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2005 USAMO Problems/Problem 4
...<math>n</math>, where <math>n</math> is a positive integer. For how many ordered 4-tuples <math>(k_1, k_2, k_3, k_4)</math> of nonnegative integers can we ...math>C_i</math>, subtract <math>1</math> from each of the cuts to obtain a set of cuts that is counted in <math>C_{i-1}</math>. For example, if <math>\{2

7 KB (1,276 words) - 20:51, 6 January 2024
Well-Ordering theorem
...red set is a [[totally ordered set]] <math>(S,\prec)</math> for which each set <math>A\subseteq S</math> has a [[minimum]] element. [[Category:Set theory]]

381 bytes (59 words) - 12:40, 2 June 2019
Permutation
...re <math>r!</math> (the [[factorial]] of <math>r</math>) permutations of a set with <math>r</math> distinct objects. ...sider permutations of [[infinite]] sets. In this case, a permutation of a set <math>S</math> is simply a [[bijection]] between <math>S</math> and itself.

3 KB (422 words) - 11:01, 25 December 2020
2006 AIME I Problems
Let set <math> \mathcal{A} </math> be a 90-element subset of <math> \{1,2,3,\ldots, Let <math> \mathcal{S} </math> be the set of real numbers that can be represented as repeating decimals of the form <

7 KB (1,173 words) - 03:31, 4 January 2023
2005 AMC 12A Problems
How many ordered triples of [[integer]]s <math>(a,b,c)</math>, with <math>a \ge 2</math>, <m ...nct numbers <math>a</math> and <math>b</math> are chosen randomly from the set <math>\{ 2, 2^2, 2^3, \ldots, 2^{25} \}</math>. What is the probability tha

13 KB (1,971 words) - 13:03, 19 February 2020
2004 AMC 12A Problems
For how many ordered pairs of positive integers <math>(x,y)</math> is <math>x+2y=100</math>? Let <math>S</math> be the set of points <math>(a,b)</math> in the coordinate plane, where each of <math>a

13 KB (1,953 words) - 00:31, 26 January 2023
2002 AMC 12A Problems
...th>\{1, 2, 3, 4, 5\}</math>, and Sergio randomly selects a number from the set <math>\{1, 2, \ldots, 10\}</math>. What is the probability that Sergio's nu ...nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?

12 KB (1,792 words) - 13:06, 19 February 2020
2005 AMC 12B Problems
...<math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set Let <math>S</math> be the set of ordered triples <math>(x,y,z)</math> of real numbers for which

12 KB (1,781 words) - 12:38, 14 July 2022
2006 AMC 12A Problems/Problem 25
How many non-[[empty set | empty]] [[subset]]s <math>S</math> of <math>\{1,2,3,\ldots ,15\}</math> h ...to choose <math>k</math> elements from an ordered <math>n</math> element [[set]] without choosing two consecutive members?

8 KB (1,405 words) - 11:52, 27 September 2022
2005 AMC 12B Problems/Problem 23
Let <math>S</math> be the set of ordered triples <math>(x,y,z)</math> of real numbers for which There are real numbers <math>a</math> and <math>b</math> such that for all ordered triples <math>(x,y,z)</math> in <math>S</math> we have <math>x^{3}+y^{3}=a

5 KB (786 words) - 11:36, 19 May 2024
Set
...satisfy. These axioms are chosen to agree with our intuitive concept of a set, on one hand, and to allow various, sometimes quite sophisticated, mathemat ...t are called the [[element]]s of the set. A common misconception is that a set can have multiple indistinct elements, such as the following: <math>\{1,4,5

11 KB (2,021 words) - 00:00, 17 July 2011
2004 AIME I Problems/Problem 12
Let <math> S </math> be the set of [[ordered pair]]s <math> (x, y) </math> such that <math> 0 < x \le 1, 0<y\le 1, </mat

2 KB (303 words) - 22:28, 11 September 2020
2004 AIME I Problems
...<math> m </math> consecutive integers whose sum is <math> 2m, </math> and set <math> B </math> consists of <math> 2m </math> consecutive integers whose s ...are on adjacent sides of the square. The midpoints of the line segments in set <math> S </math> enclose a region whose area to the nearest hundredth is <m

9 KB (1,434 words) - 13:34, 29 December 2021
1988 AIME Problems
The function <math>f</math>, defined on the set of ordered pairs of positive integers, satisfies the following properties:

6 KB (902 words) - 08:57, 19 June 2021
1993 AIME Problems
How many ordered four-tuples of integers <math>(a,b,c,d)\,</math> with <math>0 < a < b < c < .../math>, are then drawn randomly and without replacement from the remaining set of <math>997</math> numbers. Let <math>p</math> be the probability that, af

8 KB (1,275 words) - 06:55, 2 September 2021
1994 AIME Problems
For certain ordered pairs <math>(a,b)\,</math> of real numbers, the system of equations ...lution is an ordered pair <math>(x,y)\,</math> of integers. How many such ordered pairs <math>(a,b)\,</math> are there?

7 KB (1,141 words) - 07:37, 7 September 2018
1998 AIME Problems
Find the number of [[ordered pair]]s <math>(x,y)</math> of positive integers that satisfy <math>x \le 2y Let <math>n</math> be the number of ordered quadruples <math>(x_1,x_2,x_3,x_4)</math> of positive odd [[integer]]s that

7 KB (1,084 words) - 02:01, 28 November 2023
1999 AIME Problems
There is a set of 1000 switches, each of which has four positions, called <math>A, B, C</m Let <math>\mathcal{T}</math> be the set of ordered triples <math>(x,y,z)</math> of nonnegative real numbers that lie in the pl

7 KB (1,094 words) - 13:39, 16 August 2020
2000 AIME I Problems
For how many ordered pairs <math>(x,y)</math> of integers is it true that <math>0 < x < y < 10^{ ...ays and at <math>14</math> miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area

7 KB (1,204 words) - 03:40, 4 January 2023
2002 AIME I Problems
...e digits of Dick's age. Let <math>d</math> be Dick's present age. How many ordered pairs of positive integers <math>(d,n)</math> are possible? ...ct squares in the plane of the dodecagon have at least two vertices in the set <math>\{A_1,A_2,A_3,\ldots,A_{12}\}</math>?

8 KB (1,374 words) - 21:09, 27 July 2023

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