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- How many ordered triples of integers <math>(a,b,c)</math> satisfy <math> |a+b|+c = 19 </math ...ur initial assumption that <math>c\ge 0</math>. Similarly, for the second set of four solutions, we have <math>|a + b| = 78</math>, which leads to <math>4 KB (679 words) - 17:21, 26 July 2021
- We can set coordinates for the points. <math>D=(0,0), C=(6,0), B=(6,3),</math> and <m ...th>P</math>,<math>Q</math>, and <math>K</math> ordered from left to right. Set the values <math>BP=x</math>,<math>PK=x</math>,<math>BQ=4y</math> and <math8 KB (1,386 words) - 15:10, 8 October 2023
- ...rt{x^{2}+1}-\frac{1}{x+\sqrt{x^{2}+1}} </math> is a rational number is the set of all: ..., and 4 shapes (circle, hexagon, square, triangle). How many blocks in the set are different from the 'plastic medium red circle' in exactly 2 ways? (The15 KB (2,247 words) - 13:44, 19 February 2020
- ...or <math>18</math> people. If they shared, how many meals should they have ordered to have just enough food for the <math>12</math> of them? Set up the proportion <math>\frac{12\ \text{meals}}{18\ \text{people}}=\frac{x\700 bytes (106 words) - 00:54, 5 July 2013
- ...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 Consider all 1000-element subsets of the set <math> \{ 1, 2, 3, ... , 2015 \} </math>. From each such subset choose the10 KB (1,615 words) - 21:48, 13 January 2024
- ...es to the nine people so that exactly one person receives the type of meal ordered by that person. Let <math>B</math> be the set of all binary integers that can be written using exactly <math>5</math> zer10 KB (1,617 words) - 14:49, 2 June 2023
- Find the number of ordered pairs of positive integer solutions <math>(m, n)</math> to the equation <ma ...than <math>2</math> from all multiples of <math>p</math>. For example, the set of <math>10</math>-safe numbers is <math>\{ 3, 4, 5, 6, 7, 13, 14, 15, 16,7 KB (1,228 words) - 12:16, 13 March 2020
- Find the number of ordered pairs of positive integer solutions <math>(m, n)</math> to the equation <ma ...th>. So <math>20m=2000-12k</math>, where <math>k</math> is a member of the set <math>{0, 5, 10, 15...}</math>.3 KB (519 words) - 14:56, 2 June 2023
- ...that the bounding condition becomes <math>a_n \le n \cdot a_1.</math> Now set <math>b_i \equiv \frac{a_i}{a_1},</math> and since a triangle with sideleng ...base case, we see inductively that in general <math>\{s_n\}</math> is the set of the first <math>n</math> Fibonacci numbers. To show this note that if <m5 KB (930 words) - 02:17, 7 March 2018
- We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> ...> Show that for all integers <math>n\geq 3</math>, there exists a balanced set consisting of <math>n</math> points. </li>4 KB (773 words) - 08:14, 19 July 2016
- 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
