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• Math textbooks
  ...960988/artofproblems-20 Statistical Theory and Bayesian Analysis] by James O. Berger.
  7 KB (901 words) - 14:11, 6 January 2022
• 1954 AHSME Problems/Problem 46
  pair O=(0,3/8); draw(Circle(O,3/16));
  3 KB (415 words) - 18:01, 24 May 2020
• 2015 IMO Problems
  ...ath>ABC</math> has circumcircle <math>\Omega</math> and circumcenter <math>O</math>. A circle <math>\Gamma</math> with center <math>A</math> intersects
  4 KB (692 words) - 22:33, 15 February 2021
• MathCounts historical results
  * 1994: William O. Engel, [[Illinois Mathcounts]]
  6 KB (546 words) - 12:21, 13 May 2024
• USAMO historical results
  **Evan O'Dorney **Evan O'Dorney
  10 KB (1,317 words) - 08:16, 23 April 2024
• Geometric mean
  ...following situation: if <math>AB</math> is a [[chord]] of [[circle]] <math>O</math> with [[midpoint]] <math>M</math> and <math>M</math> divides the [[di
  2 KB (282 words) - 22:04, 11 July 2008
• Fermat's Little Theorem
  ...mutes the necklace in a single orbit which we can denote as <math>\mathcal{O}</math> (since the size of the orbit is a factor of <math>p</math>). Hence <center><cmath>|\mathcal{O}|=\frac{1}{|G|}\sum_{g\in G}|\text{Fix}(g)|=\frac{1}{p}\sum_{g\in C_p}|\tex
  16 KB (2,658 words) - 16:02, 8 May 2024
• Cyclic quadrilateral
  * <math>\angle A + \angle C = \angle B + \angle D = {180}^{o} </math> This property is both sufficient and necessary (Sufficient & neces
  1 KB (162 words) - 20:39, 9 March 2024
• Number theory/Advanced
  ...vertically upward as the point at infinity. We denote it by <math>\mathcal{O}</math>. ...>-axis. We may thus summarize the group law by saying <math>P+Q+R=\mathcal{O}</math> if and only if <math>P,Q</math> and <math>R</math> lie on a line.
  5 KB (849 words) - 16:14, 18 May 2021
• Divisor
  ...floor\frac N2\right\rfloor+\dots+\left\lfloor\frac NN\right\rfloor= N\ln N+O(N)}</math>
  1 KB (274 words) - 19:50, 29 August 2023
• 1988 AJHSME Problems/Problem 5
  label("O",(-2.5,0),W);
  1 KB (160 words) - 16:53, 17 December 2020
• Law of Sines
  pair A=(-1,5), B=(-4,-1), C=(4,-1), D, O; O = circumcenter(A,B,C);
  4 KB (658 words) - 16:19, 28 April 2024
• Homothety
  ...ABCD</math> and <math>A'B'C'D'</math> are homothetic with respect to <math>O</math>. label("$O$",(0,0),SW);
  3 KB (532 words) - 01:11, 11 January 2021
• 2000 AMC 12 Problems/Problem 1
  ...2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? ...clear that this is true, and in this situation, the value of <math>I + M + O</math> would be <math>18</math>. Now, we use this process on <math>2001</ma
  2 KB (276 words) - 05:25, 9 December 2023
• 1985 IMO Problems/Problem 5
  A circle with center <math>O</math> passes through the vertices <math>A</math> and <math>C</math> of the Let <math>\Omega, \Omega', \omega</math> and <math>O,O',O''</math> be the circumcircles and circumcenters of <math>AKNC, ABC, BNKM,</
  3 KB (496 words) - 13:35, 18 January 2023
• Trigonometry
  ...'''C'''osine = '''A'''djacent / '''H'''ypotenuse, and '''T'''angent = '''O'''pposite / '''A'''djacent
  8 KB (1,217 words) - 20:15, 7 September 2023
• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  pair O=(0,0),A=(-1,0),B=(0,1),C=(1,0),P=(1/2,0),Q=(1/2,sqrt(3)/2),R=foot(P,Q,O); draw(B--O--C--arc(O,C,A)--O--R--P); rightanglemark(O,P,R);
  5 KB (912 words) - 20:06, 14 March 2023
• Riemann Hypothesis
  ...nt statement of the Riemann hypothesis is that <math>\pi(x)=\mathrm{Li}(x)+O(x^{1/2}\ln(x))</math>.
  2 KB (425 words) - 12:01, 20 October 2016
• Power of a Point Theorem
  Consider a circle <math>O</math> and a point <math>P</math> in the plane where <math>P</math> is not
  5 KB (827 words) - 17:30, 21 February 2024
• Algebraic geometry
  ...topological space isomorphic to some <math>(\operatorname{Spec }A,\mathcal{O}_{\operatorname{Spec}A})</math>. ...ting an open covering <math>\{U_i\}_i</math> such that <math>(U_i,\mathcal{O}_{X|U_i})</math> is an affine scheme for every <math>i</math>.
  2 KB (361 words) - 01:59, 24 January 2020
• Mock AMC
  ! scope="row" | '''Mock AMC O'''
  51 KB (6,175 words) - 20:58, 6 December 2023
• 2006 AIME I Problems/Problem 14
  triple O=(0,0,0),T=(0,0,5),C=(0,3,0),A=(-3*3^.5/2,-3/2,0),B=(3*3^.5/2,-3/2,0); draw(T--S--B--T--C--B--S--C);draw(B--A--C--A--S,ddash);draw(T--O--M,ddash);
  6 KB (980 words) - 21:45, 31 March 2020
• 2006 AIME I Problems/Problem 13
  If a number is in the form <math>N=2^k+2^{R}O</math> where <math>O</math> is a positive odd number, <math>R<k</math>: <math>N<2^{k+1}=2^k+2^k\Longrightarrow O<2^{k-R}</math> so there are <math>2^{k-R-1}</math> numbers that satisfy thi
  10 KB (1,702 words) - 00:45, 16 November 2023
• LaTeX:Symbols
  |<math>{\oe}</math>||{\oe}||<math>{\ae}</math>||{\ae}||<math>{\o}</math>||{\o} ...<math>{\AE}</math>||{\AE}||<math>{\AA}</math>||{\AA}||<math>{\O}</math>||{\O}
  16 KB (2,324 words) - 16:50, 19 February 2024
• 2006 AMC 12B Problems
  ...s <math> A</math> and <math> B</math> are on the circle centered at <math> O</math>, and points <math> C</math> and <math> D</math> are on the circle ce pair[] O;
  13 KB (2,058 words) - 12:36, 4 July 2023
• 2005 AMC 12A Problems
  pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); dot(O);
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2000 AMC 12 Problems
  ...2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? A [[circle]] centered at <math>O</math> has [[radius]] <math>1</math> and contains the point <math>A</math>.
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2003 AMC 12B Problems
  pair O = (15*15/17,8*15/17), C = (17,0), D = (0,0), P = (25.6,19.2), Q = (25.6, 18 pair A = 2*O-C, B = 2*O-D;
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2004 AMC 12B Problems
  pair O=(0,0); path inner=Circle(O,r1), outer=Circle(O,r2);
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2006 AMC 12B Problems/Problem 15
  ...s <math> A</math> and <math> B</math> are on the circle centered at <math> O</math>, and points <math> C</math> and <math> D</math> are on the circle ce pair[] O;
  3 KB (458 words) - 16:40, 6 October 2019
• 2006 AMC 12A Problems/Problem 22
  Call the center <math>O</math>, and the two endpoints of the arc <math>A</math> and <math>B</math>,
  2 KB (343 words) - 15:39, 14 June 2023
• 2005 AIME II Problems
  ...the faces of <math> O, </math> and that the ratio of the volume of <math> O </math> to that of <math> C </math> is <math> \frac mn, </math> where <math Square <math> ABCD </math> has center <math> O, AB=900, E </math> and <math> F </math> are on <math> AB </math> with <math
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 12
  [[Square]] <math>ABCD </math> has [[center]] <math> O,\ AB=900,\ E </math> and <math> F </math> are on <math> AB </math> with <ma ...abel("\(x\)",E/2+G/2,(0,1));label("\(y\)",G/2+F/2,(0,1)); label("\(450\)",(O+G)/2,(-1,1));
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME II Problems/Problem 10
  ...the faces of <math> O, </math> and that the ratio of the volume of <math> O </math> to that of <math> C </math> is <math> \frac mn, </math> where <math
  3 KB (436 words) - 03:10, 23 September 2020
• 2005 AIME I Problems/Problem 11
  ...oints at which the "corners" of the semicircle touch the square. Let <math>O</math> be the center of the semicircle. ...of the semicircle as <math>r</math>. Draw the [[perpendicular]] from <math>O</math> to <math>AB</math>, which forms a <math>45-45-90</math> triangle. Th
  4 KB (707 words) - 11:11, 16 September 2021
• 2005 AIME II Problems/Problem 15
  label("$O$",(0,0),NW,fontsize(9)); ...icular, as <math>\angle OF_1T=\angle OF_2T</math>, this implies that <math>O, F_1, F_2</math>, and <math>T</math> are concyclic.
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2004 AIME I Problems/Problem 14
  ...-8,4), B=(0,-8,h), C=(Cxy.x,Cxy.y,0), D=(A.x,A.y,0), E=(B.x,B.y,0), O=(O.x,O.y,h); draw(circle(O,8));
  4 KB (729 words) - 01:00, 27 November 2022
• 2004 AIME I Problems/Problem 10
  ...= 36/5</math>. Since <math>\triangle AOR \sim \triangle AED</math> (<math>O</math> is the center of the circle), we find that <math>AR = 5</math> since
  5 KB (836 words) - 07:53, 15 October 2023
• 2004 AIME II Problems/Problem 12
  ...of the center circle be <math>r</math> and its center be denoted as <math>O</math>. pair A=(0,0), B=(6,0), D=(1, 24^.5), C=(5,D.y), O = (3,(r^2 + 6*r)^.5);
  3 KB (431 words) - 23:21, 4 July 2013
• 1983 AIME Problems
  pair O=(0,0), A=r*dir(45),B=(A.x,A.y-r); path P=circle(O,r);
  7 KB (1,104 words) - 12:53, 6 July 2022
• 1992 AIME Problems
  ..., <math>BB'</math>, and <math>CC'</math> are concurrent at the point <math>O^{}_{}</math>, and that <math>\frac{AO^{}_{}}{OA'}+\frac{BO}{OB'}+\frac{CO}{
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1996 AIME Problems
  In parallelogram <math>ABCD,</math> let <math>O</math> be the intersection of diagonals <math>\overline{AC}</math> and <mat
  6 KB (931 words) - 17:49, 21 December 2018
• 2000 AIME II Problems
  ..., <math>BC=14</math>, <math>CA=15</math>, and that the distance from <math>O</math> to triangle <math>ABC</math> is <math>\frac{m\sqrt{n}}k</math>, wher
  6 KB (947 words) - 21:11, 19 February 2019
• 2002 AIME II Problems
  ...th> is a right angle. A circle of radius <math>19</math> with center <math>O</math> on <math>\overline{AP}</math> is drawn so that it is tangent to <mat
  7 KB (1,177 words) - 15:42, 11 August 2023
• 1983 AIME Problems/Problem 4
  pair O=(0,0), path P=circle(O,r);
  11 KB (1,741 words) - 22:40, 23 November 2023
• 1983 AIME Problems/Problem 12
  ...distance from their intersection point <math>H</math> to the center <math>O</math> is a positive rational number. Determine the length of <math>AB</mat label("O",(0,0),NE);</asy>
  2 KB (412 words) - 18:23, 1 January 2024
• 1983 AIME Problems/Problem 15
  If we take <math>O</math> to be the center of the given circle, then this means that <math>OD< pair O = (0,0), D = (0, 5), B = (-3, 4), C = (3, 4), A = (-4, 3), EE = (4, 3);
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1984 AIME Problems/Problem 11
  ...om{8}{5}</math>. But then, we can rearrange the <math>M</math>'s and <math>O</math>'s in <math>7!/(3!4!)=\binom{7}{3}</math> ways. So then there are <ma
  7 KB (1,115 words) - 00:52, 7 September 2023
• 1984 AIME Problems/Problem 13
  On the coordinate plane, let <math>O=(0,0)</math>, <math>A_1=(3,0)</math>, <math>A_2=(3,1)</math>, <math>B_1=(21
  3 KB (473 words) - 12:06, 18 December 2018
• 1985 AIME Problems/Problem 12
  ...\frac{1}{3}O_{k-2}</cmath>. Substituting this into our equation for <math>O</math>, we have that <cmath>O_n = \frac{1}{3}O_{n-2} + \frac{2}{3}O_{n-1}</
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1985 AIME Problems/Problem 9
  pair O = (0,0), A = r*expi(pi/3); D(CR(O,r));
  5 KB (763 words) - 16:20, 28 September 2019
• 1989 AIME Problems/Problem 10
  ...c</math> are legs of right triangle <math>abc</math> with <math>\beta = 90^o</math> and <math>c=1</math>
  8 KB (1,401 words) - 21:41, 20 January 2024
• 1990 AIME Problems/Problem 14
  triple O=(0,0,0); triple O=(0,0,0);
  7 KB (1,086 words) - 08:16, 29 July 2023
• 1991 AIME Problems/Problem 12
  ...)",S,W); label("\(15\)",B/2+P/2,N);label("\(20\)",B/2+Q/2,E);label("\(O\)",O,SW); </asy></center> ...h>\triangle APS \cong \triangle CRQ</math>). Quickly we realize that <math>O</math> is also the center of the rectangle.
  8 KB (1,270 words) - 23:36, 27 August 2023
• 1991 AIME Problems/Problem 11
  ...the center of the dodecagon, which we denote <math>A, M,</math> and <math>O</math> respectively. Notice that <math>OM=1</math>, and that <math>\triangl
  4 KB (740 words) - 19:33, 28 December 2022
• 1991 AIME Problems/Problem 9
  ...ot \beta = \frac{(o+h)(a+h)}{oa} = \frac{oa +oh +ha +h^2}{oa} = 1+ \frac{h(o+a+h)}{oa} = 1+ \alpha + \beta</math>. From the information provided in the
  10 KB (1,590 words) - 14:04, 20 January 2023
• 1992 AIME Problems/Problem 14
  ..., <math>BB'</math>, and <math>CC'</math> are concurrent at the point <math>O^{}_{}</math>, and that <math>\frac{AO^{}_{}}{OA'}+\frac{BO}{OB'}+\frac{CO}{ ...bove solutions, find <math>\sum_{cyc} \frac{y+z}{x}=92</math> (where <math>O=(x:y:z)</math> in barycentric coordinates). Now letting <math>y=z=1</math>
  4 KB (667 words) - 01:26, 16 August 2023
• 2006 AMC 10B Problems/Problem 19
  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 pair O=origin, A=(1,0), C=(0,1), B=(1,1), D=(1, sqrt(3)), E=(sqrt(3), 1), point=B;
  5 KB (873 words) - 15:39, 29 May 2023
• 2006 AMC 10B Problems/Problem 24
  .... Points <math>A</math> and <math>B</math> on the circle with center <math>O</math> and points <math>C</math> and <math>D</math> on the circle with cent pair X=(-6,0), O=origin, P=(6,0), B=tangent(X, O, 2, 1), A=tangent(X, O, 2, 2), C=tangent(X, P, 4, 1), D=tangent(X, P, 4, 2);
  4 KB (558 words) - 14:38, 6 April 2024
• 1993 AIME Problems/Problem 14
  ...hen by symmetry, the other rectangle is also centered at the origin, <math>O</math>.
