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• 1990 AHSME Problems/Problem 18
  ...99,100\}</math>, and then <math>b</math> is chosen at random from the same set. The probability that the integer <math>3^a+7^b</math> has units digit <mat ...h> each power's units digit is equally probable. There are <math>16</math> ordered pairs of units digits, and three of them <math>(1,7),(7,1),(9,9)</math> hav
  987 bytes (146 words) - 11:52, 4 February 2016
• 1969 AHSME Problems
  Let a binary operation <math>\star</math> on ordered pairs of integers be defined by <math>(a,b)\star (c,d)=(a-c,b+d)</math>. Th ...values satisfying the inequality <math>\frac{x^2-4}{x^2-1}>0</math> is the set of all <math>x</math> such that:
  16 KB (2,662 words) - 14:12, 20 February 2020
• 1987 AHSME Problems
  A student recorded the exact percentage frequency distribution for a set of measurements, as shown below. How many ordered triples <math>(a, b, c)</math> of non-zero real numbers have the property t
  16 KB (2,291 words) - 13:45, 19 February 2020
• 1976 AHSME Problems
  ...ximum number of points of intersection of pairs of lines from the complete set <math>\{L_1,L_2,\dots,L_{100}\}</math> is How many distinct ordered triples <math>(x,y,z)</math> satisfy the following equations?
  17 KB (2,835 words) - 14:36, 8 September 2021
• 2014 UMO Problems
  Completely describe the set of all right triangles with positive integer-valued legs such that when fou ...with ordered lists of integers in the following way. He starts out with an ordered list
  6 KB (982 words) - 15:11, 22 October 2018
• 2016 AMC 12A Problems
  ...} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordina How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(
  15 KB (2,348 words) - 17:20, 19 January 2024
• 2015 AMC 12A Problems
  ...y four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. ...<math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For eve
  13 KB (2,117 words) - 12:33, 24 August 2023
• 2015 AMC 12B Problems
  <math>\textbf{(A)}\; \text{the set of positive integers} \\ \textbf{(B)}\; \text{the set of composite positive integers} \\
  13 KB (2,064 words) - 13:39, 1 October 2022
• 2015 AIME I Problems/Problem 9
  ...math>(a_1,a_2,a_3)</math> with <math>1 \le a_1,a_2,a_3 \le 10</math>. Each ordered triple in <math>S</math> generates a sequence according to the rule <math>a ...h>|y-x|</math> or <math>|z-y|</math> is less than or equal to 1, then that ordered triple meets the criteria.
  6 KB (1,003 words) - 20:35, 28 July 2023
• 2016 AMC 12B Problems/Problem 16
  ...math>(\text{exg}~ (10,69) ~\text{or} ~(15,46))</math> we will have several ordered pairs <math>(c,d)</math> where <math>c<d</math> ...brief testing, we find that the the other 7 factor pairs produce 7 viable ordered pairs. This means we have found <math>\boxed{\textbf{(E)}\ 7}</math> ways t
  8 KB (1,396 words) - 07:17, 31 March 2023
• 2016 AMC 12B Problems
  A set of teams held a round-robin tournament in which every team played every oth There are exactly <math>77,000</math> ordered quadruplets <math>(a, b, c, d)</math> such that <math>\gcd(a, b, c, d) = 77
  14 KB (2,037 words) - 19:09, 29 July 2023
• 2016 AIME II Problems/Problem 12
  ...complement, we wish to count the number of colorings in which at least one set of adjacent sections are the same color. There are six possible sets of adj ...>, given the bug's starting number <math>k</math>, is simply the number of ordered quintuplets of positive integers <math>(a_1, a_2, a_3, a_4, a_5)</math> tha
  20 KB (3,327 words) - 16:16, 7 June 2024
• 2006 iTest Problems
  ...ns are identical) and where a group of students is treated as an unordered set of people. Compute the remainder obtained when <math>N</math> is divided by For each positive integer <math>n</math> let <math>S_n</math> denote the set of positive integers <math>k</math> such that <math>n^k-1</math> is divisib
  31 KB (4,811 words) - 00:02, 4 November 2023
• 2017 AMC 12A Problems/Problem 25
  ...ctly 8 angles from the set <math>\{z^2,z^6\}</math> and exactly 4 from the set <math>\{z^1,z^3,z^5,z^7\}</math>. If we found a good combination where we h ...h>2a + 6b</math> is even for all <math>a + b =8, </math> and the number of ordered quadruples <math>(a_1, a_2, a_3, a_4)</math> such that <math>a_i \in \{1, 3
  18 KB (2,878 words) - 01:47, 16 December 2023
• 2017 AMC 12A Problems
  Let <math>S</math> be the set of points <math>(x,y)</math> in the coordinate plane such that two of the t How many ordered pairs <math>(a,b)</math> such that <math>a</math> is a positive real number
  15 KB (2,418 words) - 16:58, 7 November 2022
• 2018 AMC 10A Problems
  ...ance in miles to the nearest town. Which of the following intervals is the set of all possible values of <math>d</math>? How many ordered pairs of real numbers <math>(x,y)</math> satisfy the following system of eq
  14 KB (2,171 words) - 21:10, 4 November 2023
• 2017 IMO Problems/Problem 6
  ...sor of <math>x</math> and <math>y</math> is <math>1</math>. Given a finite set <math>S</math> of primitive points, prove that there exist a positive integ
  582 bytes (105 words) - 03:09, 19 November 2023
• 2018 AMC 12A Problems
  ...ance in miles to the nearest town. Which of the following intervals is the set of all possible values of <math>d</math>? ...math> such that <math>m+10<n+1</math>, both the mean and the median of the set <math>\{m, m+4, m+10, n+1, n+2, 2n\}</math> are equal to <math>n</math>. Wh
  15 KB (2,380 words) - 18:52, 7 April 2022
• Basic Programming With Python
  In the print statement, we set <math>x</math> as 2, and the function returns <math>2 + 1 = 3,</math> so th ...tion() is said to be '''called''' in the final line of the code. Again, we set <math>x</math>, our parameter, as "print_function."
  33 KB (5,277 words) - 22:14, 3 June 2023
• 2017 IMO Problems
  Let <math>\mathbb{R}</math> be the set of real numbers , determine all functions ...sor of <math>x</math> and <math>y</math> is <math>1</math>. Given a finite set <math>S</math> of primitive points, prove that there exist a positive integ
  4 KB (720 words) - 12:12, 5 August 2021

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