antioptimal box. (If the height and width weren't the same, the extra difference between them could be used to make the length longer.) Thus, let the width ...idden inside this equation and call this <math>m</math>. Now we have three equations:
  10 KB (1,554 words) - 22:26, 13 April 2024
• 2002 AMC 12P Problems/Problem 13
  Essentially, this boils down to writing <math>217</math> as a difference of squares. We know <math>217 = (7)(31)</math>, so we assume there exist po Solving this system of equations gives <math>a = 109</math> and <math>b = 108</math>. However, <math>108 > 1
  2 KB (305 words) - 20:35, 10 March 2024