  3 KB (601 words) - 09:25, 19 November 2023
• 1993 AIME Problems/Problem 13
  ...math>P</math> and <math>Q</math> be the points of tangency of circle <math>O</math> to <math>AC</math> and <math>BD</math> respectively. pair A,B,C,D,P,Q,O,X;
  8 KB (1,231 words) - 20:06, 26 November 2023
• 1994 AIME Problems/Problem 8
  ...Thus, <math>M=\left(\frac{a+b}{2}, 24\right)</math>. The vector from <math>O</math> to <math>M</math> is <math>\left[\frac{a+b}{2}, 24\right]</math>. Me
  5 KB (788 words) - 13:53, 8 July 2023
• 1994 AIME Problems/Problem 2
  Call the center of the larger circle <math>O</math>. Extend the diameter <math>\overline{PQ}</math> to the other side of
  2 KB (272 words) - 03:53, 23 January 2023
• 1995 AIME Problems/Problem 14
  Let the center of the circle be <math>O</math>, and the two chords be <math>\overline{AB}, \overline{CD}</math> and ...pi/6), D=E+48*expi(7*pi/6), A=E+30*expi(5*pi/6), C=E+30*expi(pi/6), F=foot(O,B,A);
  3 KB (484 words) - 13:11, 14 January 2023
• 1995 AIME Problems/Problem 12
  triple A, B, C, D, O, P; O = (0,0,sqrt(2*sqrt(2)));
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1996 AIME Problems/Problem 15
  In [[parallelogram]] <math>ABCD</math>, let <math>O</math> be the intersection of [[diagonal]]s <math>\overline{AC}</math> and pair B=(0,0), A=expi(pi/4), C=IP(A--A + 2*expi(17*pi/12), B--(3,0)), D=A+C, O=IP(A--C,B--D);
  5 KB (710 words) - 21:04, 14 September 2020
• 1996 AIME Problems/Problem 7
  pathpen = black; pair O = (3.5,3.5); D(O); D(arc(O,1,280,350),EndArrow(4));
  4 KB (551 words) - 11:44, 26 June 2020
• 1996 AIME Problems/Problem 4
  triple O=(0,0,0), P=(0,0,unit+unit/(r-1)); dot(P); draw(O--P); draw(O--(unit,0,0)--(unit,0,unit)--(0,0,unit)); draw(O--(0,unit,0)--(0,unit,unit)--(0,0,unit));
  2 KB (257 words) - 17:50, 4 January 2016
• 1997 AIME Problems/Problem 15
  ...math>AE=s</math> and thus circumradius <math>\frac{s}{2}</math>. Let <math>O</math> be its circumcenter. By Inscribed Angles, <math>\angle{BOD'}=2\angle
  4 KB (609 words) - 22:49, 17 July 2023
• 1997 AIME Problems/Problem 11
  Consider the rhombus <math>OABC</math> on the complex plane such that <math>O</math> is the origin, <math>A</math> represents <math>\text{cis } n^\circ</
  10 KB (1,514 words) - 14:35, 29 March 2024
• 1998 AIME Problems/Problem 4
  Let <math>O</math> stand for an odd number and <math>E</math> an even. ...<math>O</math>'s and the other two have two <math>E</math>'s and an <math>O</math> in them, respectively.) . Let's do the case <math>OOO</math>, <math>
  5 KB (917 words) - 02:37, 12 December 2022
• 1999 AIME Problems/Problem 15
  ...ely. Suppose <math>P</math> is the apex of the tetrahedron, and let <math>O</math> be the foot of the altitude from <math>P</math> to <math>\triangle A <b>Lemma:</b> The point <math>O</math> is the orthocenter of <math>\triangle ABC</math>.
  7 KB (1,169 words) - 15:28, 13 May 2024
• 1999 AIME Problems/Problem 4
  pair W=dir(225), X=dir(315), Y=dir(45), Z=dir(135), O=origin; dot(O);
  3 KB (398 words) - 13:27, 12 December 2020
• 2000 AIME I Problems/Problem 13
  Let the intersection of the highways be at the origin <math>O</math>, and let the highways be the x and y axes. We consider the case wher pair O=(0,0), B=(5,0), A=1.4*expi(atan(24/7)), C=1.4*expi(atan(7/24));
  3 KB (571 words) - 00:38, 13 March 2014
• 2001 AIME I Problems/Problem 14
  ...'s because no three o's can be adjacent, but there can be a maximum of two o's placed on the very left or right. Note that according to the [[Pigeonhole ...ixed so we count the number of ways to insert <math>19 - 10 - 9 = 0</math> o's to <math>10+1 = 11</math> spots, or <math>\binom{11}{0} = 1</math>.
  13 KB (2,298 words) - 19:46, 9 July 2020
• 2014 USAJMO Problems
  ...lateral, acute triangle with <math>\angle A=60^\circ</math>, and let <math>O</math> and <math>H</math> denote the circumcenter and orthocenter of <math>
  3 KB (600 words) - 16:42, 5 August 2023
• 2003 AIME I Problems/Problem 7
  pair O=(0,0),A=(-15,0),B=(-6,0),C=(15,0),D=(0,8);
  3 KB (490 words) - 18:13, 13 February 2021
• 2002 AIME II Problems/Problem 15
  pair[] O; O[1] = (r[1]/(2/3*sqrt(17/13)),r[1]);
  7 KB (1,182 words) - 09:56, 7 February 2022
• 2002 AIME II Problems/Problem 14
  ...ight angle]]. A [[circle]] of [[radius]] <math>19</math> with center <math>O</math> on <math>\overline{AP}</math> is drawn so that it is [[Tangent (geom Now use similarity, draw perpendicular from <math>O</math> to <math>PM</math>, name the new point <math>D</math>. Triangle <mat
  4 KB (658 words) - 19:15, 19 December 2021
• 2002 AIME II Problems/Problem 12
  Using X to represent a basket and O to represent a failure, this 'earliest' solution may be represented as:
  7 KB (1,127 words) - 13:34, 19 June 2022
• 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
• Congruent (geometry)
  ...half-ray of the straight line <math>a'</math> emanating from a point <math>O'</math> of this line. Then in the plane <math>\alpha'</math> there is one a
  10 KB (1,655 words) - 21:43, 24 March 2022
• 2005 AMC 10A Problems/Problem 12
  pair O=(0,0), A=dir(0), B=dir(60), C=dir(120), D=dir(180); draw(D--E--B--O--C--B--A,linetype("4 4"));
  2 KB (246 words) - 12:43, 13 December 2021
• Circumcenter
  ...math> is equidistant from <math>A</math> and <math>{C}</math>; hence <math>O</math> also lies on the perpendicular bisector of <math>AC</math> (and is t
  2 KB (389 words) - 14:17, 4 August 2020
• Orthocenter
  ...C=(10,0), A1 = (B+C)/2, O = circumcenter(A,B,C), G = (A+B+C)/3, H = 3*G-2*O; draw(A1--G--O--cycle);
  5 KB (829 words) - 13:11, 20 February 2024
• Chord
  ...cle]] <math>O</math> is a [[line segment]] joining two [[point]]s on <math>O</math>. pair O=origin,A=dir(135),B=dir(30);
  522 bytes (103 words) - 12:16, 18 February 2018
• 2007 iTest Problems
  \text{(O) }2007\qquad</math> \text{(O) }520\qquad
  30 KB (4,794 words) - 23:00, 8 May 2024
• 1981 IMO Problems/Problem 5
  ...[[incenter]] and the [[circumcenter]] of the triangle and the point <math>O </math> are [[collinear]]. ...h>O</math> is clearly the circumcenter of <math>O_A O_B O_C </math>, <math>O</math> is collinear with the incenter and circumcenter of <math>ABC</math>,
  2 KB (373 words) - 23:09, 29 January 2021
• 2005 AMC 10A Problems
  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 pair O=(0,0), A=dir(0), B=dir(60), C=dir(120), D=dir(180);
  14 KB (2,026 words) - 11:45, 12 July 2021
• Euler line
  ...nter]] <math>H</math>, [[centroid]] <math>G</math>, [[circumcenter]] <math>O</math>, [[nine-point center]] <math>N</math> and [[De Longchamps point | de ...ntroid of <math>\triangle O_AO_BO_C</math>. Thus, <math>\mathcal{S}(\{O_A, O, G\}) = \{A, H, G\}</math>. As a homothety preserves angles, it follows tha
  59 KB (10,203 words) - 04:47, 30 August 2023
• Nine-point circle
  ...riangle AHZ.</math> Circumcenter of <math>\triangle ABC</math> point <math>O</math> is the midpoint <math>AZ,</math> point <math>D</math> is the midpoin Point <math>O</math> is the circumcenter of <math>\triangle XYZ \implies H</math> is the
  6 KB (994 words) - 16:02, 12 March 2024
• 1985 IMO Problems
  A circle with center <math>O</math> passes through the vertices <math>A</math> and <math>C</math> of the
  3 KB (465 words) - 03:00, 29 March 2021
• 1985 IMO Problems/Problem 1
  Let <math>O</math> be the center of the circle mentioned in the problem. Let <math>T</ Let <math>O </math> be the center of the circle mentioned in the problem, and let <math
  4 KB (684 words) - 07:28, 3 October 2021
• 2001 IMO Problems/Problem 1
  ...\triangle ABC</math> issuing from the vertex <math>A</math>, and let <math>O</math> be the [[circumcenter]] of triangle <math>\triangle ABC</math>. Assu Now <math>O</math> cannot coincide with <math>Y</math> (otherwise <math>\angle A</math>
  2 KB (417 words) - 17:24, 21 July 2018
• 2003 AMC 10A Problems/Problem 25
  ...since floor function always yields an integer, and 99 is divisible by 11 w/o any remainder). After we come to this conclusion, it becomes easy to solve
  6 KB (943 words) - 20:57, 29 May 2023
• 2004 AMC 10A Problems
  pair O=origin; fill(O--Arc(O, 2, 20, 160)--cycle, mediumgray);
  15 KB (2,092 words) - 20:32, 15 April 2024
• 2003 AMC 12A Problems/Problem 1
  Sum of first <math>n</math> odd numbers, is <math>S_O=n^{2}</math>, (O standing for odd). ...heir difference is <math>|1-0|=1</math>. Similarly, take take series <math>O</math> and <math>E_2</math>. The first terms are <math>1</math> and <math>2
  3 KB (436 words) - 20:31, 28 December 2021
• Sector
  pair O=origin, A=dir(angle2), B=dir(angle1); path sector=O--B--arc(O,1,angle1,angle2)--A--cycle;
  1 KB (178 words) - 21:12, 24 April 2008
• 2005 Alabama ARML TST Problems/Problem 3
  ...ngle and <math>COM</math> is a [[median of a triangle | median]], so <math>O</math> trisects <math>CO</math> and <math>R = CO = 2OM = 2r</math>.)
  1 KB (221 words) - 19:38, 6 February 2010
• 2020 AMC 10A Problems
  pair O = (0.851, 0.461); f[2] = E--F--G--O--N--cycle;
  13 KB (1,968 words) - 18:32, 29 February 2024
• Asymptote: Useful commands and their Output
  pair A,B,C,O,I; O=circumcenter(A,B,C); // olympiad - circumcenter
  6 KB (871 words) - 21:14, 12 June 2023
• 2005 AMC 10A Problems/Problem 23
  pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); dot(O);
  6 KB (1,045 words) - 09:46, 4 April 2023
• Mock AIME 1 Pre 2005 Problems
  ...</math> is a regular heptagon inscribed in a unit circle centered at <math>O</math>. <math>l</math> is the line tangent to the circumcircle of <math>ABC
  6 KB (1,100 words) - 22:35, 9 January 2016
• Mock AIME 2 Pre 2005/Problems
  ...th>BC = 4</math>, <math>CD = 1</math>, and <math>DA = 7</math>. Let <math>O</math> and <math>P</math> denote the circumcenter and intersection of <math
  6 KB (990 words) - 15:23, 11 November 2009
• Mock AIME 3 Pre 2005 Problems
  ...th>P's</math> is a word in Zuminglish if and only if between any two <math>O's</math> there appear at least two consonants. Let <math>N</math> denote th
  7 KB (1,135 words) - 23:53, 24 March 2019
• Mock AIME 3 Pre 2005 Problems/Problem 5
  ...th>P's</math> is a word in Zuminglish if and only if between any two <math>O's</math> there appear at least two consonants. Let <math>N</math> denote th ...rm a word of length <math>n+1</math> with a <tt>CV</tt> by appending <math>O</math> to the end of a word of length <math>n</math> that ends with <tt>CC<
  5 KB (795 words) - 16:03, 17 October 2021
• Arc
  The ''measure'' of a circular arc <math>AB</math> on circle <math>O</math> is defined to be the measure of the [[central angle]] <math>\angle A
  1 KB (170 words) - 00:24, 16 December 2023
• 2007 AIME II Problems/Problem 3
  ...th>E</math> across the center of the square, which we will denote as <math>O</math>. Since <math>\angle BEA</math> and <math>\angle AOB</math> are right ...st step that <math>F</math> is a reflection of <math>E</math> across <math>O</math>, we can say that <math>EF=2EO=17\sqrt{2}</math>. This gives that <cm
  6 KB (933 words) - 00:05, 8 July 2023
• 2007 AIME II Problems/Problem 15
  ...A$",X,right,p); dot("$O_B$",Y,left+up,p); dot("$O_C$",Z,right+up,p); dot("$O$",circumcenter(X,Y,Z),right+down,p); dot("$I$",P,left+up,p); ..."$O_A$",X,right,p); dot("$O_B$",Y,left+up,p); dot("$O_C$",Z,down,p); dot("$O$",Zp,dir(-45),p+red); dot("$I$",P,left+up,p);
  11 KB (2,099 words) - 17:51, 4 January 2024
• 1989 USAMO Problems
  ...enter of the inscribed circle of triangle <math>ABC</math> and point <math>O</math> is the center of the circumscribed circle, prove that line <math>IO<
  2 KB (326 words) - 18:52, 18 July 2016
• 2007 USAMO Problems/Problem 2
  ...argement would cause it to violate the non-overlap condition. Then <math>D(O,r)</math> is tangent to at least three discs in <math>\mathcal{F}</math>. O Note that <math>r < 1/\sqrt{2}</math> because <math>D(O,r)</math> covers no grid point, and <math>(a - 3r)(b - 3r)\geq (5 - 3r)^2</
  5 KB (754 words) - 03:41, 7 August 2014
• 2007 USAMO Problems/Problem 6
  pair A=(2,8), B=(0,0), C=(13,0), I=incenter(A,B,C), O=circumcenter(A,B,C), p_a, q_a, X, Y, X1, Y1, D, E, F; real r=abs(I-foot(I,A,B)), R=abs(A-O), a=abs(B-C), b=abs(A-C), c=abs(A-B), x=(((b+c-a)/2)^2)/(r^2+4*r*R+((b+c-a)
  7 KB (1,274 words) - 15:11, 31 August 2017
• Pentagon
  # Draw circle <math>O</math> (red). ...iameter <math>AB</math> and construct a perpendicular radius through <math>O</math>.
