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1985 AIME Problems/Problem 12
For simplicity purposes, we set <math>c=\frac14,</math> which gives <cmath>Q(k)=-\frac13Q(k-1).</cmath> Thus, our desired number of paths is equivalent to the number of ordered septuples of positive integers <math>(b_1, b_2, \ldots, b_7)</math> such th

19 KB (3,128 words) - 00:53, 22 June 2024
1985 AIME Problems/Problem 10
...a</math> in the range <math>0<k<1000</math>, or <math>49\cdot12=588</math> ordered pairs <math>(a,b)</math>. If <math>a=0</math>, <math>b\neq0</math>, which includes <math>11</math> ordered pairs.

12 KB (1,859 words) - 18:16, 28 March 2022
1988 AIME Problems/Problem 8
The function <math>f</math>, defined on the set of ordered pairs of positive integers, satisfies the following properties:

4 KB (538 words) - 13:24, 12 October 2021
1993 AIME Problems/Problem 7
.../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 There is a total of <math>P(1000,6)</math> possible ordered <math>6</math>-tuples <math>(a_1,a_2,a_3,b_1,b_2,b_3).</math>

5 KB (772 words) - 09:04, 7 January 2022
1997 AIME Problems/Problem 10
...deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the followin ...\binom{27}{2} = 27*13 = 351</math> possibilities. Note, however, that each set is generated by <math>{3\choose 2} = 3</math> pairs, so we've overcounted b

3 KB (585 words) - 19:37, 25 April 2022
1998 AIME Problems/Problem 15
...for any <math>i</math> and <math>j</math>. Let <math>D_{40}</math> be the set of all dominos whose coordinates are no larger than 40. Find the length of We can draw a comparison between the domino a set of 40 points (labeled 1 through 40) in which every point is connected with

9 KB (1,659 words) - 18:35, 20 June 2024
1998 AIME Problems/Problem 13
If <math>\{a_1,a_2,a_3,\ldots,a_n\}</math> is a [[set]] of [[real numbers]], indexed so that <math>a_1 < a_2 < a_3 < \cdots < a_n ...all possible subsets of <math>\{1,2,\ldots,8\}</math>. Since the sets are ordered, a <math>9</math> must go at the end; hence we can just append a <math>9</m

2 KB (384 words) - 14:47, 14 June 2024
1998 AIME Problems/Problem 7
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 ...however note that the quadruples all need to be odd. This motivates us to set <math>x_i= 2y_i +1</math>, as for all integers <math>y_i</math>, <math>2y_i

5 KB (684 words) - 18:52, 19 June 2024
1999 AIME Problems/Problem 8
Let <math>\mathcal{T}</math> be the set of ordered triples <math>(x,y,z)</math> of nonnegative [[real number]]s that lie in th

3 KB (445 words) - 19:40, 4 July 2013
2003 AIME I Problems/Problem 9
Call the number <math>\overline{abcd}</math>. Then <math>a+b=c+d</math>. Set <math>a+b=x</math>. ...\leq k \leq 18</math>, we notice that there are <math>(18 - k) + 1</math> ordered pairs with a sum of <math>k</math>.

4 KB (696 words) - 11:55, 10 September 2023
2002 AIME II Problems/Problem 9
Let <math>\mathcal{S}</math> be the [[set]] <math>\lbrace1,2,3,\ldots,10\rbrace</math> Let <math>n</math> be the numb Thus, there are <math>3^{10}-2\cdot2^{10}+1</math> ordered pairs of sets <math>(A,B)</math>. But since the question asks for the numbe

3 KB (404 words) - 23:07, 4 May 2024
Ring
...ra]], similar to a [[group]] or a [[field]]. A ring <math>R</math> is a [[set]] of elements closed under two [[operation]]s, usually called multiplicatio ...es lead to confusion when <math>R</math> is also an [[ordered set]].) The set of invertible elements of <math>R</math> constitute a group under multiplic

6 KB (994 words) - 06:16, 8 April 2015
2005 Canadian MO Problems
Let <math>S</math> be a set of <math>n\ge 3</math> points in the interior of a circle. Let's say that an ordered triple of positive integers <math>(a,b,c)</math> is <math>n</math>-''powerf

2 KB (436 words) - 11:45, 26 December 2018
System of equations
A '''system of equations''' is a set of [[equation]]s which share the same [[variable]]s. Below is an example o Find the ordered pair <math>(x,y)</math> for which

5 KB (784 words) - 23:27, 30 July 2020
2007 iTest Problems
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahe

30 KB (4,794 words) - 23:00, 8 May 2024
Least upper bound
Given a [[subset]] <math>S</math> in some larger [[partially ordered set]] <math>R</math>, a '''least upper bound''' or '''supremum''', for <math>S< ...th>S</math> is said to be '''complete''' if any [[empty set | nonempty]] [[set|subset]] of <math>S</math> that is [[bounded]] above has a supremum.

1,011 bytes (177 words) - 14:08, 5 March 2022
2004 AMC 10A Problems
...t of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the ...y collection of condiments. How many different kinds of hamburgers can be ordered?

15 KB (2,092 words) - 20:32, 15 April 2024
Ordered pair
...in which we have <math>\{2, 3\} = \{3, 2\}</math>. In general, we say two ordered pairs, <math>(x, y)</math> and <math>(a, b)</math> are the same if and only ...notion of an ordered pair can be naturally extended to that of an [[tuple|ordered tuple]].

1 KB (179 words) - 20:40, 28 February 2020
2020 AMC 10A Problems
...ed quadruple of not necessarily distinct integers, each one of them in the set <math>\{0,1,2,3\}.</math> For how many such quadruples is it true that <mat

13 KB (1,968 words) - 18:32, 29 February 2024
Greatest lower bound
Given a [[subset]] <math>S</math> in some larger [[partially ordered set]] <math>R</math>, a '''greatest lower bound''' or '''infimum''' for <math>S

597 bytes (109 words) - 13:55, 5 March 2022

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