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- Let the ordered triple <math>(a,b,c)</math> denote that <math>a</math> songs are liked by A ...e number of ways to rearrange <math>AB, BJ, AJ</math>, and a song from the set <math>\{N, A, B, J, AB, BJ, AJ\}</math>.8 KB (1,462 words) - 11:33, 21 April 2024
- Let <math>S</math> be the set of the first <math>2n</math> positive integers, and let <math>R</math> be a ...ath> for some <math>4 \leq k \leq n</math> such that there exists no other set of four consecutive integers in the permutation with sum greater than <math15 KB (2,452 words) - 03:03, 4 July 2020
- ...nd sunrise were correct, but the sunset was wrong. When did the sun really set? Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice e13 KB (1,835 words) - 08:51, 8 March 2024
- Let <math>S</math> be the set of positive integers <math>n</math> for which <math>\tfrac{1}{n}</math> has ...at either <math>0 < a - b \le 9</math> or <math>b - a > 9</math>. How many ordered triples <math>(x,y,z)</math> of elements of <math>S</math> have the propert14 KB (2,206 words) - 19:31, 15 May 2024
- ...at either <math>0 < a - b \le 9</math> or <math>b - a > 9</math>. How many ordered triples <math>(x,y,z)</math> of elements of <math>S</math> have the propert Consider the 1st set of conditions for <math>x, y, z</math>. We get that there are5 KB (885 words) - 10:14, 29 October 2023
- For <math>b_1,b_2<6</math>, there is a total of 7 ordered pairs that satisfy the condition. Namely, Since we know that <math>a_i<5</math>, for each of the ordered pairs <math>(b_1,b_2)</math>, there is respectively one and only one soluti10 KB (1,623 words) - 15:44, 31 August 2022
- Let <math>N</math> be the number of ordered triples <math>(A,B,C)</math> of integers satisfying the conditions (d) each ordered triple <math>(A,B,C)</math> and each ordered triple <math>(b,a,c)</math> form arithmetic sequences. Find <math>N</math>.4 KB (661 words) - 23:14, 26 May 2023
- ...ts in the plane, no three of which lie on the same line. At most how many ordered triples of points <math>(A,B,C)</math> in <math>R</math> exist such that <m ...tit{n-unsound}</math> rationals. The sum of all the elements in the union set <math>S_2\cup S_3\cup\cdots\cup S_{14}</math> is <math>\frac mn</math>, whe7 KB (1,173 words) - 21:04, 7 December 2018
- Let the number of ordered tuples of positive odd integers <math> \left( x_1, x_2, \cdots, x_{42} \rig Define a ''T-Polyomino'' to be a set of 4 cells in a grid that form a T, as shown below. Dai wants to place T-Po8 KB (1,336 words) - 09:10, 30 May 2020
- ...lled the origin. Each point in the coordinate plane can be specified by an ordered pair of numbers, (x,y). The cordinate system is organized to 4 quadrants. I ...xis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the pe4 KB (591 words) - 01:23, 25 January 2016
- ...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> hav987 bytes (146 words) - 11:52, 4 February 2016
- 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
- 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 t16 KB (2,291 words) - 13:45, 19 February 2020
- ...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
- 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 list6 KB (982 words) - 15:11, 22 October 2018
- ...} 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
- ...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 eve13 KB (2,117 words) - 12:33, 24 August 2023
- <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
- ...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
- ...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 t8 KB (1,396 words) - 07:17, 31 March 2023