  1 KB (191 words) - 09:59, 6 June 2022
• 2007 AMC 12B Problems/Problem 3
  The point <math>O</math> is the center of the circle circumscribed about triangle <math>ABC</ ...^o</math> and <math>\angle OBC=30^o</math>. Therefore, <math>\angle ABC=50^o</math>, or <math>\mathrm{(D)}</math>.
  905 bytes (130 words) - 10:39, 27 February 2022
• Circumradius
  pair O, A, B, C, D; O=(0,0);
  4 KB (729 words) - 16:52, 19 February 2024
• Dodecagon
  pair O = (dir(360/12*0)+dir(360/12*6))/2; label("O",O,S);
  1 KB (219 words) - 13:08, 15 June 2018
• 2007 AMC 12A Problems/Problem 12
  Case 1: <math>odd-odd</math> (which must be <math>o \cdot o-o \cdot o</math>). The probability for this to occur is <math>\left(\frac 12\right)^4 ...any kind), and (<math>e \cdot e-o \cdot e</math>) with its reverse, (<math>o \cdot e-e \cdot e</math>).
  3 KB (445 words) - 08:59, 24 March 2023
• 2007 AMC 12A Problems/Problem 25
  ...e balls there are at least two dividers. So for example, o | | o | | o | | o | | represents <math>{1,4,7,10}</math>. ...e arrangment o | | o | | o | | | o | corresponds to the arrangment o o o | o |. Notice that there is no longer any restriction on consectutive numbers.
  9 KB (1,461 words) - 23:07, 27 January 2024
• 2005 AMC 12A Problems/Problem 15
  pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); dot(O);
  14 KB (1,970 words) - 17:02, 18 August 2023
• 2006 AIME II Problems/Problem 12
  path O = CP((0,-2),A); pair G = OP(A--F,O); D(MP("A",A,N,s)--MP("B",B,W,s)--MP("C",C,E,s)--cycle);D(O);
  6 KB (1,033 words) - 02:36, 19 March 2022
• 1998 PMWC Problems
  label("O",(0,0),SE); ...acute angle formed by the hour hand and the minute hand at <math>6</math> o'clock <math>75</math> minutes?
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2001 PMWC Problems
  pair A,B,C,D,E,F,G,O; O=(A+B+C+D)/4;
  4 KB (641 words) - 21:24, 21 April 2014
• 1971 Canadian MO Problems
  ...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
  3 KB (519 words) - 08:58, 13 September 2012
• 1999 USAMO Problems/Problem 5
  ...s 2000</math> grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes t ...quare surrounded by two empty squares. In both cases, the player places an O in that square. In both situations, it is easy to verify that the first pla
  2 KB (433 words) - 13:35, 4 July 2013
• 1989 USAMO Problems/Problem 4
  ...enter of the inscribed circle of triangle <math>ABC</math> and point <math>O</math> is the center of the circumscribed circle, prove that line <math>IO< Consider the lines that pass through the circumcenter <math>O</math>. Extend <math>AO</math>, <math> BO</math>, <math>CO</math> to <math>
  2 KB (290 words) - 19:11, 18 July 2016
• 2007 AMC 12B Problems
  The point <math>O</math> is the center of the circle circumscribed about triangle <math>ABC</ ...math>AB=BC=CD=DE=EA=2</math> and <math>\angle ABC=\angle CDE=\angle DEA=90^o</math>. The plane of <math>\triangle ABC</math> is parallel to <math>\overl
  12 KB (1,814 words) - 12:58, 19 February 2020
• 1960 IMO Problems/Problem 6
  ...be the angle with arms <math>AO</math> and <math>AA_1</math>, where <math>O</math> is the center of the circle; <math>0 < \theta < \frac{\pi}{4}</math> Knowing <math>\angle A_1 A H_1 = 2 \times \angle A_1 A O = 2 \theta</math>, and using the formula for a double-angle tangent, we can
  7 KB (1,214 words) - 18:49, 29 January 2018
• Asymptote: Useful functions
  ...ons, you can replace <code>line l1, line l2</code> with <code>pair A, pair O, pair B</code>, which would mark the angle <math>\angle AOB</math>. The fun pair O = (0,0);
  4 KB (646 words) - 21:18, 26 March 2024
• Excircle
  ...ertices <math>B,C</math> of triangle <math>\triangle ABC</math>. Let <math>O</math> be the circumcenter <math>\triangle ABC</math>. Prove that <cmath>OA
  5 KB (843 words) - 03:02, 1 July 2020
• 2001 USAMO Problems/Problem 2
  ...ion for OA */ pair Ca = 2C-A, Cb = bisectorpoint(Ca,C,B), OA = IP(A--A+10*(O-A),C--C+50*(Cb-C)), D2 = D(MP("D_2",foot(OA,B,C))), Fa=2B-A, Ga=2C-A, F=MP(
  11 KB (2,091 words) - 08:35, 16 November 2017
• 1992 USAMO Problems/Problem 2
  ...h>M_k = (1, \tan k^\circ)</math> in the coordinate plane with origin <math>O=(0,0)</math>, for integers <math>0 \le k \le 89</math>. pair O=(0,0), a=expi(0), b=expi(1/6), c=expi(2/6), d=expi(3/6), y=expi(32/30), z=
  4 KB (628 words) - 07:41, 19 July 2016
• De Longchamps point
  Let <math>H, O,</math> and <math>L</math> be orthocenter, circumcenter, and De Longchamps ...point of intersection of the heights <math>H</math> with respect to <math>O.</math>
  10 KB (1,780 words) - 09:23, 17 November 2022
• 2000 AMC 12 Problems/Problem 17
  A circle centered at <math>O</math> has radius <math>1</math> and contains the point <math>A</math>. The pair O=(0,0);
  6 KB (979 words) - 12:50, 17 July 2022
• 2000 AMC 12 Problems/Problem 24
  ...eral, it follows that <math>MB=18/{\pi}</math>, and <math>OA</math> (where O is the center of the circle) is <math>36/{\pi}-r</math>. By the Pythagorea
  2 KB (263 words) - 19:59, 18 April 2024
• 2007 AMC 10A Problems
  ...and <math>B</math> each have radius <math>2</math>, as shown. Point <math>O</math> is the midpoint of <math>\overline{AB}</math>, and <math>OA = 2\sqrt pair O=(0,0);
  13 KB (2,058 words) - 17:54, 29 March 2024
• 1995 AHSME Problems
  Two rays with common endpoint <math>O</math> form a <math>30^\circ</math> angle. Point <math>A</math> lies on one ...d <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math>, and chord <math>\ov
  17 KB (2,387 words) - 22:44, 26 May 2021
• 1995 AHSME Problems/Problem 28
  We let <math>O</math> be the center, <math>\overline{A_1AA_2}</math>, <math>\overline{B_1B Let <math>O</math> be the center of the circle, <math>H</math> be the midpoint of <math
  3 KB (509 words) - 22:56, 5 December 2023
• 1951 AHSME Problems
  At <math> 2: 15</math> o'clock, the hour and minute hands of a clock form an angle of: <math>AB</math> is a fixed diameter of a circle whose center is <math>O</math>. From <math>C</math>, any point on the circle, a chord <math>CD</mat
  23 KB (3,641 words) - 22:23, 3 November 2023
• 1966 AHSME Problems
  <math>AB</math> is the diameter of a circle centered at <math>O</math>. <math>C</math> is a point on the circle such that angle <math>BOC</ Five points <math>O,A,B,C,D</math> are taken in order on a straight line with distances <math>O
  19 KB (3,159 words) - 22:10, 11 March 2024
• 2003 AMC 12B Problems/Problem 22
  pair O = (15*15/17,8*15/17), C = (17,0), D = (0,0), P = (25.6,19.2), Q = (25.6, 18 pair A = 2*O-C, B = 2*O-D;
  4 KB (551 words) - 14:17, 23 June 2022
• 2003 AMC 12B Problems/Problem 21
  <math>O</math> is the point that satisfies the system of equations: <math>\begin{ca ..., <math>y = - \frac52</math>, <math>x = \frac{5 \sqrt{3}}{2}</math>, <math>O = (\frac{5 \sqrt{3}}{2}, - \frac52)</math>
  2 KB (372 words) - 09:38, 1 September 2022
• 2007 AMC 10A Problems/Problem 24
  ...and <math>B</math> each have radius <math>2</math>, as shown. Point <math>O</math> is the [[midpoint]] of <math>\overline{AB}</math>, and <math>OA = 2\ label("$O$",(0,0),dir(270));
  3 KB (439 words) - 15:39, 3 June 2021
• 1998 AHSME Problems/Problem 26
  label("\(O\)",circumcenter(A,B,C),SW);
  3 KB (391 words) - 14:30, 5 July 2013
• 1995 USAMO Problems
  Given a nonisosceles, nonright triangle <math>\, ABC, \,</math> let <math>\, O \,</math> denote the center of its circumscribed circle, and let <math>\, A
  3 KB (540 words) - 13:31, 4 July 2013
• 2004 AMC 12B Problems/Problem 19
  pair P=(D+G)/2, Q=(D+H)/2, R=(B+E)/2, T=(A+H)/2, O=(E+G)/2; dot(E);dot(F);dot(G);dot(H);dot(O);
  3 KB (520 words) - 19:12, 20 November 2023
• 2007 AMC 10B Problems/Problem 11
  Let <math>\triangle ABC</math> have vertex <math>A</math> and center <math>O</math>, with foot of altitude from <math>A</math> intersecting <math>BC</ma pair O=circumcenter(A,B,C);
  5 KB (851 words) - 22:02, 26 July 2021
• Isogonal conjugate
  ...nt <math>O</math> be called isogonals with respect to the pair <math>(\ell,O).</math> ...h the line <math>\ell</math> and the origin coincides with the point <math>O,</math> then the isogonals define the equations <math>y = \pm kx,</math> an
  54 KB (9,416 words) - 08:40, 18 April 2024
• 2008 AMC 10A Problems
  Points <math>A</math> and <math>B</math> lie on a circle centered at <math>O</math>, and <math>\angle AOB = 60^\circ</math>. A second circle is internal
  14 KB (2,138 words) - 15:08, 18 February 2023
• 2008 AMC 12A Problems
  Points <math>A</math> and <math>B</math> lie on a circle centered at <math>O</math>, and <math>\angle AOB = 60^\circ</math>. A second circle is internal ...math>y</math> axis, and one on the positive <math>z</math> axis. Let <math>O</math> be the origin. What is the volume of <math>OABC</math>?
  13 KB (2,025 words) - 13:56, 2 February 2021
• 2008 AMC 12A Problems/Problem 13
  ...ts <math>A</math> and <math>B</math> lie on a [[circle]] centered at <math>O</math>, and <math>\angle AOB = 60^\circ</math>. A second circle is internal pair O=(0,0), A=(3,0), B=(3/2,3/2*3^.5), C=(3^.5,1), D=(3^.5,0), F=(1.5*3^.5,1.5),
  4 KB (630 words) - 20:32, 4 June 2021
• 2008 AMC 12A Problems/Problem 18
  ...math>y</math>-axis, and one on the positive <math>z</math>-axis. Let <math>O</math> be the [[origin]]. What is the volume of [[tetrahedron]] <math>OABC< ..."A",(8,0),(1,0));label("B",(0,10),(0,1));label("C",(-3,-4),(-1,-1));label("O",(0,0),(1,1));
  2 KB (302 words) - 04:51, 16 January 2023
• 2008 AMC 12A Problems/Problem 20
  pair O=incenter(A,C,D), P=incenter(B,C,D); dot(O);dot(P);
  6 KB (951 words) - 16:31, 2 August 2019
• 2008 AMC 12A Problems/Problem 22
  Let one of the mats be <math>ABCD</math>, and the center be <math>O</math> as shown: label("\(O\)",(0.00,-0.10),E);
  10 KB (1,518 words) - 17:00, 16 May 2023
• 2008 AMC 10B Problems
  ...th>OA</math> is rotated <math>90^\circ</math> counterclockwise about <math>O</math>. What are the coordinates of the image of <math>A</math>?
  12 KB (1,838 words) - 16:52, 7 October 2022
• 2008 AMC 12B Problems
  On circle <math>O</math>, points <math>C</math> and <math>D</math> are on the same side of di pair O = (0,0);
  14 KB (2,199 words) - 13:43, 28 August 2020
• 2008 AMC 12B Problems/Problem 4
  On circle <math>O</math>, points <math>C</math> and <math>D</math> are on the same side of di pair O = (0,0);
  1 KB (183 words) - 22:35, 10 June 2017
• 2008 AMC 12B Problems/Problem 9
  ...erline{AB}</math>, and <math>O</math> the center of the circle. Then <math>O</math>, <math>C</math>, and <math>D</math> are collinear, and since <math>D pair O = (0,0), A=(5,0), B = IP(p,CR(A,6)), C = IP(p,CR(A,3)), D=IP(A--B,O--C);
  2 KB (319 words) - 13:48, 15 February 2021
• 2008 AIME I Problems/Problem 14
  pair O = (0,0), B = O - (9,0), A= O + (9,0), C=A+(18,0), T = 9 * expi(-1.2309594), P = foot(A,C,T); draw(Circle(O,9)); draw(B--C--T--O); draw(A--P); dot(A); dot(B); dot(C); dot(O); dot(T); dot(P);
  8 KB (1,333 words) - 00:18, 1 February 2024
• Mock AIME 5 Pre 2005 Problems
  ...and such that <math>\frac {j + k}{j - k} = \frac {o}{p}</math>, find <math>o + p.</math>
  6 KB (909 words) - 07:27, 12 October 2022
• Mock AIME 1 2007-2008 Problems/Problem 14
  ...math>AB = BC</math>. Let <math>D</math> be the intersection of <math>\odot O</math> and <math>OC</math> such that <math>CD = \frac {1}{18}</math> and <m pair O = (0,0), A=(r,0), B= A*dir(112.02432), C=2*B-A;
  2 KB (375 words) - 15:23, 25 November 2009
• Mock AIME 1 2007-2008 Problems
  ...math>AB = BC</math>. Let <math>D</math> be the intersection of <math>\odot O</math> and <math>OC</math> such that <math>CD = \frac {1}{18}</math> and <m
  6 KB (992 words) - 14:15, 13 February 2018
• 2008 AIME II Problems/Problem 9
  ...>xy</math>-plane, <math>r</math> be the length <math>OP</math> where <math>O</math> is the origin, and <math>\theta</math> be the inclination of OP to t Since <math>R+R^{7}=O, R^2+R^6=O, R^3+R^5=O, I+R^4=O</math>, so we have <math>R^6\begin{pmatrix}5\\0\end{pmatrix}+(-R^6-R^7)\beg
  5 KB (725 words) - 22:37, 28 January 2024
• 2008 AIME II Problems/Problem 11
  Let the incenter be O and the altitude from A to <math>\overline{BC}</math> be T. Note that by AA
  6 KB (1,065 words) - 20:12, 9 August 2022
• 2005 USAMO Problems/Problem 3
  path O=circumcircle(B1,C1,P); pair Q=IntersectionPoint(O,B--C,1);
  6 KB (1,117 words) - 01:17, 11 October 2021
• 2008 AMC 10B Problems/Problem 10
  Let the center of the circle be <math>O</math>, and let <math>D</math> be the intersection of <math>\overline{AB}</ pair O = (0,0), A=(5,0), B = IP(p,CR(A,6)), C = IP(p,CR(A,3)), D=IP(A--B,O--C);
  1 KB (247 words) - 12:36, 7 June 2021
• 2008 AMC 10B Problems/Problem 14
  ...th>OA</math> is rotated <math>90^\circ</math> counterclockwise about <math>O</math>. What are the coordinates of the image of <math>A</math>? <!-- don't Rotate this triangle <math>90^\circ</math> counterclockwise around <math>O</math>, and you will find that <math>A</math> will end up in the second qua
  3 KB (444 words) - 04:45, 21 August 2023
• 2001 Iran NMO (Round 3) Problems/Problem 5
  path O=circumcircle(A,B,C); pair M=IntersectionPoint(A--Ia,O,1);
  3 KB (437 words) - 15:47, 27 April 2008
• 2008 USAMO Problems/Problem 2
  ...le OFA = 90^{\circ}</math>, as this would immediately prove that <math>A,P,O,F,N</math> are concyclic. ...and <math>OD</math> bisects exterior <math>\angle FDE</math>, making <math>O</math> the <math>F</math>-excentre of <math>\triangle FED</math>. This impl
  20 KB (3,565 words) - 11:54, 1 May 2024
• 1966 AHSME Problems/Problem 8
  Let <math>O</math> be the center of the circle of radius <math>10</math> and <math>P</m ...base <math>AB</math>. The height of <math>\triangle OAB</math> from <math>O</math> to <math>AB</math> is <math>\sqrt {\overline{OB}^2 - (\frac{\overlin
  2 KB (256 words) - 01:45, 26 June 2016
• 1995 AHSME Problems/Problem 26
  ...d <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math>, and chord <math>\ov pair O=origin, A=(-5,0), B=(5,0), C=(0,5), D=(0,-5), F=5*dir(40), E=intersectionpo
  2 KB (359 words) - 20:01, 23 January 2017
• 1995 AHSME Problems/Problem 18
  Two rays with common endpoint <math>O</math> forms a <math>30^\circ</math> angle. Point <math>A</math> lies on on Triangle <math>OAB</math> has the property that <math>\angle O=30^{\circ}</math> and <math>AB=1</math>.
