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• 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

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