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• 2005 IMO Shortlist Problems/A1
  ...\frac{|z|^n -1}{|z|-1} = \sum_{i=0}^{n-1}|\pm z|^i \ge \left| \sum_{i=0}^{n-1} \pm z^i \right|
  2 KB (346 words) - 07:56, 14 May 2024
• 2007 IMO Shortlist Problems/A1
  <center><math>\max\{|x_i-a_i|:1\leq i\leq n\}\geq \frac{d}{2}</math>.</center> ..._j:i\le j\le n\}</math>, all <math>d_i</math> can be expressed as <math>a_p-a_q</math>, where <math>1\le p\le i\le q \le n</math>.
  3 KB (734 words) - 04:11, 26 March 2024
• 2011 IMO Shortlist Problems/A1
  500 bytes (91 words) - 05:44, 28 October 2013
• 2011 PuMAC Problems/Algebra Problem A1
  Find the sum of the coefficients of the polynomial <math>(63x-61)^4</math>. Therefore, the sum of the coefficients of <math>(63x-61)^4</math> is <math>(63-61)^4 = 2^4 = \boxed{16}</math>
  609 bytes (83 words) - 19:15, 6 August 2023
• 2011 PuMAC Problems/Geometry Problem A1
  40 bytes (6 words) - 00:07, 13 March 2023