  1 KB (167 words) - 13:59, 5 July 2013
• 1999 AHSME Problems/Problem 29
  pair O = (0,0), A = expi(pi/2), B = expi(7 * pi/6), C= expi(11 * pi / 6); D(CR(O, 1)); D(A--B--C--A); D(CR((0,-3/4),1/4));
  3 KB (522 words) - 11:39, 3 October 2023
• New York MathCounts
  * [[Home Educators Enrichment Group]] Coach Mary O'Keeffe.
  1 KB (138 words) - 09:58, 3 May 2010
• 1988 IMO Problems/Problem 1
  ...dii <math>R</math> and <math>r</math> (<math>R>r</math>) with center <math>O</math>. Fix <math>P</math> on the small circle and consider the variable c
  3 KB (545 words) - 11:32, 30 January 2021
• 2003 USAMO Problems/Problem 4
  pair B=origin, A=dir(theta), C=A+(rotate(78)*0.8*A), O=IP(CR(B,r),CR(A,r)); path c=CR(O,r);
  5 KB (820 words) - 02:39, 10 January 2023
• 2001 IMO Shortlist Problems/G2
  ...gle <math>ABC</math> issuing from the vertex <math>A</math>, and let <math>O</math> be the circumcenter of triangle <math>ABC</math>. Assume that <math>
  677 bytes (95 words) - 16:06, 17 March 2022
• Millennium Problems
  ...in polynomial time (as a function of the input, often expressed using big-O notation) can also be ''solved'' in polynomial time. The set <math>P</math> ...nt statement of the Riemann hypothesis is that <math>\pi(x)=\mathrm{Li}(x)+O(x^{1/2}\ln(x))</math>.
  13 KB (1,969 words) - 17:57, 22 February 2024
• Carnot's Theorem
  ...>O</math> to the sides (i.e., signed lengths of the pedal lines from <math>O</math>) is: pair a,b,c,O,i,d,f,g;
  4 KB (723 words) - 01:45, 18 February 2021
• 1995 IMO Problems/Problem 1
  ...<math>AC</math> is line <math>AM</math>. Also, the radical axis of <math>O</math> and the circle with diameter <math>BD</math> is line <math>DN</math>
  5 KB (847 words) - 19:03, 12 October 2021
• 2000 USAMO Problems/Problem 5
  Let the [[circumcenter]] of <math>\triangle ABC</math> be <math>O</math>, and let the center of <math>\omega_k</math> be <math>O_k</math>. <m <math>O</math> is the intersection of the perpendicular bisectors of <math>\overlin
  3 KB (609 words) - 09:52, 20 July 2016
• 2008 iTest Problems
  ...o hours it's left stuck to the wall. One morning, at around <math>9</math> o' clock, Tony sticks the spider to the wall in the living room three feet ab
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2007 AMC 12B Problems/Problem 25
  ...math>AB=BC=CD=DE=EA=2</math> and <math>\angle ABC=\angle CDE=\angle DEA=90^o</math>. The plane of <math>\triangle ABC</math> is parallel to <math>\overl Now, note that <math>\angle CDE=\angle DEA=90^o</math>. This means that there exists some vector <math>DE</math> parallel t
  3 KB (470 words) - 19:46, 17 July 2023
• 2005 AMC 8 Problems
  label("$O$", (0,0), SW); ...a semicircle of area <math> 2\pi</math>. The circle has its center <math> O</math> on hypotenuse <math> \overline{AB}</math> and is tangent to sides <m
  13 KB (1,821 words) - 22:18, 5 December 2023
• Spiral similarity
  ...ABC</math> to <math>\triangle A'B'C', T(ABC) = A'B'C'.</math> Denote <math>O</math> the center of <math>T.</math> ...this center is point <math>O,</math> so these circles contain point <math>O</math>. Similarly for another circles.
  28 KB (4,863 words) - 00:29, 16 December 2023
• Van Aubel's Theorem
  pair A, B, C, D, O, P, Q, R, SS; O = (0,0) ;
  2 KB (410 words) - 14:01, 4 March 2023
• 2001 USAMO Problems/Problem 4
  ...90^\circ</math>, either <math>O</math> is on side <math>BC</math> or <math>O</math> and <math>A</math> are on opposite sides of line <math>BC</math>. In
  8 KB (1,470 words) - 22:24, 18 June 2022
• Neutralization
  ...h> ions from the acid (proton donor) react with <math>OH^-</math> or <math>O^{2-}</math> ions from the base (proton acceptor) to form water. The cations
  358 bytes (65 words) - 20:10, 23 January 2017
• 2002 AMC 10B Problems
  Using the letters <math>A</math>, <math>M</math>, <math>O</math>, <math>S</math>, and <math>U</math>, we can form five-letter "words"
  10 KB (1,540 words) - 22:53, 19 December 2023
• 2002 AMC 10B Problems/Problem 9
  Using the letters <math>A</math>, <math>M</math>, <math>O</math>, <math>S</math>, and <math>U</math>, we can form five-letter "words" Let <math>A = 1</math>, <math>M = 2</math>, <math>O = 3</math>, <math>S = 4</math>, and <math>U = 5</math>. Then counting backw
  1 KB (178 words) - 04:20, 4 November 2022
• 2004 AMC 10B Problems
  pair O=(0,0); path inner=Circle(O,r1), outer=Circle(O,r2);
  13 KB (1,988 words) - 20:19, 15 May 2024
• 1999 AHSME Problems/Problem 23
  pair O=(0,0), E=dir(0), NE=dir(60), NW=dir(120); draw(O -- (O+E) -- (O+E+4*NE) -- (O+E+4*NE+2*NW) -- (O-3*E+4*NE+2*NW));
  6 KB (949 words) - 17:48, 19 March 2020
• 1973 USAMO Problems
  ...of a regular tetrahedron <math>ABCD</math>. Prove that angle <math>PAQ<60^o</math>.
  2 KB (273 words) - 18:53, 3 July 2013
• 1974 USAMO Problems/Problem 3
  ...rically opposite A, and let <math>Q</math> be the plane through the center O of the sphere perpendicular to <math>BB'</math> and passing through the mid
  2 KB (360 words) - 23:13, 18 July 2016
• 1975 USAMO Problems/Problem 2
  ...f points A, B, C, and D, respectively, with respect to an arbitrary origin O. Let us also for simplicity define <math>a^2 = a \cdot a = ||a||^2</math>,
  6 KB (1,096 words) - 23:07, 26 August 2017
• 1960 IMO Problems/Problem 7
  ...XY is our axis of symmetry and it intersects with CD at a point O. Point O is our origin of reference whose coordinates are (0,0). Let our point P be on the axis of symmetry at z distance from the origin O.
  2 KB (410 words) - 15:25, 23 March 2020
• 1962 IMO Problems/Problem 6
  ...ry triangle <math>ABC</math>. Let <math>ABC</math> have circumcenter <math>O</math> and incenter <math>I</math>. Extend <math>AI</math> to meet the circ
  2 KB (308 words) - 06:29, 16 December 2023
• 2000 AMC 10 Problems
  ...= 2001</math>. What is the largest possible value of the sum <math>I + M + O</math>?
  14 KB (2,035 words) - 21:57, 2 May 2024
• 1976 USAMO Problems/Problem 2
  ...X=r*expi(pi/3), X1=r*expi(-pi/12), Y=r*expi(4*pi/3), Y1=r*expi(11*pi/12), O=(0,0), P, P1; draw(pica, circ1);draw(pica, B--A--P--Y--X);dot(pica,P^^O);
  5 KB (848 words) - 23:41, 6 July 2020
• 1976 USAMO Problems/Problem 4
  ...ahedron <math>PABC</math> (i.e., <math>\angle APB=\angle BPC=\angle CPA=90^o</math>) is <math>S</math>, determine its maximum volume.
  2 KB (358 words) - 23:15, 18 July 2016
• 1976 USAMO Problems
  ...X=r*expi(pi/3), X1=r*expi(-pi/12), Y=r*expi(4*pi/3), Y1=r*expi(11*pi/12), O=(0,0), P, P1; draw(pica, circ1);draw(pica, B--A--P--Y--X);dot(pica,P^^O);
  3 KB (510 words) - 19:01, 3 July 2013
• 1999 AHSME Problems/Problem 21
  Let <math>\circ{O}</math> be the circumcircle of <math>\triangle{XYZ}</math> in the problem,
  2 KB (396 words) - 18:47, 7 September 2023
• 2000 AMC 10 Problems/Problem 13
  label("O",(0,-.35),N);
  2 KB (308 words) - 21:32, 27 July 2022
• 1986 AJHSME Problems
  ...er. Which of the following statements about the whole number <math>(\text{o}^2+\text{no})</math> is always true? ...have their centers on <math>\text{AC}</math> and just touch at <math>\text{O}</math>, the center of the large circle. If each small circle has radius <
  14 KB (2,054 words) - 15:41, 8 August 2020
• 1986 AJHSME Problems/Problem 17
  ...er. Which of the following statements about the whole number <math>(\text{o}^2+\text{no})</math> is always true? ...of two odd numbers is odd. Since <math>\text{o}</math> is odd, <math>\text{o}^2</math> will also be odd. And adding two odd numbers makes an even numbe
  3 KB (457 words) - 15:02, 4 April 2021
• 1995 USAMO Problems/Problem 3
  Given a nonisosceles, nonright triangle <math>ABC,</math> let <math>O</math> denote its circumcenter, and let <math>A_1, \, B_1,</math> and <math '''LEMMA 1: ''' In <math>\triangle ABC</math> with circumcenter <math>O</math>, <math>\angle OAC = 90 - \angle B</math>.
  2 KB (364 words) - 01:42, 19 April 2024
• 1986 AJHSME Problems/Problem 23
  ...have their centers on <math>\text{AC}</math> and just touch at <math>\text{O}</math>, the center of the large circle. If each small circle has radius < pair A=(-2,0), O=origin, C=(2,0);
  2 KB (277 words) - 21:32, 3 July 2013
• 2004 AMC 10B Problems/Problem 22
  pair A, B, C, D, E, F, I, O; O = (B + C)/2;
  5 KB (700 words) - 13:46, 6 April 2024
• 2004 AMC 10B Problems/Problem 25
  pair O=(0,0), A=(0,1), B=(0,-1); path bigc1 = Circle(A,2), bigc2 = Circle(B,2), smallc = Circle(O,1);
  4 KB (703 words) - 22:37, 2 November 2022
• 2004 AMC 12B Problems/Problem 10
  pair O=(0,0); path inner=Circle(O,r1), outer=Circle(O,r2);
  2 KB (340 words) - 14:35, 23 April 2023
• 2004 AMC 12B Problems/Problem 14
  ...th>AC</math> that goes through <math>M</math>. Let them intersect at <math>O</math>. It follows that quadrilateral <math>MONC</math> is a square with si
  7 KB (1,083 words) - 22:41, 23 November 2020
• 2001 AMC 12 Problems/Problem 17
  </asy>Let <math>O=(0,0)</math>, <math>A=(0,2)</math>, <math>B=(4,0)</math>, <math>C=(2\pi+1,0
  5 KB (792 words) - 15:23, 30 November 2021
• 2009 AMC 10A Problems/Problem 7
  ...sest and less than 4 is 10/3, our answer. 38% fat is a lot of fat in milk :O
  1 KB (180 words) - 08:05, 19 July 2022
• 2002 AMC 12A Problems/Problem 18
  label("$O$", (0,0), SW ); ...le with the radii (6 and 9). This, as well as the two vertical angles near O, prove triangles <math>D_2QO</math> and <math>D_1PO</math> similar by AA, w
  3 KB (485 words) - 03:13, 1 September 2023
• 2009 AMC 10B Problems
  Points <math>A</math> and <math>C</math> lie on a circle centered at <math>O</math>, each of <math>\overline{BA}</math> and <math>\overline{BC}</math> a
  15 KB (2,262 words) - 00:53, 18 June 2021
• 2009 AMC 10B Problems/Problem 16
  Points <math>A</math> and <math>C</math> lie on a circle centered at <math>O</math>, each of <math>\overline{BA}</math> and <math>\overline{BC}</math> a pair B=(0,0), A=(3,0), C=3*dir(60), O=intersectionpoint( C -- (C+3*dir(-30)), A -- (A+3*dir(90)) );
  4 KB (576 words) - 19:59, 25 November 2023
• 2009 AMC 10B Problems/Problem 22
  Extend <math>RN</math> to intersect <math>PQ</math> at <math>O</math>: label("$O$",(0,1),N);
  12 KB (1,868 words) - 03:36, 30 September 2023
• 1988 AJHSME Problems
  label("O",(-2.5,0),W);
  14 KB (1,872 words) - 15:23, 17 January 2023
• Mock AIME 1 Pre 2005 Problems/Problem 10
  ...</math> is a regular heptagon inscribed in a unit circle centered at <math>O</math>. <math>l</math> is the line tangent to the circumcircle of <math>ABC
  540 bytes (96 words) - 00:28, 23 December 2023
• 2009 AIME I Problems
  Consider the set of all triangles <math>OPQ</math> where <math>O</math> is the origin and <math>P</math> and <math>Q</math> are distinct poi
  7 KB (1,152 words) - 02:24, 23 July 2021
• 2009 AIME I Problems/Problem 4
  pair O=extension(A,C,B,D); dot(A);dot(B);dot(C);dot(D);dot(O);dot(M);dot(NN);dot(P);
  7 KB (1,117 words) - 00:23, 9 January 2023
• 2009 AIME I Problems/Problem 11
  Consider the set of all triangles <math>OPQ</math> where <math>O</math> is the origin and <math>P</math> and <math>Q</math> are distinct poi ...e can calculate the height of <math>\triangle OPQ</math> from vertex <math>O</math> (the origin) to be:
  8 KB (1,319 words) - 15:01, 16 August 2020
• 2009 AIME I Problems/Problem 12
  ...th> and <math>DB=\dfrac{35^{2}}{37}</math> by similar triangles. Let <math>O</math> be the center of <math>\omega</math>; notice that <cmath>\tan(\angle Let <math>O</math> be center of the circle and <math>P</math>,<math>Q</math> be the two
  12 KB (1,970 words) - 22:53, 22 January 2024
• 2009 AIME I Problems/Problem 15
  ...CPB = 150</math>, so <math>\angle PBC = \angle PCB = 15</math>. Let <math>O</math> be the foot of the perpendicular from <math>P</math> to line <math>B
  6 KB (1,048 words) - 19:35, 2 January 2023
• Prime counting function
  ...show that <math>\pi(x) - </math> <math>\int^n_2 \frac{1}{log x} =\mathcal{O}(x^{\frac{1}{2}+\epsilon})</math>. This second statement is equivalent to t
  1 KB (238 words) - 13:45, 13 August 2015
• 1989 AJHSME Problems
  ...ngle between the hour hand and the minute hand on a clock that reads seven o'clock? label("S",(8.5,0),S); label("O",(9.5,0),S); label("N",(10.5,0),S); label("D",(11.5,0),S);
  13 KB (1,765 words) - 11:55, 22 November 2023
• 2008 IMO Problems/Problem 6
  Let B be the top vertex of triangle ABC, O and K are the centers of the incircles of triangles ABC and ADC with radii E be the foot of the projection of O to AB.
