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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

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