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• 1995 AIME Problems
  ...o sides of square <math>S_{i+1},</math> are the perpendicular bisectors of two adjacent sides of square <math>S_{i+2}.</math> The total area enclosed by ...<math>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.
  6 KB (1,000 words) - 00:25, 27 March 2024
• 1998 AIME Problems
  Given that <math>A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,</math> find <math>|A_{19} + A_{20} + \cdots + A_{98}|.</math> Except for the first two terms, each term of the sequence <math>1000, x, 1000 - x,\ldots</math> is o
  7 KB (1,084 words) - 02:01, 28 November 2023
• 2003 AIME I Problems
  ...ion bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green r ...math> \mathcal{S}, </math> she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.
  6 KB (965 words) - 16:36, 8 September 2019
• 2002 AIME II Problems
  ...e1,2,3,\ldots,10\rbrace</math> Let <math>n</math> be the number of sets of two non-empty disjoint subsets of <math>\mathcal{S}</math>. (Disjoint sets are ...math> radians are <math>\frac{m\pi}{n-\pi}</math> and <math>\frac{p\pi}{q+\pi}</math>, where <math>m</math>, <math>n</math>, <math>p</math>, and <math>q<
  7 KB (1,177 words) - 15:42, 11 August 2023
• 2003 AIME II Problems
  ...gers is 6 times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of <math>N</math>. ...e log. The number of cubic inches in the wedge can be expressed as <math>n\pi</math>, where n is a positive integer. Find <math>n</math>.
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1983 AIME Problems/Problem 15
  The adjoining figure shows two intersecting chords in a circle, with <math>B</math> on minor arc <math>AD< ...ally, the circle <math>P</math> can intersect the chord <math>BC</math> at two points, one point, or they may not have a point of intersection. By the pro
  20 KB (3,497 words) - 15:37, 27 May 2024
• 1984 AIME Problems/Problem 12
  <cmath>f(x) = \sin \frac{\pi x}{10}\sin \frac{\pi (x-4)}{10}</cmath> ...tice that it starts at x=0, then it goes to x=5, x=10, etc... each f() has two possible x values, but we are only counting the total number of x values so
  3 KB (588 words) - 14:37, 22 July 2020
• 1985 AIME Problems/Problem 12
  Two solutions follow from here: ...}</math> and add them up before dividing by <math>3^7</math>. Here we have two ways to proceed:
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1985 AIME Problems/Problem 9
  ...ha + \beta</math> [[radian]]s, respectively, where <math>\alpha + \beta < \pi</math>. If <math>\cos \alpha</math>, which is a [[positive]] [[rational num pair O = (0,0), A = r*expi(pi/3);
  5 KB (763 words) - 16:20, 28 September 2019
• 1987 AIME Problems/Problem 14
  Two solutions follow from here: ...eta</math> is the argument of <math>N</math> such that <math>0\leq\theta<2\pi.</math>
  7 KB (965 words) - 10:42, 12 April 2024
• 1988 AIME Problems/Problem 14
  Solving these two equations, we find <math>x = \frac{-3x' + 4y'}{5}</math> and <math>y = \fra Now to find the equation of the hyperbola, we multiply the two expressions together to get one side of the equation: <math>(3x-4y)(4x+3y)=
  4 KB (700 words) - 17:21, 3 May 2021
• 1989 AIME Problems/Problem 6
  ...s the [[straight]] path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meetin pair A=(0,0),B=(10,0),C=6*expi(pi/3);
  6 KB (980 words) - 15:08, 14 May 2024
• 1990 AIME Problems/Problem 12
  ...method involves drawing a triangle connecting the center of the 12-gon to two vertices of the 12-gon. Since the distance from the center to a vertex of To simplify the two nested radicals, add them, and call the sum <math>x</math>:
  6 KB (906 words) - 13:23, 5 September 2021
• 1990 AIME Problems/Problem 10
  ...8</math> and <math>48</math> is <math>144</math>, so define <math>n = e^{2\pi i/144}</math>. We can write the numbers of set <math>A</math> as <math>\{n^ ...</math> are relatively prime, and by the [[Chicken McNugget Theorem]], for two relatively prime integers <math>a,b</math>, the largest number that cannot
  3 KB (564 words) - 04:47, 4 August 2023
• 1991 AIME Problems/Problem 15
  .../math> for <math>1 \le i \le n</math> and <math>0 \le \theta_{i} < \frac {\pi}{2}</math>. We then have that ...e can see that it is basically calculating distance between the origin and two points. Therefore we can assume every <math>\sqrt{(2k-1)^2+a_k^2}</math> to
  4 KB (658 words) - 16:58, 10 November 2023
• 1991 AIME Problems/Problem 11
  ...The sum of the areas of the twelve disks can be written in the from <math>\pi(a-b\sqrt{c})</math>, where <math>a,b,c^{}_{}</math> are positive integers a dot((cos(i*pi/6), sin(i*pi/6)));
  4 KB (740 words) - 17:46, 24 May 2024
• 1991 AIME Problems/Problem 4
  ...^{}_{}</math> satisfy the [[equation]] <math>\frac{1}{5}\log_2 x = \sin (5\pi x)</math>? ...h>y = 0</math>, there is exactly <math>1</math> touching point between the two functions: <math>\left(\frac{1}{5},0\right)</math>. When <math>y < 0</math>
  2 KB (300 words) - 16:01, 26 November 2019
• 1992 AIME Problems/Problem 10
  ...area of the two circles and then subtracting out their overlap. There are two methods of finding the area of overlap: ...of those three graphs is <math>40^2-(200\pi + 400) \Rightarrow 1200 - 200\pi \approx 571.68</math>
  2 KB (323 words) - 12:05, 16 July 2019
• 2006 AMC 10B Problems/Problem 19
  <math> \mathrm{(A) \ } \frac{\pi}{3}+1-\sqrt{3}\qquad \mathrm{(B) \ } \frac{\pi}{2}(2-\sqrt{3}) ...rm{(D) \ } \frac{\pi}{6}+\frac{\sqrt{3}+1}{2}\qquad \mathrm{(E) \ } \frac{\pi}{3}-1+\sqrt{3} </math>
  5 KB (873 words) - 15:39, 29 May 2023
• 1994 AIME Problems/Problem 15
  ...points of <math>\triangle ABC\,</math> can be written in the form <math>q\pi-r\sqrt{s},\,</math> where <math>q, r,\,</math> and <math>s\,</math> are pos ...of the locus of <math>P</math> (shaded region below) is simply the sum of two [[segment]]s of the circles. If we construct the midpoints of <math>M_1, M_
  4 KB (717 words) - 22:20, 3 June 2021

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