  1 KB (283 words) - 01:15, 19 November 2023
• 2009 AIME II Problems/Problem 10
  ...B=(0,0), A=(5,0), C=(0,13), E=(-5,0), O = incenter(E,C,A), D=IP(A -- A+3*(O-A),E--C); D(A--B--C--cycle); D(A--D--C); D(D--E--B, linetype("4 4")); MP("5 Let <math>O</math> be the intersection of <math>BC</math> and <math>AD</math>. By the [
  9 KB (1,415 words) - 13:56, 18 December 2022
• 2009 AIME II Problems/Problem 13
  Let <math>O</math> be the midpoint of <math>A</math> and <math>B</math>. Assume <math>C <i><b>Lemma 1:</b> A chord <math>ab</math> of a circle with center <math>O</math> and radius <math>r</math> has length <math>2r\sin\left(\dfrac{\angle
  8 KB (1,279 words) - 20:27, 17 May 2024
• Radical axis
  ...to circle <math>\omega</math> (with radius <math>r</math> and center <math>O</math>), which shall thereafter be dubbed <math>pow(P, \omega)</math>, is d
  10 KB (1,797 words) - 02:05, 24 October 2023
• 2009 AIME II Problems/Problem 15
  Let <math>O</math> be the center of the circle. Define <math>\angle{MOC}=t</math>, <mat
  11 KB (1,849 words) - 19:43, 2 January 2023
• 1968 AHSME Problems/Problem 35
  MP("O",(0,0),S); ...inches, chord <math>EF</math> is parallel to chord <math>CD</math>. <math>O</math>,<math>G</math>,<math>H</math>,<math>J</math> are collinear, and <mat
  3 KB (597 words) - 01:52, 16 August 2023
• 1989 AJHSME Problems/Problem 10
  ...le]] between the hour hand and the minute hand on a clock that reads seven o'clock?
  683 bytes (100 words) - 23:57, 4 July 2013
• 1989 AJHSME Problems/Problem 19
  label("S",(8.5,0),S); label("O",(9.5,0),S); label("N",(10.5,0),S); label("D",(11.5,0),S);
  2 KB (255 words) - 03:28, 28 November 2019
• 1990 AJHSME Problems/Problem 19
  Let <math>p</math> be a person seated and <math>o</math> is an empty seat
  727 bytes (119 words) - 01:59, 25 November 2020
• 1965 IMO Problems
  ...OAB</math> with acute angle <math>AOB</math>. Through a point <math>M \neq O</math> perpendiculars are drawn to <math>OA</math> and <math>OB</math>, the
  3 KB (497 words) - 12:39, 29 January 2021
• 1965 IMO Problems/Problem 5
  ...OAB</math> with acute angle <math>AOB</math>. Through a point <math>M \neq O</math> perpendiculars are drawn to <math>OA</math> and <math>OB</math>, the Let <math>O(0,0),A(a,0),B(b,c)</math>.
  1 KB (269 words) - 00:23, 9 December 2022
• 2009 IMO Problems/Problem 2
  Let <math>ABC</math> be a triangle with circumcentre <math>O</math>. The points <math>P</math> and <math>Q</math> are interior points of dot("O", (50, 38), N);
  2 KB (280 words) - 01:16, 19 November 2023
• 2009 IMO Problems
  Let <math>ABC</math> be a triangle with circumcentre <math>O</math>. The points <math>P</math> and <math>Q</math> are interior points of
  3 KB (509 words) - 09:23, 10 September 2020
• Schroeder-Bernstein Theorem
  ...there is an open interval <math>I</math> satisfying <math>r\in I\subseteq O</math>. Show that the following sets are equal in cardinality: *<math>\{O\subset\mathbb{R}\mid O\text{ is open}\}</math>
  4 KB (805 words) - 13:09, 20 February 2024
• 1992 AJHSME Problems
  label("O",(16.5,5.8),N);
  17 KB (2,346 words) - 13:36, 19 February 2020
• 2010 AMC 10B Problems
  A circle is centered at <math>O</math>, <math>\overline{AB}</math> is a diameter and <math>C</math> is a po ...th> is a chord on the circle, <math>OA = 4\sqrt{3}</math>, and point <math>O</math> is outside <math>\triangle ABC</math>. What is the side length of <m
  12 KB (1,817 words) - 22:44, 22 December 2020
• Characteristic polynomial
  ...\bold{T}^1 + v_2 \cdot \bold{T}^2 + \cdots + v_n \cdot \bold{T}^n = \bold{O}.</cmath> ...lambda \bold{v}</math>, then <math>\lambda I \bold{v} - A \bold{v} = \bold{O}</math>. But then, the column vectors of <math>\lambda I - A</math> are lin
  19 KB (3,412 words) - 14:57, 21 September 2022
• Linearly independent
  ...not all equal to zero such that <cmath>c_1v_1 + c_2v_2 + \cdots + c_nv_n = O.</cmath> Otherwise, the vectors are said to be [[linearly dependent]]. ...mbda_1c_iv_i = O,\qquad L\left(\sum c_iv_i\right) = \sum \lambda_ic_iv_i = O.</cmath> Subtracting the two equations yields <cmath>\sum (\lambda_1-\lambd
  2 KB (300 words) - 23:35, 16 March 2010
• 2010 AIME I Problems/Problem 13
  label("$O$",(129.36332,5.00321),NE/2); ...the semicircle is also the midpoint of <math>AB</math>. Let this point be O. Let <math>h</math> be the length of <math>AD</math>.
  10 KB (1,418 words) - 23:05, 20 October 2021
• Mock AIME 2 2010 Answer Key
  ...center of the circle, and let points E and F to be the perpendiculars from O to AC and BC, respectively. First, by power of a point, we have <math>CD \c We shall prove that in a circle <math>\mathcal{C}</math> with center <math>O</math>, radius <math>R</math>, a chord <math>BC</math> with midpoint <math>
  36 KB (6,214 words) - 20:22, 13 July 2023
• 2007 IMO Problems/Problem 1
  ...who is a horrible proof writer, so please fix the proof for me. Thank you. O, also the formatting.
  4 KB (768 words) - 23:15, 31 March 2010
• 2010 AIME II Problems/Problem 14
  Let <math>O</math> be the [[circumcenter]] of <math>ABC</math> and let the intersection Denote <math>E</math> the projection of <math>O</math> onto <math>CD</math>. Now <math>CD = CP + DP = 3</math>. By the [[Py
  10 KB (1,507 words) - 00:31, 19 November 2023
• 2010 AIME II Problems/Problem 9
  pair M,N,O,P,Q,R; O=extension(C,J,D,K);
  5 KB (857 words) - 22:22, 27 August 2023
• 2010 AIME II Problems/Problem 11
  ...ac-toe board: five <math>1</math>'s (or X's) and four <math>0</math>'s (or O's).
  6 KB (1,057 words) - 01:58, 8 January 2023
• 2010 USAMO Problems
  ...d <math>RS</math> is half the size of <math>\angle XOZ</math>, where <math>O</math> is the midpoint of segment <math>AB</math>.
  3 KB (525 words) - 13:44, 4 July 2013
• 2010 USAJMO Problems
  ...d <math>RS</math> is half the size of <math>\angle XOZ</math>, where <math>O</math> is the midpoint of segment <math>AB</math>.
  3 KB (538 words) - 13:55, 16 June 2020
• 2010 USAMO Problems/Problem 1
  <math>O</math> is the midpoint of segment <math>AB</math>. // Semi-circle: centre O, radius r, diameter A--B.
  13 KB (2,242 words) - 03:10, 21 May 2024
• 2010 USAMO Problems/Problem 2
  ...math>. This answer seems to disagree, though, because the worst case <math>O</math> efficiency is <math>n^2</math>, not <math>\frac{n(n-1)(n-2)}{6}</mat
  5 KB (823 words) - 19:20, 3 October 2017
• 1978 USAMO Problems
  ...Also, give a Euclidean construction (straight edge and compass) for <math>O</math>.
  2 KB (372 words) - 19:06, 3 July 2013
• 2010 AMC 10B Problems/Problem 6
  A circle is centered at <math>O</math>, <math>\overline{AB}</math> is a diameter and <math>C</math> is a po ...s half of its central angle, we will try a different approach. Since <math>O</math> is the center, <math>OC</math> and <math>OA</math> are radii and the
  2 KB (260 words) - 17:00, 1 August 2022
• 1982 AHSME Problems/Problem 9
  pair A=origin, O=(10,0), B=(3,0), N=(0,5), C=(3,5), P=(5,0), D=(1,1), G=(9,1), F=(1,0); draw(A--O, linewidth(0.4));
  2 KB (315 words) - 12:30, 19 October 2020
• 1996 AJHSME Problems
  Figure <math>OPQR</math> is a square. Point <math>O</math> is the origin, and point <math>Q</math> has coordinates (2,2). What pair O,P,Q,R,T;
  13 KB (1,880 words) - 13:35, 19 February 2020
• 1973 IMO Problems
  Point <math>O</math> lies on line <math>g</math>; <math>\overrightarrow{OP_1}, \overright
  3 KB (602 words) - 15:44, 29 January 2021
• 1966 AHSME Problems/Problem 6
  <math>AB</math> is the diameter of a circle centered at <math>O</math>. <math>C</math> is a point on the circle such that angle <math>BOC</
  1 KB (164 words) - 12:42, 28 January 2020
• 1999 AMC 8 Problems/Problem 15
  ...etters. The first is chosen from the set {C,H,L,P,R}, the second from {A,I,O}, and the third from {D,M,N,T}.
  3 KB (516 words) - 14:50, 21 December 2022
• 2010 AMC 10B Problems/Problem 20
  ...nt of tangency with <math>GK</math> and draw another line connecting <math>O</math> to <math>G</math>. Note that because triangle <math>BGA</math> is eq Call the radius of circle <math>O</math> <math>r</math>.
  3 KB (483 words) - 18:41, 4 May 2024
• 2010 AMC 10B Problems/Problem 19
  ...th> is a chord on the circle, <math>OA = 4\sqrt{3}</math>, and point <math>O</math> is outside <math>\triangle ABC</math>. What is the side length of <m ...}=\sqrt{48} < \sqrt{156}, A</math> must be in the interior of circle <math>O.</math>
  3 KB (443 words) - 18:34, 4 May 2024
• 2010 AMC 10B Problems/Problem 16
  pair O=(0,0); path outer=Circle(O,r);
  3 KB (544 words) - 20:54, 24 March 2024
• 2011 AMC 12A Problems
  ...and <math>AC \geq AB</math>. Let <math>H</math>, <math>I</math>, and <math>O</math> be the orthocenter, incenter, and circumcenter of <math>\triangle AB
  13 KB (1,994 words) - 13:52, 3 July 2021
• 2011 AMC 12A Problems/Problem 13
  Let <math>O</math> be the incenter of <math>\triangle{ABC}</math>. Because <math>\overl Let <math> O </math> be the incenter. <math> AO </math> is the angle bisector of <math>
  4 KB (683 words) - 03:12, 23 January 2023
• 2011 AMC 12A Problems/Problem 15
  ...th> be the pyramid with <math> ABCD </math> as the square base. Let <math> O </math> and <math> M </math> be the center of square <math> ABCD </math> an Notice that <math> \triangle EOM </math> has a right angle at <math> O </math>. Since the hemisphere is tangent to the triangular face <math> ABE
  2 KB (376 words) - 23:14, 5 January 2024
• 2011 AMC 12A Problems/Problem 25
  ...and <math>AC \geq AB</math>. Let <math>H</math>, <math>I</math>, and <math>O</math> be the orthocenter, incenter, and circumcenter of <math>\triangle AB ...\circ + \tfrac 12\angle A = 120^\circ .</cmath>Thus the points <math>B, C, O, I</math>, and <math>H</math> are all on a circle.
  6 KB (1,046 words) - 13:05, 28 June 2022
• 2011 AMC 12A Problems/Problem 24
  ...ath>\angle BCD=180^{\circ}-\theta</math>. Let the circle have center <math>O</math> and radius <math>r</math>. Note that <math>OHD</math>, <math>OGC</ma ...S</math> also equals <math>x</math>. Let the center of the circle be <math>O</math>. Observe that <math>AO</math> bisects angle <math>\angle A</math>, s
  4 KB (753 words) - 18:58, 2 June 2022
• 1958 AHSME Problems
  ...math> \overline{DE}</math> are equal chords of a circle with center <math> O</math>. Arc <math> AB</math> is a quarter-circle. Then the ratio of the are MP("O",(0,0),N);MP("C",(-10,0),W);MP("D",(10,0),E);;MP("E",(0,10),N);
  25 KB (3,872 words) - 14:21, 20 February 2020
• 1993 AHSME Problems
  ...one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock, then <math>\frac{q}{t}=</math>
  20 KB (2,814 words) - 08:15, 27 June 2021
• 2021 AIME II Problems/Problem 5
  pair A, B, O, P, Q, C1, C2, D; O = midpoint(A--B);
  14 KB (2,269 words) - 00:43, 2 January 2023
• 2011 AMC 12B Problems/Problem 6
  ...the center of the circle be <math>O</math>. The radii extending from <math>O</math> to the points of tangency <math>B</math> and <math>C</math> are both
  2 KB (370 words) - 13:35, 26 January 2021
• 2011 AMC 12B Problems/Problem 20
  ...umcircle of <math>\triangle BAC</math> to <math>O.</math> Therefore, <math>O</math> lies on the circumcircle of <math>\triangle BDE.</math> Similarly, i ...line to <math>OX</math> passing through <math>X,</math>, therefore, <math>O</math> coincides with <math>X.</math>
  8 KB (1,200 words) - 19:31, 7 August 2023
• 2001 AMC 10 Problems/Problem 19
  ...ill be chocolate, and the third will be powdered. For example, <math> O|OO|O </math> represents one glazed, two chocolate, and one powdered. We have six
  2 KB (335 words) - 17:44, 11 April 2024
• 2001 AMC 10 Problems/Problem 20
  ...ug in <math>s</math> and use the equation <math>2000-2s = o </math>, <math>o</math> being the length of a side of the octagon, to derive the value of a
  3 KB (460 words) - 09:59, 1 March 2024
• 2011 AIME I Problems/Problem 4
  ...>\angle MON \implies MN^2 = 2R^2(1 - \cos{2\angle MCN})</math> where <math>O</math> is the center of <math>\omega</math>. Now, the circumradius <math>R<
  6 KB (1,068 words) - 18:52, 2 August 2023
• 2011 AIME I Problems/Problem 10
  By symmetry, we can assume W/O LOG that the location of vertex A is vertex <math>v_0</math>.
