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2004 AIME II Problems/Problem 9
...ression. Let <math> a_n </math> be the greatest term in this sequence that is less than <math>1000</math>. Find <math> n+a_n. </math> ...th>. This happens with <math>f(7)f(8) = 29 \cdot 33 = 957</math>, and this is the <math>2(8) = 16</math>th term of the sequence.

3 KB (537 words) - 18:03, 25 May 2024
2004 AIME II Problems/Problem 8
...the prime factorization of <math>2004^{2004}</math> is <math>2^{4008}\cdot 3^{2004}\cdot 167^{2004}</math>. ...ample, the number of divisors of <math>2004=2^2\cdot 3^1\cdot 167^1</math> is <math>(2+1)(1+1)(1+1)=12</math>.

2 KB (353 words) - 18:08, 25 November 2023
2004 AIME II Problems/Problem 7
...th> CF=3 </math> are given. The perimeter of rectangle <math> ABCD </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relativ pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C;

9 KB (1,501 words) - 05:34, 30 October 2023
2004 AIME II Problems/Problem 6
...t the end of the process are in the [[ratio]] <math> 3: 2: 1, </math>what is the least possible total for the number of bananas? ...c{11}{24}b_3</math>, and the third monkey got <math>\frac{1}{8}b_1 + \frac{3}{8}b_2 + \frac{1}{12}b_3</math>.

6 KB (950 words) - 14:18, 15 January 2024
2004 AIME II Problems/Problem 5
...urther behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the <math>800</math> worke ...0}{800}(60)=\frac{150}{8}</math>. The train then has <math>60-15-\frac{50}{3}-\frac{150}{8}=230/24</math> minutes left to travel 250 miles, and doing th

4 KB (592 words) - 19:02, 26 September 2020
2004 AIME II Problems/Problem 4
...ath>n</math>-digit number, for a total of <math>(2^1 - 2) + (2^2 - 2) + (2^3 -2) + (2^4 - 2) = 22</math> such numbers (or we can list them: <math>AB, BA ...s we can form, for a total of <math>(2^0 - 1) + (2^1 - 1) + (2^2 - 1) + (2^3 - 1) = 11</math> such numbers (or we can list them: <math>A0, A00, A0A, AA0

3 KB (508 words) - 01:16, 19 January 2024
2004 AIME II Problems/Problem 3
...gruent]] 1-cm [[cube (geometry) | cube]]s [[face]] to face. When the block is viewed so that three of its faces are visible, exactly <math>231</math> of ...s close together as possible, which occurs when the smaller block is <math>3 \times 7 \times 11</math>. Then the extra layer makes the entire block <ma

2 KB (377 words) - 11:53, 10 March 2014
2004 AIME II Problems/Problem 2
...ability]] that they get the same color combination, irrespective of order, is <math> m/n, </math> where <math> m </math> and <math> n </math> are [[relat ...c{28}{153}</math>. So the probability that they both pick two red candies is <math>\frac{9}{38} \cdot \frac{28}{153} = \frac{14}{323}</math>. The same

2 KB (330 words) - 13:42, 1 January 2015
2004 AIME II Problems/Problem 1
...y [[prime]]. Find the [[remainder]] when the product <math> abcdef </math> is divided by 1000. .../math>; the rest of the area of the circle is then equal to <math>\frac{2}{3}r^2\pi</math>.

2 KB (329 words) - 23:20, 4 July 2013
2004 AIME II Problems
...re of any prime. Find the remainder when the product <math> abcdef </math> is divided by 1000. ...obability that they get the same color combination, irrespective of order, is <math> m/n, </math> where <math> m </math> and <math> n </math> are relativ

9 KB (1,410 words) - 05:05, 20 February 2019
1983 AIME Problems
== Problem 3 == What is the product of the real roots of the equation <math>x^2 + 18x + 30 = 2 \sqr

7 KB (1,104 words) - 03:13, 27 May 2024
2019 AIME I Problems/Problem 2
...9, 20</math> distinct from <math>J</math>. The value of <math>B - J</math> is at least <math>2</math> with a probability that can be expressed in the for ...because <math>B \ne J</math>, so the probability that <math>B-J < 0</math> is <math>\frac{1}{2}</math> by symmetry.

5 KB (830 words) - 22:15, 28 December 2023
1984 AIME Problems
..._{98}</math> if <math>a_1</math>, <math>a_2</math>, <math>a_3\ldots</math> is an arithmetic progression with common difference 1, and <math>a_1+a_2+a_3+\ ...sitive multiple of <math>15</math> such that every digit of <math>n</math> is either <math>8</math> or <math>0</math>. Compute <math>\frac{n}{15}</math>.

6 KB (933 words) - 01:15, 19 June 2022
1986 AIME Problems
What is the sum of the solutions to the equation <math>\sqrt[4]{x} = \frac{12}{7 - == Problem 3 ==

5 KB (847 words) - 15:48, 21 August 2023
1987 AIME Problems
An ordered pair <math>(m,n)</math> of non-negative integers is called "simple" if the addition <math>m+n</math> in base <math>10</math> re What is the largest possible distance between two points, one on the sphere of radi

6 KB (869 words) - 15:34, 22 August 2023
1988 AIME Problems
...rder -- the correct five buttons. The sample shown below has <math>\{1, 2, 3, 6, 9\}</math> as its combination. Suppose that these locks are redesigned == Problem 3 ==

6 KB (902 words) - 08:57, 19 June 2021
1989 AIME Problems
== Problem 3 == Suppose <math>n_{}^{}</math> is a positive integer and <math>d_{}^{}</math> is a single digit in base 10. Find <math>n_{}^{}</math> if

7 KB (1,045 words) - 20:47, 14 December 2023
1990 AIME Problems
The [[increasing sequence]] <math>2,3,5,6,7,10,11,\ldots</math> consists of all [[positive integer]]s that are ne Find the value of <math>(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}</math>.

6 KB (870 words) - 10:14, 19 June 2021
1991 AIME Problems
...overline {AB}</math> of length 4 and <math>\overline {CB}</math> of length 3. Divide <math>\overline {AB}</math> into 168 congruent segments with points == Problem 3 ==

7 KB (1,106 words) - 22:05, 7 June 2021
1992 AIME Problems
...n its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are == Problem 3 ==

8 KB (1,117 words) - 05:32, 11 November 2023

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