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2010 AMC 12B Problems/Problem 15
...s less than <math>20</math> are there exactly two distinct elements in the set <math>\{i^x, (1+i)^y, z\}</math>, where <math>i=\sqrt{-1}</math>? ...ath>i^{x}=(1+i)^{y}\neq z</math>, we have <math>5\cdot 1\cdot 19=95</math> ordered triples.

2 KB (360 words) - 17:29, 26 May 2023
1993 AHSME Problems
How many ordered pairs <math>(m,n)</math> of positive integers are solutions to Let <math>S</math> be the set of points on the rays forming the sides of a <math>120^{\circ}</math> angle

20 KB (2,814 words) - 08:15, 27 June 2021
2021 AIME II Problems/Problem 6
.../math> denote the number of elements in <math>S</math>. Find the number of ordered pairs <math>(A,B)</math> such that <math>A</math> and <math>B</math> are (n ...<math>X</math>, <math>Y</math>, <math>Z</math>, <math>W</math> is an empty set.

8 KB (1,364 words) - 01:02, 29 January 2024
2011 AIME II Problems/Problem 14
...e <math>(i, j, k)</math>. The <math>n</math>th position is defined by this ordered triple where <math>i</math> is <math>n \mod 2</math>, <math>j</math> is <ma The ordered triple (or position) in which 1 can be placed has 2 options for i, 3 for j,

10 KB (1,581 words) - 16:55, 27 May 2024
2011 AIME II Problems/Problem 13
...degrees is rather friendly in terms of ordered-pair representation! We can set <math>A=(0, 12)</math>, <math>B=(12,12)</math>, <math>C=(12, 0)</math>, <ma

13 KB (2,055 words) - 05:25, 9 September 2022
List
...ts''' in [[Python]] are used to store multiple objects in a single ordered set. Unlike [[tuple]]s, lists are mutable, so entries can be added, removed, o ...ents of a list can be changed with a simple reassignment. For example, to set the i'th element of myList to x, you would use

3 KB (470 words) - 12:27, 9 September 2021
Sequence (Python)
In [[Python]], '''sequence''' is the generic term for an ordered set. There are several types of sequences in Python, the following three are t

3 KB (443 words) - 16:53, 19 February 2024
Tuple
...ples''' in [[Python]] are used to store multiple items in a single ordered set. However, unlike [[list]]s, tuples are ''immutable'' - they can't be chang

1 KB (196 words) - 17:18, 19 April 2011
2011 USAMO Problems/Problem 6
Let <math>A</math> be a set with <math>|A| = 225</math>, meaning that <math>A</math> has 225 elements. ...9 elements in common. Since <math>\textstyle \binom{10}2 = 45,</math> each set is in 45 triples and thus will have 45 elements. We can now throw in 60 mor

7 KB (1,209 words) - 12:50, 25 August 2023
1997 AJHSME Problems/Problem 14
.... What is the difference between the largest and smallest integers in the set? When these numbers are ordered in ascending order, 5, the median, falls right in the middle, which is the

949 bytes (150 words) - 22:29, 17 December 2020
1997 AHSME Problems/Problem 28
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
2016 AMC 10A Problems/Problem 19
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 <math

8 KB (1,386 words) - 15:10, 8 October 2023
1989 AHSME Problems
...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? (The

15 KB (2,247 words) - 13:44, 19 February 2020
2004 AMC 8 Problems/Problem 3
...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
2015 AIME I Problems
...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 the

10 KB (1,615 words) - 21:48, 13 January 2024
2012 AIME I Problems
...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> zer

10 KB (1,617 words) - 14:49, 2 June 2023
2012 AIME II Problems
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
2012 AIME II Problems/Problem 1
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
2012 USAMO Problems/Problem 1
...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 <m

5 KB (930 words) - 02:17, 7 March 2018
2015 IMO Problems/Problem 1
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

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