  6 KB (980 words) - 03:14, 5 May 2020
• 2011 AIME I Problems/Problem 14
  pair A,B,C,D,E,F,G,H,M,N,O,O2,P,W,X,Y,Z; O=(E+F)/2;
  8 KB (1,344 words) - 18:39, 9 February 2023
• 2011 AIME II Problems
  A circle with center <math>O</math> has radius 25. Chord <math>\overline{AB}</math> of length 30 and cho
  8 KB (1,301 words) - 08:43, 11 October 2020
• 2011 AIME II Problems/Problem 10
  A [[circle]] with center <math>O</math> has radius 25. [[Chord]] <math>\overline{AB}</math> of length 30 and ...hen, by Power of a Point on P with respect to the circle with center <math>O</math>, we have that
  11 KB (1,720 words) - 03:12, 18 December 2023
• 1999 AMC 8 Problems
  ...e degree measure of the smaller angle formed by the hands of a clock at 10 o clock? label("O",(8,-2),S);
  17 KB (2,394 words) - 19:51, 8 May 2023
• Mock Geometry AIME 2011 Problems
  ...math>A_1,A_2</math> respectively, that <math>OA_1=OA_2,</math> where <math>O</math> is the origin, and that the radius of <math>\omega_2</math> is <math ...C</math> is on a semicircle with diameter <math>AB</math> and center <math>O.</math> Circle radius <math>r_1</math> is tangent to <math>OA,OC,</math> an
  8 KB (1,349 words) - 19:10, 14 June 2022
• 2011 USAJMO Problems/Problem 5
  Let <math>O</math> be the center of the circle, and let <math>X</math> be the intersect Let <math>O</math> be the center of the circle, and let <math>M</math> be the midpoint
  4 KB (717 words) - 17:00, 14 April 2024
• 2011 USAJMO Problems/Problem 3
  ...> with <math>\overleftrightarrow{HF}</math>. Say this ray intersects <math>O</math> in a point <math>B</math> besides <math>F</math>, and let <math>q</m ...th>X</math> be the point diametrically opposite to <math>B</math> on <math>O</math>. Also let <math>\overline{HB}</math> intersect <math>q</math> at <ma
  15 KB (2,593 words) - 13:37, 29 January 2021
• 2011 AMC 10B Problems
  pair O=(0,0), E=(-3,0), B=(3,0); path outer=Circle(O,r);
  13 KB (2,090 words) - 18:05, 7 January 2021
• 2011 AMC 10B Problems/Problem 17
  pair O=(0,0), E=(-3,0), B=(3,0); path outer=Circle(O,r);
  5 KB (699 words) - 04:53, 21 January 2023
• 2007 AMC 10B Problems
  The point <math>O</math> is the center of the circle circumscribed about <math>\triangle ABC, <math>\textbf{(A) } g \qquad\textbf{(B) } h \qquad\textbf{(C) } o \qquad\textbf{(D) } s \qquad\textbf{(E) } t</math>
  15 KB (2,297 words) - 12:57, 19 February 2020
• 2007 AMC 10B Problems/Problem 4
  The point <math>O</math> is the center of the circle circumscribed about <math>\triangle ABC,
  928 bytes (140 words) - 12:19, 4 July 2013
• 2007 AMC 10B Problems/Problem 19
  pair O=(0,0); path circleO=Circle(O,r);
  5 KB (723 words) - 10:58, 27 October 2021
• 2007 AMC 10B Problems/Problem 9
  <math>\textbf{(A) } g \qquad\textbf{(B) } h \qquad\textbf{(C) } o \qquad\textbf{(D) } s \qquad\textbf{(E) } t</math>
  2 KB (280 words) - 13:06, 4 June 2021
• Asymptote: 3D graphics
  label("$o$",(0,0,0),NW); label("$o$",(0,0,0),NW);
  4 KB (568 words) - 10:54, 8 October 2021
• Projective geometry
  ...math>O</math>, and any fifth line <math>m</math> not passing through <math>O</math>.
  4 KB (720 words) - 15:03, 26 January 2015
• 2002 AMC 10B Problems/Problem 24
  pair O=(0,0), A=(0,-20), B=(0,-10), C=(10sqrt(3),-10); path ferriswheel=Circle(O,r);
  3 KB (559 words) - 02:44, 8 February 2024
• Olympiad books
  ...thematical Challenges'' - '''Edward J. Barbeau, Murray S. Klamkin, William O. J. Moser'''. ...Book (Selected Problems and Theorems of Elementary Mathematics)'' - '''D. O. Shklarsky, N. N. Chentzov, I. M. Yaglom'''.
  17 KB (2,261 words) - 00:30, 22 April 2024
• Block walking
  label("O", (1,3), NW); label("O", (1,3), NW);
  9 KB (1,228 words) - 21:47, 26 February 2022
• Mock Amc 12
  ! scope="row" | '''Mock AMC O'''
  18 KB (2,206 words) - 19:41, 24 December 2020
• 1999 AMC 8 Problems/Problem 2
  ...e degree measure of the smaller angle formed by the hands of a clock at 10 o'clock?
  1 KB (209 words) - 21:45, 15 January 2024
• 1999 AMC 8 Problems/Problem 4
  label("O",(8,-2),S);
  1 KB (196 words) - 21:02, 14 April 2017
• 2015 IMO Problems/Problem 4
  ...ath>ABC</math> has circumcircle <math>\Omega</math> and circumcenter <math>O</math>. A circle <math>\Gamma</math> with center <math>A</math> intersects
  3 KB (502 words) - 23:58, 5 October 2015
• 1984 AHSME Problems/Problem 18
  ...ed [[circle]]. Now, remove the coordinate system. Let the origin be <math> O </math>, the y-intercept of the line be <math> A </math>, the x-intercept o label("$O$",(0,0),WNW);
  3 KB (446 words) - 18:21, 4 June 2021
• 1984 AHSME Problems/Problem 29
  ...nt to finding the maximum slope of a line passing through the origin <math>O</math> and intersecting the circle. The steepest such line is tangent to th
  4 KB (614 words) - 20:09, 12 September 2022
• 1967 AHSME Problems
  ...>\overline{AC}</math> and <math>\overline{BD}</math> intersecting at <math>O</math>, <math>\overline{BO}=4</math>, <math>\overline{OD}=6</math>, <math>\
  20 KB (3,108 words) - 14:14, 20 February 2020
• Simson line
  Let <math>O,O_0,</math> and <math>O_1</math> be the circumcenters of triangles <math>\t
  4 KB (638 words) - 06:38, 26 April 2023
• 2003 AMC 8 Problems
  ...\tabcolsep}{0.5mm}\begin{array}{cccc}&T & W & O\\ + &T & W & O\\ \hline F& O & U & R\end{array} </math> If <math>T = 7</math> and the letter <math>O</math> represents an even number, what is the only possible value for <math
  16 KB (2,236 words) - 12:02, 19 February 2024
• 1966 IMO Problems/Problem 3
  ...be <math>ABCD</math>, and the center of its circumscribed sphere be <math>O</math>. Construct a new regular tetrahedron, <math>WXYZ</math>, such that t <cmath>OA + OB + OC + OD = \sum \textrm{Distances from }O\textrm{ to faces of }WXYZ</cmath>
  2 KB (360 words) - 15:51, 11 December 2022
• 1984 AHSME Problems/Problem 4
  ...=origin, E=(3,0), F=(10,0), G=(12,0), H=(12,1), A=(0,1), B=(4,1), C=(9,1), O=circumcenter(B,C,F); draw(Circle(O, abs(O-C)));
  2 KB (367 words) - 16:28, 2 January 2021
• 1999 AMC 8 Problems/Problem 8
  Six squares are colored, front and back, (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white). They are hinged together a label("O",(3.5,1.3),N);
  1 KB (176 words) - 00:34, 5 July 2013
• 2009 AMC 8 Problems
  ...(0,1), B=A+1*dir(60), C=(1,1), D=(1,0), E=D+1*dir(-72), F=E+1*dir(-144), G=O+1*dir(-108); draw(O--A--B--C--D--E--F--G--cycle);
  18 KB (2,551 words) - 18:46, 27 February 2024
• 2000 AMC 8 Problems/Problem 19
  ...</math>. These squares share side <math>\overline{AO}</math>, where <math>O</math> is the center of the large semicircle. ...>\overline{BD}</math>. Then draw <math>\overline {CO}</math>, where <math>O</math> is the center of the semicircle. You have two quarter circles on to
  2 KB (383 words) - 16:58, 12 January 2024
• 1996 AJHSME Problems/Problem 17
  Figure <math>OPQR</math> is a square. Point <math>O</math> is the origin, and point <math>Q</math> has coordinates (2,2). What pair O,P,Q,R,T;
  2 KB (257 words) - 11:20, 22 March 2015
• 1996 AJHSME Problems/Problem 21
  ...must add either <math>E+O+O</math>, or <math>E + E + E</math>, where <math>O</math> represents an odd number, and <math>E</math> represents an even numb
  1 KB (240 words) - 19:43, 26 May 2021
• 1992 AJHSME Problems/Problem 11
  label("O",(16.5,5.8),N);
  2 KB (226 words) - 00:09, 5 July 2013
• 1997 AHSME Problems
  A circle with center <math>O</math> is tangent to the coordinate axes and to the hypotenuse of the <math pair O = (2.33,2.33);
  17 KB (2,590 words) - 13:38, 19 February 2020
• 1997 AHSME Problems/Problem 19
  A circle with center <math>O</math> is tangent to the coordinate axes and to the hypotenuse of the <math pair O = (2.33,2.33);
  3 KB (422 words) - 19:00, 9 August 2015
• 2010 AMC 8 Problems/Problem 23
  ...o of the combined areas of the two semicircles to the area of circle <math>O</math>? dot((0,0),ds); label("$O$",(-0.24,-0.35),NE*lsf); dot((1.41,1.41),ds); dot((-1.4,1.43),ds); dot((1.4
  2 KB (298 words) - 00:29, 18 December 2023
• 2009 AMC 8 Problems/Problem 9
  ...(0,1), B=A+1*dir(60), C=(1,1), D=(1,0), E=D+1*dir(-72), F=E+1*dir(-144), G=O+1*dir(-108); draw(O--A--B--C--D--E--F--G--cycle);
  1 KB (239 words) - 08:24, 30 May 2023
• 1979 USAMO Problems/Problem 2
  ...of radius one in space with <math>N=(0,0,1)</math> and sphere center <math>O=(0,0,0)</math>
  6 KB (1,013 words) - 22:09, 21 November 2023
• 1979 USAMO Problems/Problem 4
  ...</math> go to the circles <math>(O_1),(O_2)</math> passing through <math>P,O</math> and the line <math>QR</math> cuts <math>(O_1),(O_2)</math> again at pair O = (0,0), A = (14,28), Q = (20,40), B = (16,0), R = (25,0), P = (23,16);
  3 KB (568 words) - 12:24, 11 March 2018
• 1996 AHSME Problems
  draw(box(O, (4,4,3)));
  15 KB (2,343 words) - 13:39, 19 February 2020
• 1996 AHSME Problems/Problem 11
  Let line segment <math>AB = 2</math>, and let it be tangent to circle <math>O</math> at point <math>P</math>, with radius <math>OP = 2</math>. Let <math ...ngle at <math>P</math>, because <math>AB</math> is tangent to circle <math>O</math> at point <math>P</math>, and <math>OP</math> is a radius.
  2 KB (266 words) - 14:07, 5 July 2013
• 1996 AHSME Problems/Problem 19
  pair O=(0,0); dot(A^^B^^C^^D^^E^^F^^O);
  3 KB (540 words) - 21:32, 10 July 2017
• 1996 AHSME Problems/Problem 20
  Let <math>O</math> be the center of the circle at <math>(6,8)</math>.
  2 KB (327 words) - 13:45, 30 September 2021
• 1996 AHSME Problems/Problem 25
  ...}{3}*8 = \frac{40}{3}</math>. Note that the horizontal distance from <math>O</math> to the origin is <math>7</math>, and the horizontal distance from K
  9 KB (1,441 words) - 17:51, 22 October 2023
• 1996 AHSME Problems/Problem 28
  draw(box(O, (4,4,3)));
  3 KB (545 words) - 10:21, 16 September 2022
• 2003 AMC 8 Problems/Problem 14
  ...{\tabcolsep}{0.5mm}\begin{array}{cccc}&T & W & O\\ +&T & W & O\\ \hline F& O & U & R\end{array} </math> If T = 7 and the letter O represents an even number, what is the only possible value for W?
