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2004 AIME I Problems
The digits of a positive integer <math> n </math> are four consecutive integers in decreasi ...he absolute value of the difference between the greatest element of <math> A </math> and the greatest element of <math> B </math> is 99. Find <math> m.

9 KB (1,434 words) - 13:34, 29 December 2021
2004 AIME II Problems/Problem 14
Consider a string of <math> n </math> <math> 7 </math>'s, <math> 7777\cdots77, </math> ...</math>. Then the question is asking for the number of values of <math>n = a + 2b + 3c</math>.

11 KB (1,857 words) - 21:55, 19 June 2023
2004 AIME II Problems/Problem 12
...k </math> and <math> p </math> are [[relatively prime]]. Find <math> k+m+n+p. </math> pair A=(0,0), B=(6,0), D=(1, 24^.5), C=(5,D.y), O = (3,(r^2 + 6*r)^.5);

3 KB (431 words) - 23:21, 4 July 2013
2004 AIME II Problems/Problem 10
...> and <math> q </math> are relatively prime positive integers. Find <math> p+q. </math> A positive integer <math>n</math> has exactly two 1s in its binary representa

8 KB (1,283 words) - 19:19, 8 May 2024
2004 AIME II Problems
...a, b, c, d, e, </math> and <math> f </math> are positive integers, <math> a </math> and <math> e </math> are relatively prime, and neither <math> c </m A jar has 10 red candies and 10 blue candies. Terry picks two candies at rand

9 KB (1,410 words) - 05:05, 20 February 2019
1983 AIME Problems
...th> and <math>z</math> all exceed <math>1</math> and let <math>w</math> be a positive number such that <math>\log_xw=24</math>, <math>\log_y w = 40</mat ...ue taken by <math>f(x)</math> for <math>x</math> in the [[interval]] <math>p \leq x\leq15</math>.

7 KB (1,104 words) - 03:13, 27 May 2024
1984 AIME Problems
...of <math>\triangle ABC</math> such that when lines are drawn through <math>P</math> parallel to the sides of <math>\triangle ABC</math>, the resulting s pair A=(0,0),B=(12,0),C=(4,5);

6 KB (933 words) - 01:15, 19 June 2022
1986 AIME Problems
The pages of a book are numbered <math>1_{}^{}</math> through <math>n_{}^{}</math>. When t ...r point <math>P</math> is then drawn, and segments are drawn through <math>P</math> parallel to the sides of the triangle. If these three segments are o

5 KB (847 words) - 15:48, 21 August 2023
1987 AIME Problems
...er we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the p ...r which <math>[a,b] = 1000</math>, <math>[b,c] = 2000</math>, and <math>[c,a] = 2000</math>.

6 KB (869 words) - 15:34, 22 August 2023
1988 AIME Problems
...</math>, let <math>f_n(k) = f_1(f_{n - 1}(k))</math>. Find <math>f_{1988}(11)</math>. Let <math>\frac{m}{n}</math>, in lowest terms, be the probability that a randomly chosen positive divisor of <math>10^{99}</math> is an integer mult

6 KB (902 words) - 08:57, 19 June 2021
1989 AIME Problems
Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be dra ...ose <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
...neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. ...}</math> be a regular <math>r~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that ea

6 KB (870 words) - 10:14, 19 June 2021
1991 AIME Problems
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerat Suppose <math>r^{}_{}</math> is a real number for which

7 KB (1,106 words) - 22:05, 7 June 2021
1992 AIME Problems
A positive integer is called ascending if, in its decimal representation, the ...she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math>0.500</math>. During the weekend, s

8 KB (1,117 words) - 05:32, 11 November 2023
1993 AIME Problems
...n for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went :(a) the winner caught <math>15</math> fish;

8 KB (1,275 words) - 06:55, 2 September 2021
1994 AIME Problems
...\,</math> consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence ...circle of radius 20. Square <math>ABCD\,</math> is constructed with <math>A\,</math> and <math>B\,</math> on the larger circle, <math>\overline{CD}\,</

7 KB (1,141 words) - 07:37, 7 September 2018
1995 AIME Problems
...object reaches <math>(2,2)</math> in six or fewer steps. Given that <math>p</math> can be written in the form <math>m/n,</math> where <math>m</math> an ...us <math>9</math>. The circle of radius <math>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the le

6 KB (1,000 words) - 00:25, 27 March 2024
1996 AIME Problems
...mn, or diagonal is the same value. The figure shows four of the entries of a magic square. Find <math>x</math>. ...at <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer?

6 KB (931 words) - 17:49, 21 December 2018
1997 AIME Problems
...he two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah sho ...ny of which could be zero. It is desired to approximate <math>r</math> by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The

7 KB (1,098 words) - 17:08, 25 June 2020
1998 AIME Problems
...Extend <math>\overline{DA}</math> through <math>A</math> to a point <math>P,</math> and let <math>\overline{PC}</math> meet <math>\overline{AB}</math> ...egative]] term encountered. What positive integer <math>x</math> produces a sequence of maximum length?

7 KB (1,084 words) - 02:01, 28 November 2023

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