Page text matches

• Joining an ARML team
  ...as well as practices of previous years' team rounds. Please email Xinke Guo-Xue at xinkeguoxue@gmail.com, or message Xinke's AoPS account "hurdler", if ...room 2112, on Thursdays at 7pm. For more information, e-mail Eric Brattain-Morrin at [mailto:eric.brattain@gmail.com eric.brattain@gmail.com] and visit
  22 KB (3,532 words) - 10:25, 27 September 2024
• ARML historical results
  |[[West-Windsor-Plainsboro ARML]] (WWP^2 A1) |[[West Windsor-Plainsboro ARML]] (WWP A1)
  20 KB (2,642 words) - 20:23, 1 June 2024
• Georgia ARML
  1 KB (186 words) - 10:52, 4 June 2012
• 2004 AIME I Problems/Problem 14
  pair Cxy = 8*expi((3*pi)/2-CE/8); label("$4$",D/2+A/2,(-1,0));label("$x$",C/2+B/2,(1,0.5));label("$20-x$",0.7*A+0.3*B,(1,0.5));
  4 KB (729 words) - 00:00, 27 November 2022
• 1985 AIME Problems/Problem 9
  == Solution 1== <!-- Images obsoleted Image:1985_AIME-9.png, Image:1985_AIME-9a.png by asymptote --> ...[arc]] and therefore the same central angle. Thus, by the given, we can re-arrange our chords into a [[triangle]] with the circle as its [[circumcircle
  5 KB (763 words) - 15:20, 28 September 2019
• Ring
  * <math>a(b-c)= ab-ac \\ (b-c)a = ba - ca </math> (double [[distributive property |distributivity]] of m ...nct elements of <math>R</math> for which <math>xy=xz</math>, then <math>x(y-z)=0</math>.
  6 KB (994 words) - 05:16, 8 April 2015
• Mock AIME 1 2006-2007 Problems/Problem 15
  *[[Mock AIME 1 2006-2007 Problems/Problem 14 | Previous Problem]] *[[Mock AIME 1 2006-2007]]
  3 KB (414 words) - 12:45, 19 February 2016
• IMO Problems and Solutions
  * [[1959 IMO Problems/Problem 2 | Problem 2]] proposed by Constantin Ionescu-Tiu, Romania * [[1987 IMO Problems/Problem 5 | Problem 5]] proposed by Hans-Dietrich Gronau, East Germany
  35 KB (4,036 words) - 07:06, 4 August 2024
• Orthocenter
  ...0), C=(10,0), A1 = (B+C)/2, O = circumcenter(A,B,C), G = (A+B+C)/3, H = 3*G-2*O; ...triangles <math>AGH</math>, <math>A'GO</math> are [[similar]] by side-angle-side similarity. It follows that <math>AH</math> is parallel to <math>OA'</
  5 KB (828 words) - 13:09, 18 October 2024
• Asymptote: Useful commands and their Output
  ...mages the forums will show you the Asymptote source code, which can be copy-pasted.
  6 KB (871 words) - 20:14, 12 June 2023
• Science Olympiad
  '''Science Olympiad''' is a national team-based science competition primarily for middle high school students, though ...on its team. Division C is generally for high school students in grades 10-12, though ninth graders are allowed to compete.
  3 KB (443 words) - 10:27, 25 May 2023
• 2001 PMWC Problems
  F=O+(G-O)+(E-O); draw((xstart,A.y)--(xstart,A.y-len));
  4 KB (641 words) - 20:24, 21 April 2014
• 2007 AMC 10B Problems/Problem 18
  ...t(2)), C=(-1-sqrt(2),1+sqrt(2)), D=(-1-sqrt(2),-1-sqrt(2)), E=(1+sqrt(2),-1-sqrt(2)); ...2sqrt(2),1+sqrt(2)), C1=(0,1+sqrt(2)), D1=(0,-1-sqrt(2)), E1=(2+2sqrt(2),-1-sqrt(2));
  2 KB (385 words) - 13:17, 4 June 2021
• 2008 AIME I Problems/Problem 15
  real r = (51^0.5-17^0.5)/200, h=867^0.25/100; ...--A1--B); draw((A1+(F-A1)*factor)--(A1+(F-A1)*factor+(B-A1)*factor)--(A1+(B-A1)*factor));
  6 KB (1,044 words) - 12:28, 3 September 2024
• 1974 USAMO Problems/Problem 5
  pair RR=R+Q-P, MM= rotate(-60,Q)*M; pair RR=R+Q-P, MM= rotate(-60,Q)*M;
  7 KB (1,221 words) - 17:57, 3 July 2013
• 2008 iTest Problems/Problem 99
  ...math>, and for each <math>i \geq 2</math>, let <math>P_i = \text{clip}(P_{i-1})</math>. This iterative clipping process approaches a limiting shape <mat ...-cycle); D(B--D); D(MP("A_1",A1,NW) -- MP("A_2",A2)); D(D+2(C-D)/3 -- B+2(C-B)/3); D(4B/9 -- (A+D)/3+2*((A+B)/3 - (A+D)/3)/3, d); D(4D/9 -- (A+D)/3 + ((
  4 KB (685 words) - 13:39, 7 October 2017
• 2001 AMC 12 Problems/Problem 16
  Pretty clearly, if we choose each sock-shoe pair individually, they will automatically align themselves. https://youtu.be/D1n-qEPdm6g
  3 KB (492 words) - 05:07, 30 May 2024
• 2010 IMO Problems/Problem 6
  <cmath>a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.</cmath> :<math>\text{To this end, let’s define an n-type to be a vector T = ⟨t1,...,ts⟩ of nonnegative integers such that}</
  4 KB (786 words) - 07:46, 12 March 2024
• 2013 IMO Problems/Problem 3
  ...of triangle <math>ABC</math>. Prove that triangle <math>ABC</math> is right-angled. ...I_c}</math> be <math>M_a</math>, which lies on <math>(ABC)</math>, the nine-point circle of <math>\triangle I_aI_bI_c</math>; analogously define <math>M
  3 KB (553 words) - 08:41, 17 January 2016
• 2007 AMC 10B Problems
  ...s the average of <math>a</math> and <math>c.</math> How many different five-digit numbers satisfy all these properties? ...t(2)), C=(-1-sqrt(2),1+sqrt(2)), D=(-1-sqrt(2),-1-sqrt(2)), E=(1+sqrt(2),-1-sqrt(2));
  15 KB (2,297 words) - 11:57, 19 February 2020

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