  1 KB (177 words) - 19:56, 15 April 2023
• 2003 AMC 8 Problems/Problem 25
  ...side <math>\overline{BC}</math>, point <math>A</math> coincides with <math>O</math>, the center of square <math>WXYZ</math>. What is the area of <math>\ pair Z=origin, W=(0,10), X=(10,10), Y=(10,0), O=(5,5), B=(-4,8), C=(-4,2), A=(-13,5);
  2 KB (387 words) - 19:58, 15 April 2023
• 2003 AMC 8 Problems/Problem 18
  pair m=(282,411), n=(147,451), o=(103,437), p=(31,373); draw(o--oval);
  2 KB (250 words) - 22:17, 5 January 2024
• 2009 PUMaC Problems
  ...four consecutive vertices of an 18-sided regular polygon with center <math>O</math>. Let <math>P</math> be the midpoint of <math>AC</math> and <math>Q</ ...=(O+D)/2; D(D("O",O,NE)--D("A",A,W)--D("B",B,SW)--D("C",C,S)--D("D",D,SE)--O--D("P",P,1.6*dir(95))--D("Q",Q,NE)); D(A--C); D(A--(A+dir(start-360/n))/2,
  25 KB (4,154 words) - 16:27, 2 September 2011
• 1985 AHSME Problems
  In a circle with center <math>O</math>, <math>AD</math> is a diameter, <math>ABC</math> is a chord, <math>B ...O=origin, A=dir(35), C=dir(155), D=dir(215), B=intersectionpoint(dir(125)--O, A--C);
  17 KB (2,488 words) - 03:26, 20 March 2024
• 1985 AHSME Problems/Problem 11
  ...e vowels and consonants separately. There are <math>2</math> vowels (<math>O</math> and <math>E</math>), giving <math>2! = 2</math> choices for the firs
  1 KB (164 words) - 20:40, 19 March 2024
• 1985 AHSME Problems/Problem 22
  In a circle with center <math>O</math>, <math>AD</math> is a diameter, <math>ABC</math> is a chord, <math>B ...O=origin, A=dir(35), C=dir(155), D=dir(215), B=intersectionpoint(dir(125)--O, A--C);
  2 KB (277 words) - 00:19, 20 March 2024
• 2010 AMC 8 Problems
  ...o of the combined areas of the two semicircles to the area of circle <math>O</math>? dot((0,0),ds); label("$O$",(-0.24,-0.35),NE*lsf); dot((1.41,1.41),ds); dot((-1.4,1.43),ds); dot((1.4
  18 KB (2,768 words) - 21:05, 9 January 2024
• 2011 AMC 8 Problems
  ...F=(2,0), G=(3,0), H=(1,4), I=(2,4), J=(3,4), K=(0,-2), L=(4,-2), M=(0,-6), O=(0,-4), P=(4,-4), Q=(2,-2), R=(2,-6), T=(6,4), U=(10,0), V=(10,4), Z=(10,2) draw(C--H--(1,0)--A--cycle,linewidth(1.6)); draw(M--O--Q--R--cycle,linewidth(1.6)); draw(A_1--V--Z--cycle,linewidth(1.6)); draw(G
  16 KB (2,371 words) - 17:34, 9 January 2024
• 1973 Canadian MO Problems
  ...</math> are fixed points on a given circle not collinear with center <math>O</math> of the circle, and if <math>XY</math> is a variable diameter, find t
  4 KB (540 words) - 18:23, 8 October 2014
• Barycentric coordinates
  ...AC (b^2(x - z) + y(a^2 - c^2) =0) \implies</math> its BC coordinates <math>O = (a^2S_A : b^2S_B : c^2S_C).</math> ...conjugate with respect to <math>\triangle ABC</math> with the point <math>O \implies H =\left( \frac {1}{S_A} : \frac {1}{S_B} : \frac {1}{S_C}\right
  25 KB (5,067 words) - 22:15, 31 March 2024
• Mock Geometry AIME 2011 Problems/Problem 11
  ...C</math> is on a semicircle with diameter <math>AB</math> and center <math>O.</math> Circle radius <math>r_1</math> is tangent to <math>OA,OC,</math> an label("$O$",(0,0),S);
  3 KB (432 words) - 14:12, 2 January 2012
• 2011 AMC 8 Problems/Problem 7
  ...F=(2,0), G=(3,0), H=(1,4), I=(2,4), J=(3,4), K=(0,-2), L=(4,-2), M=(0,-6), O=(0,-4), P=(4,-4), Q=(2,-2), R=(2,-6), T=(6,4), U=(10,0), V=(10,4), Z=(10,2) draw(C--H--(1,0)--A--cycle,linewidth(1.6)); draw(M--O--Q--R--cycle,linewidth(1.6)); draw(A_1--V--Z--cycle,linewidth(1.6)); draw(G
  4 KB (557 words) - 00:30, 4 January 2023
• Mock Geometry AIME 2011 Problems/Problem 10
  ...math>A_1,A_2</math> respectively, that <math>OA_1=OA_2,</math> where <math>O</math> is the origin, and that the radius of <math>\omega_2</math> is <math ...)</math> for some <math>x</math>. Using the distance formula between <math>O</math> and <math>C_1</math> yields <math>C_1O=\sqrt{(7-0)^2+(1-0)^2}=\sqrt{
  2 KB (324 words) - 22:28, 26 December 2022
• 2012 AMC 12A Problems
  Circle <math>C_1</math> has its center <math>O</math> lying on circle <math>C_2</math>. The two circles meet at <math>X</
  14 KB (2,197 words) - 13:34, 12 August 2020
• 2012 AMC 12A Problems/Problem 12
  pair O=(0,0); path inner=Circle(O,a);
  3 KB (574 words) - 20:42, 3 January 2020
• 2012 AMC 12A Problems/Problem 16
  Circle <math>C_1</math> has its center <math>O</math> lying on circle <math>C_2</math>. The two circles meet at <math>X</ pair O, Z;
  9 KB (1,496 words) - 02:40, 2 October 2022
• 2012 AMC 12A Problems/Problem 18
  ...bisectors of triangle <math>ABC</math> are concurrent at the center <math>O</math>(also <math>I</math>) of circle <math>C</math>. Let <math>x=QB</math> Since the radius of circle <math>O</math> is perpendicular to <math>BC</math> at <math>R</math>, we have by th
  4 KB (717 words) - 19:07, 28 July 2021
• 2012 AMC 12A Problems/Problem 22
  ...,1), H=(-1,1), I=(1,0), J=(2,1), K=(1,2), L=(0,1), M=(-0.5,0.5), N=(-1,2), O=(-0.5,2.5), P=(0,3), Q=(1.5,2.5), R=(1,2), S=(1.5,0.5), T=(0,1); ...,1), H=(-1,1), I=(1,0), J=(2,1), K=(1,2), L=(0,1), M=(-0.5,0.5), N=(-1,2), O=(-0.5,2.5), P=(0,3), Q=(1.5,2.5), R=(1,2), S=(1.5,0.5), T=(0,1);
  5 KB (940 words) - 17:13, 4 April 2020
• 2012 AMC 12B Problems/Problem 8
  ...know that the number of desired meal plans is <math>\frac{\text{\# plans w/o restrictions}}{4} = \frac{4 \cdot 3^6}{4} = 3^6 = 729</math>, choice <math>
  5 KB (802 words) - 20:13, 9 September 2020
• 2015 AIME I Problems
  pair A,B,C,D,E,F,G,H,I,O; O=(0,0);
  10 KB (1,615 words) - 21:48, 13 January 2024
• 2012 AIME I Problems
  pair M = 6.4*dir(54), N = 6.4*dir(126), O = 6.4*dir(198), P = 6.4*dir(270), L = 6.4*dir(342); pair[] dotted = {A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P};
  10 KB (1,617 words) - 14:49, 2 June 2023
• 2012 AIME I Problems/Problem 7
  pair M = 6.4*dir(54), N = 6.4*dir(126), O = 6.4*dir(198), P = 6.4*dir(270), L = 6.4*dir(342); pair[] dotted = {A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P};
  6 KB (1,058 words) - 01:49, 25 November 2023
• 2012 AIME I Problems/Problem 13
  pair O = origin; c3 = CR(O,3);
  11 KB (1,889 words) - 20:42, 25 January 2023
• 2007 iTest Problems/Problem 15
  \text{(O) }2007\qquad</math> pair O=(0,0), A=(0,100), B=(-50,13.397), C=(50,13.397), D=(50,-86.603), E=(-50,-86
  3 KB (539 words) - 21:01, 29 July 2018
• Mock AIME II 2012 Problems
  ...h>A</math> be a point outside circle <math>\Omega</math> with center <math>O</math> and radius <math>9</math> such that the tangents from <math>A</math>
  7 KB (1,274 words) - 21:16, 8 March 2021
• Mock AIME II 2012 Problems/Problem 8
  ...h>A</math> be a point outside circle <math>\Omega</math> with center <math>O</math> and radius <math>9</math> such that the tangents from <math>A</math>
  1 KB (184 words) - 03:11, 5 April 2012
• Mock AIME II 2012 Problems/Problem 13
  ...</math> be the center of the octahedron. The plane must pass through <math>O</math> in order to bisect the area of the octahedron. We see that the cross .../math>. The altitude of <math>\Delta ABC</math>, the segment joining <math>O</math> and the midpoint of <math>BC</math>, and <math>AO</math> make a righ
  4 KB (628 words) - 19:32, 27 December 2012
• Mock AIME I 2012 Problems
  ...h>O</math> for which the circle of radius <math>1</math> centered at <math>O</math> intersects <math>ABC</math> exactly <math>3</math> times. The points
  7 KB (1,309 words) - 11:13, 8 April 2012
• 1998 USAMO Problems/Problem 2
  pair O,A,B,C,D,E,F,DEb,CFb,Fo,M; O=(0,0);
  4 KB (656 words) - 17:26, 20 June 2019
• 2012 USAJMO Problems/Problem 1
  pair A, B, C, O, P, Q, R, S; O = extension(P, P + rotate(90)*(A - P), Q, Q + rotate(90)*(A - Q));
  4 KB (613 words) - 20:50, 19 December 2023
• 1951 AHSME Problems/Problem 12
  At <math> 2: 15</math> o'clock, the hour and minute hands of a clock form an angle of:
  698 bytes (97 words) - 12:20, 5 July 2013
• 2015 IMO Problems/Problem 1
  <math>K</math> with centre <math>O</math>, and let <math>O</math>, <math>A_i</math> and <math>A_j</math> form an equilateral triangle.
  4 KB (773 words) - 08:14, 19 July 2016
• 2002 Romanian NMO Problems
  ...triangles <math>OXY</math> and <math>OZT</math> are right-angled at <math>O</math> and isosceles.
  10 KB (1,695 words) - 10:03, 10 May 2012
• Install Eclipse
  Save your work (CTRL+o) and close Nano (CTRL+x).
  8 KB (1,222 words) - 17:47, 9 October 2014
• 1974 AHSME Problems
  pair O = (0,0); pair T = dir(90); dot(O);
  15 KB (2,151 words) - 14:04, 19 February 2020
• 1974 AHSME Problems/Problem 23
  pair O = (0,0); pair T = dir(90); dot(O);
  2 KB (337 words) - 12:44, 5 July 2013
• 1974 AHSME Problems/Problem 25
  ...h> Q </math>. Lines <math> DP </math> and <math> CQ </math> meet at <math> O </math>. If the area of parallelogram <math> ABCD </math> is <math> k </mat label("O", (3.4,1.75));</asy>
  2 KB (232 words) - 12:44, 5 July 2013
• 2006 SMT/General Problems
  ...<math> A_1, A_2, \cdots </math> are placed on a circle with center <math> O </math> such that <math> \angle OA_nA_{n+1}=35^\circ </math> and <math> A_n
  10 KB (1,477 words) - 16:02, 27 May 2012
• 2006 SMT/Advanced Topics Problems/Problem 7
  ...an even coordinate. Therefore, we have four cases: <math> (o,o); (o,e); (e,o); (e,e) </math>. Let <math> a </math> be the number of coordinates in the f
  2 KB (297 words) - 21:38, 27 May 2012
• 1954 AHSME Problems
  ...BC</math>, <math>AB=AC</math>, <math>\angle A=40^\circ</math>. Point <math>O</math> is within the triangle with <math>\angle OBC \cong \angle OCA</math> ...a given external point <math>P</math> to a given circle with center <math>O</math> and radius <math>r</math>, is:
  23 KB (3,535 words) - 16:29, 24 April 2020
• 1955 AHSME Problems
  In circle <math>O</math> chord <math>AB</math> is produced so that <math>BC</math> equals a r pair O=origin, D=dir(195), A=dir(150), B=dir(30), C=B+1*dir(0);
  22 KB (3,509 words) - 21:29, 31 December 2023
• Dirichlet's Theorem
  ...ubstack{p\leq x\\ p\equiv a\mod m}}\frac{1}{p}=\frac{1}{\phi(m)}\log\log x+O(1)
  805 bytes (138 words) - 12:19, 30 May 2019
• 1971 Canadian MO Problems/Problem 8
  Let <math>O</math> be the center of the pentagon. We know that <math>\angle AOB=72^{\ci
  2 KB (363 words) - 22:38, 11 March 2016
• 1998 USAMO Problems/Problem 6
  pair A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U; O = extension(K,D,C,N);
  5 KB (871 words) - 18:59, 10 May 2023
• 1999 USAMO Problems
  ...s 2000</math> grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes t
  3 KB (465 words) - 13:35, 4 July 2013
• 1977 USAMO Problems/Problem 4
  ...tors of A, B, C, D be a, b, c, d with respect to an arbitrary origin <math>O</math>. Define the square of a vector <math>a^2 = a \cdot a</math>. Thus, <
  4 KB (614 words) - 18:53, 12 July 2023
• 1978 USAMO Problems/Problem 2
  ...Also, give a Euclidean construction (straight edge and compass) for <math>O</math>. ...se let AA' and BB' meet at O. Then triangles OAB and OA'B' are similar, so O must represent the same point. So assume A'B' is not parallel to AB.
  4 KB (712 words) - 21:57, 12 November 2023
• 2008 AMC 8 Problems/Problem 25
  pair O=origin; pair P=O+8*dir(d);
  1 KB (227 words) - 20:38, 2 January 2023
• 2008 AMC 8 Problems
  pair O=origin; pair P=O+8*dir(d);
  14 KB (2,035 words) - 15:23, 26 January 2024
• 2012 AMC 8 Problems
  pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; O=(0,5);
  13 KB (1,835 words) - 08:51, 8 March 2024
• 2012 AMC 8 Problems/Problem 5
  pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; O=(0,5);
  2 KB (208 words) - 20:16, 3 January 2024
• 2005 AMC 8 Problems/Problem 25
  pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2); draw(circle(o, 2.5));
  2 KB (288 words) - 00:53, 16 November 2023
• 2005 AMC 8 Problems/Problem 13
  Notice that <math>AF + DE = BC</math>, so <math>DE=4</math>. Let <math>O</math> be the intersection of the extensions of <math>AF</math> and <math>D
  1 KB (216 words) - 21:46, 2 January 2023
• 2005 AMC 8 Problems/Problem 17
  label("$O$", (0,0), SW);
  1 KB (160 words) - 13:34, 19 October 2020
• 2005 AMC 8 Problems/Problem 23
  ...a semicircle of area <math> 2\pi</math>. The circle has its center <math> O</math> on hypotenuse <math> \overline{AB}</math> and is tangent to sides <m <asy>pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2);
  1 KB (198 words) - 23:19, 17 December 2023
• Thales' theorem
  3. <math>AC</math> is a diameter to circle O with radius 5. If B is on O and <math>AB = 6</math>, then find <math>BC</math>. ...math>BC</math> such that <math>AD</math> is tangent to O. If the radius of O is 5 and <math>AD = 24</math>, find <math>AB</math>.
  2 KB (334 words) - 00:42, 5 June 2021
• 2013 AMC 10A Problems/Problem 20
  Let <math>O</math> be the center of the square and <math>C</math> be the intersection o
  4 KB (701 words) - 17:55, 23 July 2021
• 2013 AMC 12B Problems/Problem 24
  ...<math>B</math> to <math>NX.</math> On the coordinate plane, position <math>O</math> at <math>(0, 0)</math>, and make <math>NX</math> lie on the x-axis.
