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• Fermat's Little Theorem
  ...] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to ...}</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
• 2005 AIME II Problems
  A game uses a deck of <math> n </math> different cards, where <math> n </math> is an inte ...e indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one r
  7 KB (1,119 words) - 20:12, 28 February 2020
• 2005 AIME II Problems/Problem 9
  ...al to <math>1000</math> is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true for all real <math> t </math>? ...math>n</math>. So, we'd like to somehow convert our given expression into a form from which we can apply De Moivre's Theorem.
  6 KB (1,154 words) - 02:30, 11 January 2024
• 1984 IMO Problems/Problem 4
  Let <math>ABCD</math> be a convex quadrilateral with the line <math>CD</math> being tangent to the cir ...^\prime</math> which also satisfies the condition. Suppose there was such a <math>T</math>. Like before, define the points <math>E,M,N</math> for quad
  4 KB (750 words) - 22:49, 29 January 2021
• 2010 AIME I Problems/Problem 13
  [[Rectangle]] <math>ABCD</math> and a [[semicircle]] with diameter <math>AB</math> are coplanar and have nonoverl label("$A$",(-16.43287,-9.3374),NE/2);
  10 KB (1,418 words) - 22:05, 20 October 2021
• 2010 PUMaC Problems
  == Division A == Assume that <math>f(a+b) = f(a) + f(b) + ab</math>, and that <math>f(75) - f(51) = 1230</math>. Find <mat
  22 KB (3,694 words) - 22:58, 3 June 2022
• 1954 AHSME Problems
  <math>\textbf{(A)}\ y^2-5\sqrt{y^2-25} \qquad \textbf{(B)}\ -y^2 \qquad \textbf{(C)}\ y^2 \\ <math>\textbf{(A)}\ 4 \text{ and }1 \qquad \textbf{(B)}\ \text{only }1 \qquad \textbf{(C)}\
  23 KB (3,534 words) - 18:16, 8 November 2024
• 1954 AHSME Problems/Problem 28
  ...h> and <math>\frac{r}{t}=\frac{9}{14}</math>, the value of <math>\frac{3mr-nt}{4nt-7mr}</math> is: <math> \textbf{(A)}\ -5\frac{1}{2}\qquad\textbf{(B)}\ -\frac{11}{14}\qquad\textbf{(C)}\ -1\fr
  1 KB (235 words) - 00:31, 28 February 2020
• 2017 AIME I Problems/Problem 5
  A rational number written in base eight is <math>\underline{ab} . \underline{ First, note that the first two digits will always be a positive number. We will start with base twelve because of its repetition.
  6 KB (961 words) - 21:03, 18 January 2024
• Ajax
  ...a post. We submit, and something interesting shows up in the network tab. A text named <code>ajax.php</code> shows up. we click on (the first one), and -H "user-agent: Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/89.0.4389.1
  3 KB (539 words) - 14:39, 18 April 2021
• 2020 FMC 12A Problems
  ...identical bottles. Given that Amy spent <math>3</math> more dollars to buy a standard pack of soda in <math>2020</math> than in <math>2019, </math> what <math>\textbf{(A)}\ 1.6 \qquad\textbf{(B)}\ 1.65 \qquad\textbf{(C)}\ 1.7 \qquad\textbf{(D)}\
  15 KB (2,388 words) - 12:24, 9 June 2020
• 2020 USOJMO Problems/Problem 2
  ...<math>CA</math> at points <math>P</math> and <math>Q</math>, respectively. A point <math>R</math> is chosen such that <math>PR = PA</math> and <math>QR Call a point <i>good</i> if it is a possible location for <math>R</math>.
  7 KB (1,437 words) - 18:14, 6 October 2023
• Wood2009
  1. Wood2009 is a kid. He likes Aops. EDIt: Of course everybody likes Aops 7. He has a lot of friends. Edit: Breh
  3 KB (452 words) - 09:21, 17 November 2022
• 2022 AIME I Problems/Problem 14
  ...h> into two polygons of equal perimeter. Let <math>\triangle ABC</math> be a triangle where <math>BC = 219</math> and <math>AB</math> and <math>AC</math ...mple angle chasing reveals the condition is now equivalent to <math>\angle A=120^\circ</math>.
  16 KB (2,730 words) - 01:56, 4 January 2023
• 2020 INMO Problems/Problem 5
  ...r <math>n \geqslant 3</math> is called frameable if it is possible to draw a regular polygon with <math>n</math> sides all whose vertices lie on these l (a) Show that <math>3, 4, 6</math> are frameable.
  3 KB (607 words) - 02:12, 22 September 2023
• 2021 IMO Problems/Problem 4
  ...Gamma</math> be a circle with centre <math>I</math>, and <math>ABCD</math> a convex quadrilateral such that each of The extension of <math>BA</math> beyond <math>A</math> meets <math>\Omega</math> at <math>X</math>, and the extension of <m
  7 KB (1,196 words) - 09:30, 18 June 2023
• DVI exam
  ...State University named after M.V. Lomonosov. The first four problems have a standard level. A right circular cylinder is located so that the circle of its upper base tou
  48 KB (8,214 words) - 07:59, 30 July 2024
• 2024 USAJMO Problems/Problem 3
  ...k</math> is a positive integer. Prove that some term of the sequence <math>a(n)</math> is divisible by <math>p^k</math>. ...>m</math> such that <math>p^k|a(m)</math>, we must have that <math>p^{k+1}|a(m+2)</math>.
  2 KB (469 words) - 20:42, 19 July 2024