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2004 AIME II Problems/Problem 1
...b, c, d, e, </math> and <math> f </math> are [[positive integer]]s, <math> a </math> and <math> e </math> are [[relatively prime]], and neither <math> c ...f the radius (since the chord [[bisect]]s it), and the radius. Thus, it is a <math>30^\circ</math> - <math>60^\circ</math> - <math>90^\circ</math> [[tri

2 KB (329 words) - 23:20, 4 July 2013
2004 AIME II Problems
{{AIME Problems|year=2004|n=II}} ...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

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 ...hat of <math>BC</math> is <math>2</math> cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B<

7 KB (1,104 words) - 03:13, 27 May 2024
2019 AIME I Problems/Problem 2
...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. ...2</math> elements are adjacent. Using the well-known formula <math>\dbinom{n-k+1}{k}</math>, there are <math>\dbinom{20-2+1}{2} = \dbinom{19}{2} = 171</

5 KB (830 words) - 22:15, 28 December 2023
1984 AIME Problems
...h>n</math> is either <math>8</math> or <math>0</math>. Compute <math>\frac{n}{15}</math>. A point <math>P</math> is chosen in the interior of <math>\triangle ABC</math

6 KB (933 words) - 01:15, 19 June 2022
1986 AIME Problems
...]] <math>n</math> for which <math>n^3+100</math> is [[divisible]] by <math>n+10</math>? The pages of a book are numbered <math>1_{}^{}</math> through <math>n_{}^{}</math>. When t

5 KB (847 words) - 15:48, 21 August 2023
1987 AIME Problems
.../math> of non-negative integers is called "simple" if the addition <math>m+n</math> in base <math>10</math> requires no carrying. Find the number of sim ...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

6 KB (869 words) - 15:34, 22 August 2023
1988 AIME Problems
...ts of <math>k</math>. For <math>n \ge 2</math>, let <math>f_n(k) = f_1(f_{n - 1}(k))</math>. Find <math>f_{1988}(11)</math>. Suppose that <math>|x_i| < 1</math> for <math>i = 1, 2, \dots, n</math>. Suppose further that

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
...etc. If the candidate went <math>\frac{n^{2}}{2}</math> miles on the <math>n^{\mbox{th}}_{}</math> day of this tour, how many miles was he from his star ...many contestants caught <math>n\,</math> fish for various values of <math>n\,</math>.

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 ..., where <math>m\,</math> and <math>n\,</math> are integers. Find <math>m + n\,</math>.

7 KB (1,141 words) - 07:37, 7 September 2018
1995 AIME Problems
...> and <math>n</math> are relatively prime positive integers. Find <math>m-n.</math> ...h> and <math>n</math> are relatively prime positive integers, find <math>m+n.</math>

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
...and <math>n</math> are relatively prime positive integers. Find <math>m + n.</math> ...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

7 KB (1,098 words) - 17:08, 25 June 2020
1998 AIME Problems
...ath>n</math> are [[relatively prime]] [[positive integer]]s. Find <math>m+n.</math> ...allelogram]]. Extend <math>\overline{DA}</math> through <math>A</math> to a point <math>P,</math> and let <math>\overline{PC}</math> meet <math>\overli

7 KB (1,084 words) - 02:01, 28 November 2023
1999 AIME Problems
...nd <math>n_{}</math> are relatively prime positive integers. Find <math>m+n.</math> ...all positive integers <math>n</math> for which <math>n^2-19n+99</math> is a perfect square.

7 KB (1,094 words) - 13:39, 16 August 2020
2000 AIME I Problems
{{AIME Problems|year=2000|n=I}} ...he least positive integer <math>n</math> such that no matter how <math>10^{n}</math> is expressed as the product of any two positive integers, at least

7 KB (1,204 words) - 03:40, 4 January 2023

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