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• Fermat's Little Theorem
  ...a hard time following the rest of this article). This theorem is credited to [[Pierre de Fermat]]. ...}</math> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>.
  16 KB (2,660 words) - 22:42, 28 August 2024
• 1986 AIME Problems/Problem 3
  ...n y}{1-\tan x \tan y} = \frac{25}{1-\tan x \tan y}</math>. We only need to find <math>\tan x \tan y</math>. ...x \sin y}{\cos x \cos y} = \tan x \tan y = \frac{5}{6}</math>. Plugging in to the earlier formula, we have <math>\tan(x+y) = \frac{25}{1-\frac{5}{6}} = \
  3 KB (545 words) - 22:44, 12 October 2023
• 1988 AIME Problems/Problem 14
  Find the product <math>bc</math>. ...Both points lie on a line that is [[perpendicular]] to <math>y=2x</math>, so the slope of <math>\overline{PP'}</math> is <math>\frac{-1}{2}</math>. Thus
  4 KB (700 words) - 16:21, 3 May 2021
• 1988 AIME Problems/Problem 9
  Find the smallest positive integer whose [[perfect cube|cube]] ends in <math>888 A little bit of checking tells us that the units digit must be 2. Now our cub
  6 KB (893 words) - 07:15, 2 February 2023
• 1989 AIME Problems/Problem 11
  ...floor</math> is the [[floor function|greatest integer]] less than or equal to <math>x</math>.) ..., [[without loss of generality]], let <math>x=1</math>. Now, we would like to maximize the quantity
  5 KB (851 words) - 17:01, 28 December 2022
• 1996 AIME Problems/Problem 6
  ...am, where <math>m</math> and <math>n</math> are relatively prime integers. Find <math>m+n</math>. No more than 1 team can win or lose all games, so at most one team can win all games and at most one team can lose all games.
  3 KB (461 words) - 23:33, 15 May 2024
• 1998 AIME Problems/Problem 4
  .../math> and <math>n</math> are [[relatively prime]] [[positive integer]]s. Find <math>m+n.</math> In order for a player to have an odd sum, he must have an odd number of odd tiles: that is, he can e
  5 KB (917 words) - 01:37, 12 December 2022
• 2003 AIME I Problems/Problem 3
  ...{S}, </math> she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list. Order the numbers in the set from greatest to least to reduce error: <math>\{34, 21, 13, 8, 5, 3, 2, 1\}.</math>
  2 KB (317 words) - 23:09, 8 January 2024
• 2001 AIME II Problems/Problem 7
  ...line{UV}</math> is perpendicular to <math>\overline{PQ}</math> and tangent to <math>C_{1}</math>. Let <math>C_{2}</math> be the inscribed circle of <math ..., UV</math> lie respectively on the lines <math>y = 60, x = 60</math>, and so <math>RS = 60, UQ = 30</math>.
  7 KB (1,112 words) - 01:15, 26 December 2022
• 2000 AIME II Problems/Problem 7
  ...frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}</math></center> find the [[floor function|greatest integer]] that is less than <math>\frac N{100 So, <math>N=\frac{262124}{19}=13796</math> and <math>\left\lfloor \frac{N}{100
  2 KB (281 words) - 11:09, 5 April 2024
• Asymptote: Advanced Configuration
  ...rs commonly would like to do with their figures: make the images output in a different format, and include Asymptote code directly into LaTeX files. ...ly make your images in pdf format rather than eps format, with the help of a program called '''GhostScript'''.
  12 KB (1,931 words) - 12:53, 26 January 2020
• 2007 AIME I Problems/Problem 14
  A [[sequence]] is defined over [[non-negative]] integral indexes in the follo Find the [[floor function|greatest integer]] that does not exceed <math>\frac{a_
  13 KB (2,185 words) - 22:48, 31 July 2024
• 2007 AIME I Problems/Problem 15
  ...</math> is an [[integer]] not divisible by the [[square]] of a [[prime]]. Find <math>r</math>. ...\cdot y + (x - 2)(x - 5) + 2(x - y)) + 14\sqrt{3}</math>. This simplifies to <math>\frac{x^2\sqrt{3}}{4} = \frac{\sqrt{3}}{4}(5y + x^2 - 7x + 10 + 2x -
  4 KB (673 words) - 21:14, 6 August 2022
• LaTeX:Layout
  ...trying out each of the examples as you go. It takes almost no time at all to just copy-paste, compile, and view the results. ...his allows you to use the same code in many source files by just including a single line in each source file.
  29 KB (4,929 words) - 12:26, 12 September 2024
• 2002 AMC 10A Problems/Problem 25
  pair A,B,C,D; A=(0,0);
  8 KB (1,308 words) - 06:05, 19 December 2022
• How should I prepare?
  ...roblem-solving skills. Depending on your current abilities, you will want to start out with different practice problems, different books, If you don't feel like going too deep and want a straightforward answer, here it is:
  15 KB (2,218 words) - 19:19, 9 October 2024
• 2008 IMO Problems/Problem 3
  ...many positive integers <math>n</math> such that <math>n^{2} + 1</math> has a prime divisor greater than <math>2n + \sqrt {2n}</math>. ...<math>a+bi</math> and multiply it by a "twice as small" <math>c+di</math> to get <math>n+i</math>. The rest is just making up the little details.
  5 KB (1,002 words) - 00:09, 19 November 2023
• 2009 AMC 12A Problems/Problem 25
  The first two terms of a sequence are <math>a_1 = 1</math> and <math>a_2 = \frac {1}{\sqrt3}</math>. <math>\textbf{(A)}\ 0\qquad \textbf{(B)}\ 2 - \sqrt3\qquad \textbf{(C)}\ \frac {1}{\sqrt3}\q
  7 KB (994 words) - 15:57, 9 July 2024
• 2009 AIME I Problems/Problem 12
  ...<math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. ...angle ABI</math> is equal to <cmath>2(x+AD+DB)=2(x+37).</cmath> It remains to compute <math>\dfrac{2(x+37)}{37}=2+\dfrac{2}{37}x</math>.
  13 KB (2,170 words) - 22:35, 3 September 2024
• 2010 AMC 12A Problems/Problem 21
  <math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \ In order to find <math>\sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c}</math> we must first expand out
  6 KB (1,056 words) - 16:45, 26 August 2024

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