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• 2001 AIME II Problems/Problem 6
  ...ath>EFGH</math>. By the [[Pythagorean Theorem]], the radius of <math>\odot O = OC = a\sqrt{2}</math>. ...MP("E",E,SW)--MP("F",F,NW)--MP("G",G,NE)--MP("H",H,SE)--cycle); D(CP(D(MP("O",(0,0))), A));
  4 KB (772 words) - 19:31, 6 December 2023
• 2000 AIME II Problems/Problem 12
  ..., <math>BC=14</math>, <math>CA=15</math>, and that the distance from <math>O</math> to <math>\triangle ABC</math> is <math>\frac{m\sqrt{n}}k</math>, whe Let <math>D</math> be the foot of the [[perpendicular]] from <math>O</math> to the plane of <math>ABC</math>. By the [[Pythagorean Theorem]] on
  3 KB (532 words) - 13:14, 22 August 2020
• 2000 AIME II Problems/Problem 10
  ...drawing the lines from <math>O</math> tangent to the sides and from <math>O</math> to the vertices of the quadrilateral, four pairs of congruent [[righ
  2 KB (399 words) - 17:37, 2 January 2024
• Centroid
  pair O=(A+B)/2; D(MP("M",M,dir(270)));D(MP("N",N,D(N)));D(MP("O",O,D(O)));D(M);
  3 KB (612 words) - 22:32, 25 February 2024
• Radius
  pair O=origin, P=dir(30); D(O--P);
  929 bytes (156 words) - 22:49, 5 January 2023
• 2006 AMC 10B Problems
  A circle of radius <math>2</math> is centered at <math>O</math>. Square <math>OABC</math> has side length <math>1</math>. Sides <mat label("$O$",(0,0),SW);
  14 KB (2,059 words) - 01:17, 30 January 2024
• University of South Carolina High School Math Contest/1993 Exam/Problems
  ...ath> respectively. <math>CN</math> and <math>AM</math> intersect at <math>O</math>. If the length of <math>CQ</math> is 4, then what is the length of
  14 KB (2,102 words) - 22:03, 26 October 2018
• Mock AIME 1 2006-2007 Problems/Problem 14
  ...at <math>T</math>. Let <math>P</math> be the point such that circle <math>O</math> is the incircle of <math>\triangle APB</math>. Construct <math>M</ma Let O be the centre of the incircle, and <math>r</math> be the inradius.
  3 KB (541 words) - 17:32, 22 November 2023
• 1977 Canadian MO Problems/Problem 2
  ...math>A</math> be a fixed interior point of the circle different from <math>O.</math> Determine all points <math>P</math> on the circumference of the cir ...> to meet the circle at point <math>C</math>. It is now evident that <math>O</math> is the midpoint of <math>AC</math>, <math>X</math> is the midpoint o
  2 KB (365 words) - 23:28, 21 September 2014
• Mock AIME 1 2006-2007/Problems
  ...at <math>T</math>. Let <math>P</math> be the point such that circle <math>O</math> is the incircle of <math>\triangle APB</math>. Construct <math>M</ma
  8 KB (1,355 words) - 14:54, 21 August 2020
• Mock AIME 2 2006-2007 Problems/Problem 7
  ...th <math>17\cdot 2\pi=34\pi</math>. Let the vertex of this sector be <math>O</math>. The problem is then reduced to finding the shortest distance betwee
  1 KB (231 words) - 18:10, 10 July 2014
• Mock AIME 2 2006-2007 Problems/Problem 12
  ...C= m\angle DBC </math> and <math>\frac{[ADB]}{[ABC]}=\frac12.</math> <math>O</math> is defined to be the intersection of the diagonals of <math>ABCD</ma
  2 KB (311 words) - 10:53, 4 April 2012
• 2005 AMC 10A Problems/Problem 9
  Three tiles are marked <math>X</math> and two other tiles are marked <math>O</math>. The five tiles are randomly arranged in a row. What is the probabil ...!}=10</math> distinct arrangements of three <math>X</math>'s and two <math>O</math>'s.
  764 bytes (112 words) - 12:01, 13 December 2021
• 1977 Canadian MO Problems
  ...math>A</math> be a fixed interior point of the circle different from <math>O.</math> Determine all points <math>P</math> on the circumference of the cir
  3 KB (560 words) - 19:23, 10 March 2015
• 1971 Canadian MO Problems/Problem 1
  ...of a circle such that <math>DE=3</math> and <math>EB=5 .</math> Let <math>O</math> be the center of the circle. Join <math>OE</math> and extend <math>O
  680 bytes (114 words) - 21:38, 9 July 2019
• 1959 IMO Problems/Problem 4
  ...pick the one closer to N. Draw circle around this new point going through O and M. The intersection of the two circles is the desired third vertex of t
  6 KB (939 words) - 17:31, 15 July 2023
• 2006 Romanian NMO Problems/Grade 8/Problem 1
  ...ath>, and <math>OO''D'</math> are congruent. Thus, <math>O''A'=O''B'=O''C'=O''D'</math> and <math>A'B'C'D'</math> is cyclic.
  3 KB (509 words) - 23:22, 15 August 2012
• 2006 Romanian NMO Problems/Grade 9/Problem 3
  ...>O</math> at <math>C</math>. Finally, extend <math>CP</math> to meet <math>O</math> at <math>D</math> and we are done! ...en <PAD=x+30. Then PD=PA so se need to prove that ODA is equilateral where O is the center of ABCD. However, since <DAP=<DPA=x+30 DP=AP and so ABPD is a
  6 KB (1,080 words) - 19:28, 21 September 2014
• 2005 Canadian MO Problems/Problem 3
  ...ints <math>Q_i</math> on any of the line segments <math>OV_i</math> (<math>O</math> is the center), where <math>OQ_i < 1 - \frac{\sqrt{3}}{2},</math> th
  2 KB (460 words) - 13:35, 9 June 2011
• P versus NP
  ...akes to solve a problem as a function of input, usually expressed with big-O notation) and [[space]] (how much memory it takes to solve a problem). In s ...ath>\text{TIME}(f(n))</math> is the set of languages decidable by an <math>O(f(n))</math>-time deterministic Turing machine.
  6 KB (1,104 words) - 15:11, 25 October 2017

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