  11 KB (1,876 words) - 00:08, 12 October 2023
• 1982 USAMO Problems/Problem 5
  ...an be flattened to the plane <math>P'</math> through <math>A</math>, <math>O</math>, <math>O_1</math> and <math>O_2</math>. ...math>X = S \cap O O_1, \quad Y = S_1 \cap S_2 \neq A, \quad Z = S \cap O O_2, \quad M = O_1 Y \cap AO, \quad N = O_2 Y \cap AO</math>
  3 KB (524 words) - 16:08, 11 July 2018
• 2013 AIME I Problems
  ...h> be a triangle with <math>\angle P = 75^o</math> and <math>\angle Q = 60^o</math>. A regular hexagon <math>ABCDEF</math> with side length 1 is drawn i
  9 KB (1,580 words) - 13:07, 24 February 2024
• 1991 AHSME Problems/Problem 10
  Let <math>O</math> be the center of the circle, and let the chord passing through <math pair O, P, Q, R;
  2 KB (387 words) - 14:27, 23 June 2021
• 2013 AIME II Problems
  ...be a point at a distance <math>4 + \sqrt{13}</math> from the center <math>O</math> of the circle. Let <math>B</math> be the point on the circle nearest
  8 KB (1,402 words) - 12:17, 13 March 2020
• 2013 AIME II Problems/Problem 8
  ...e midpoint of <math>BC</math>, <math>X</math>. Now, draw a line from <math>O</math> to the midpoint of <math>AB</math>, <math>Y</math>. Clearly, <math>\ ...CD</math> upwards until they meet at point <math>G</math>. Let point <math>O</math> be the center of the hexagon. By the <math>AA</math> postulate, <mat
  8 KB (1,357 words) - 09:23, 11 March 2024
• 2013 AIME II Problems/Problem 10
  ...be a point at a distance <math>4 + \sqrt{13}</math> from the center <math>O</math> of the circle. Let <math>B</math> be the point on the circle nearest label("$O$", (0,0), S);label("$B$", B, SW);label("$A$", A, S);
  5 KB (823 words) - 17:57, 29 December 2023
• 2013 USAJMO Problems/Problem 5
  Let us use coordinates. Let O, the center of the circle, be (0,0). WLOG the radius of the circle is 1, so
  7 KB (1,250 words) - 18:05, 1 October 2021
• Dilation
  Now, choose a center of dilation, and name it <math>O</math>. Choose an arbitrary factor of dilation <math>k</math> ...b,c\dots,n</math> dilated by a factor <math>k</math> around a center <math>O</math> results in the vertices <math>a',b',c'\dots,n'</math> such that <mat
  2 KB (273 words) - 18:09, 12 September 2013
• Rotation
  ...le <math>ABC</math> <math>60^{\circ}</math> clockwise around a point <math>O</math>, also known as the '''center of rotation'''. ...ath> such that the angle formed is <math>60^{\circ}</math>, and <math>AO=A'O</math>. Do this for points <math>B</math> and <math>C</math>, to get the ne
  3 KB (432 words) - 23:22, 13 January 2021
• CSS: Pseudo-class and Pseudo-element
  <li class="b">o</li>
  15 KB (2,197 words) - 22:49, 24 July 2023
• Mock AIME 6 2006-2007 Problems
  ...ifferent sizes that do not intersect. The smaller circle has center <math>O</math>. Label the intersection of their common external tangents <math>P</
  7 KB (1,173 words) - 21:04, 7 December 2018
• 2014 AIME II Problems/Problem 11
  .... Let <math>O</math> be the foot of altitude <math>RO</math> and set <math>O</math> as the origin. Now we notice special right triangles! In particular, pair a, o, d, r, e, m, cm, c,p;
  6 KB (1,059 words) - 18:24, 20 January 2024
• 2014 AIME II Problems/Problem 10
  ...{i \theta_w}</math> and likewise for <math>z</math>. Consider circle <math>O</math> with the origin as the center and radius 2014 on the complex plane. ...cubed when <math>w</math> is cubed. Thus <math>w</math> must lie on <math>O</math>, since its the cube of its modulus, and thus its modulus, must be eq
  6 KB (1,045 words) - 13:08, 21 January 2024
• 2014 AIME II Problems/Problem 14
  ...math>B</math> to <math>AC</math> and label the point of intersection <math>O</math>. We will use this point later in the problem. As above, get to <math>AP=HM</math>. As in the figure, let <math>O</math> be the foot of the perpendicular from <math>B</math> to <math>AC</ma
  11 KB (1,442 words) - 19:28, 21 October 2023
• Symmetry
  Denote <math>O</math> the circumcenter of <math>ABCDEF,</math> <math>\ell \cap m = O, \alpha</math> the smaller angle between lines <math>\ell</math> and <math>
  19 KB (3,292 words) - 13:04, 13 May 2024
• ASIA TEAM Problems
  ...1</math> for the first time in <math>r</math> jumps. In particular, <math>o(1)=0</math> because Kelvin is already at lily pad <math>1</math>. ...), <math>C</math> is the intersection of <math>PM</math> with circle <math>O</math> such that <math>C</math> lies between <math>P</math> and <math>M</ma
  10 KB (1,710 words) - 23:23, 10 January 2020
• Mock AIME I 2015 Problems
  ...<math>AB=13</math>, <math>BC=14</math>, and <math>CA=15</math>. Let <math>O</math> denote its circumcenter and <math>H</math> its orthocenter. The cir
  9 KB (1,463 words) - 14:48, 12 February 2017
• 1966 AHSME Problems/Problem 28
  Five points <math>O,A,B,C,D</math> are taken in order on a straight line with distances <math>O
  764 bytes (146 words) - 02:33, 15 September 2014
• 1966 AHSME Problems/Problem 31
  MP("O'", (0,0), W); MP("O", (-2,2), W);
  2 KB (355 words) - 18:52, 22 April 2024
• 1966 AHSME Problems/Problem 35
  Let <math>O</math> be an interior point of triangle <math>ABC</math>, and let <math>s_1
  727 bytes (119 words) - 17:39, 18 May 2019
• 1966 AHSME Problems/Problem 38
  ...math>BC</math> and <math>AB</math>, respectively, intersect in point <math>O</math>. <math>P</math> is the midpoint of side <math>AC</math>, and <math>M
  2 KB (415 words) - 20:21, 3 April 2023
• 1966 AHSME Problems/Problem 40
  MP("O", (0,0), S);MP("A", (-10,0), W);MP("B", (10,0), E);MP("C", (10,10), E);MP(" In this figure <math>AB</math> is a diameter of a circle, centered at <math>O</math>, with radius <math>a</math>. A chord <math>AD</math> is drawn and ex
  1 KB (209 words) - 21:43, 23 September 2014
• 1974 IMO Problems/Problem 2
  Let <math>w = (O,R)</math> the circumscribed circle of <math>\triangle ABC</math>, and <math
  3 KB (553 words) - 15:56, 29 January 2021
• 1962 AHSME Problems/Problem 25
  Let <math>O</math> be the center of the circle and <math>E</math> be the point of tange
  1 KB (216 words) - 21:22, 25 November 2017
• 1962 AHSME Problems/Problem 39
  ...ath>OC</math> have length <math>2x</math>. Note that the median from <math>O</math> to <math>AB</math> in <math>\triangle AOB</math> is equal to <math>x ...h>. We can then use Steward's* to find the length of the median from <math>O</math>, since we know the median cuts <math>AB</math> into segments each of
  2 KB (430 words) - 12:48, 19 July 2023
• 1951 AHSME Problems/Problem 46
  <math>AB</math> is a fixed diameter of a circle whose center is <math>O</math>. From <math>C</math>, any point on the circle, a chord <math>CD</mat pair O=(0,0), A=(-1,0), B=(1,0), C=(-0.5,0.5sqrt(3)),
  2 KB (283 words) - 17:26, 1 January 2014
• 2013 AMC 8 Problems
  triple A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P; O = (11,9,5);
  15 KB (2,162 words) - 20:05, 8 May 2023
• 2013 AMC 8 Problems/Problem 18
  triple A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P; O = (11,9,5);
  3 KB (412 words) - 22:30, 18 December 2023
• 1967 AHSME Problems/Problem 32
  ...with diagonals <math>AC</math> and <math>BD</math>, intersecting at <math>O</math>, <math>BO=4</math>, <math>OD = 6</math>, <math>AO=8</math>, <math>OC label("$O$", (-1.34,1.98), NE * labelscalefactor);
  4 KB (560 words) - 18:56, 17 December 2023
• 1968 AHSME Problems
  Let <math>O</math> be the intersection point of medians <math>AP</math> and <math>CQ</m ..., and in <math>8</math> minutes more they are again equidistant from <math>O</math>. Then the ratio of <math>A</math>'s speed to <math>B</math>'s speed
  16 KB (2,571 words) - 14:13, 20 February 2020
• 2014 AMC 10A Problems/Problem 4
  The only possible arrangement is <math>\text{B}-\text{O}-\text{Y}-\text{R}</math> <math>\text{B}-\text{O}-\text{R}-\text{Y}</math>
  3 KB (532 words) - 17:28, 5 May 2024
• 1994 AHSME Problems/Problem 17
  pair A=(0,0),B=(0,s),C=(8,s),D=(8,0),O=(4,sqrt(2)),X; D(CR(O,2));
  2 KB (287 words) - 03:07, 28 May 2021
• 1994 AHSME Problems/Problem 29
  ...math> is the central angle in radians. Call the center of the circle <math>O</math>. Then <math>\angle{BOC} = 1</math> radian because the minor arc <mat label("O", (0,0), NE);
  3 KB (395 words) - 09:19, 28 September 2023
• 1983 IMO Problems/Problem 2
  ...us define center of <math>u</math> as <math>O</math>. Then <math>(B, C, P, O)</math> are concyclic.
  7 KB (1,267 words) - 23:35, 29 January 2021
• 2014 USAJMO Problems/Problem 2
  ...teral, acute triangle with <math>\angle A=60^{\circ}</math>, and let <math>O</math> and <math>H</math> denote the circumcenter and orthocenter of <math> pair A,B,C,O,H,P,Q,i1,i2,i3,i4;
  7 KB (1,197 words) - 19:40, 28 February 2023
• 2014 USAMO Problems/Problem 6
  ...bound, and <cmath> \sum_p 1 < r \sim O \left( \frac{N^2}{\ln N} \right) < o(N^2) </cmath> via Prime Number Theorem. Hence the sum in question is certa
  2 KB (361 words) - 11:55, 25 June 2020
• 2014 USAMO Problems/Problem 5
  ...ath>(ABC)</math>. Note that <math>(O_1)</math> is the reflection of <math>(O)</math> across <math>AC</math>, so <math>AO=AO_1</math>. Additionally so <math>Y</math> lies on <math>(O)</math>. Now since <math>XO,OO_1,XO_1</math> are perpendicular to <math>AB,
  3 KB (530 words) - 12:08, 27 March 2022
• 1960 AHSME Problems
  ...or a clock to strike <math>6</math> o'clock beginning at <math>6:00</math> o'clock precisely. ...ormly spaced, how long, in seconds, does it take to strike <math>12</math> o'clock?
  21 KB (3,242 words) - 21:27, 30 December 2020
• 2014 AMC 12A Problems/Problem 17
  Let <math>O</math> be the center of any of the top spheres (you choose!). We have <math
  4 KB (602 words) - 02:42, 13 June 2022
• 2014 AMC 10B Problems/Problem 17
  ...X-1</math> must be <math>4e + 4 \cdot 31 = 4(e+31) = 4o</math> where <math>o</math> is an odd number
  7 KB (988 words) - 18:47, 11 August 2023
• 2014 AMC 10B Problems/Problem 19
  ...<math>C</math>. Let <math>H</math> be the foot of the altitude from <math>O</math> to <math>\overline{BC}</math>. Then we have the following diagram. pair A,O,B,C,H;
  2 KB (383 words) - 19:44, 28 April 2021
• 2014 AMC 10B Problems/Problem 23
  pair O = (0,s); draw(shift(O)*scale(s)*unitcircle);
  6 KB (1,060 words) - 19:15, 11 August 2023
• 2014 AMC 10B Problems/Problem 24
  pair O, A, B, C, D, E; O=origin;
  7 KB (1,133 words) - 12:46, 12 March 2022
• 2014 AMC 12B Problems/Problem 24
  pair O,A,B,C,D,E0,F; O=origin;
  7 KB (1,037 words) - 08:09, 1 January 2023
• 2014 AMC 12B Problems/Problem 19
  pair O = (0,s); draw(shift(O)*scale(s)*unitcircle);
  4 KB (703 words) - 16:24, 9 September 2022
• 2014 AIME I Problems/Problem 7
  pair O = (0,0); draw(A--B--O--cycle);
  5 KB (782 words) - 20:25, 10 October 2023
• 2014 AIME I Problems/Problem 13
  ...("$F$",F,dir(270)); dot("$E$",E,dir(180)); dot("$G$",G,dir(0)); dot("$Q$",O,dir(-90)); dot("$R$",R,dir(-180)); dot("$S$",S,dir(0)); dot("$T$",T,dir(90)
  9 KB (1,404 words) - 21:07, 13 October 2023
• 2014 AIME I Problems/Problem 15
  Let <math>O</math> be the midpoint of <math>DE.</math> <math>\angle EBF = 90^\circ \imp <math>O</math> is the center of the circle <math>BDGFE.</math>
  10 KB (1,643 words) - 22:30, 28 January 2024
• 1993 AHSME Problems/Problem 17
  ...one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock, then <math>\frac{q}{t}=</math>
  2 KB (301 words) - 03:42, 12 December 2018
• 1993 AHSME Problems/Problem 25
  ...ide <math>O</math> down an equal distance on the bottom ray to point <math>O'</math>. ...<math>m \angle O = 60^\circ</math>, therefore <math>\angle A \cong \angle O</math>. By our construction of moving the points the same distance, we hav
  2 KB (375 words) - 19:13, 29 July 2019
• 1973 AHSME Problems
  ...\textbf{(D)}\ x \in \left( - \frac32,\frac72\right) \qquad \textbf{(E)}\ \O \text{ (empty})</math> ...llel to the unit radius <math>OR</math> of the circle with center at <math>O</math>. Chords <math>MP</math>, <math>PQ</math>, and <math>NR</math> are ea
  18 KB (2,788 words) - 13:55, 20 February 2020
• 1968 AHSME Problems/Problem 7
  Let <math>O</math> be the intersection point of medians <math>AP</math> and <math>CQ</m
  498 bytes (74 words) - 01:52, 16 August 2023
• 1968 AHSME Problems/Problem 32
  ..., and in <math>8</math> minutes more they are again equidistant from <math>O</math>. Then the ratio of <math>A</math>'s speed to <math>B</math>'s speed
  739 bytes (119 words) - 01:53, 16 August 2023
• 1990 AHSME Problems/Problem 28
  ...math>BC=110</math>, <math>CD=130</math>, and <math>DA=90</math>. Let <math>O</math> be the center of the incircle. Draw in the radii from the center of
  2 KB (335 words) - 11:51, 5 October 2019
• 1969 AHSME Problems/Problem 15
  ...AB</math> is drawn with length equal to <math>r</math> (units). From <math>O</math>, a perpendicular to <math>AB</math> meets <math>AB</math> at <math>M pair O = (0,0);
  2 KB (250 words) - 18:29, 21 June 2018
• 1969 AHSME Problems/Problem 31
  ...<math>OABC</math> be a unit square in the <math>xy</math>-plane with <math>O(0,0),A(1,0),B(1,1)</math> and <math>C(0,1)</math>. Let <math>u=x^2-y^2</mat MP("O",(0,0),SW);
  2 KB (377 words) - 17:24, 20 June 2018
• 1969 AHSME Problems
  ...AB</math> is drawn with length equal to <math>r</math> (units). From <math>O</math>, a perpendicular to <math>AB</math> meets <math>AB</math> at <math>M ...<math>OABC</math> be a unit square in the <math>xy</math>-plane with <math>O(0,0),A(1,0),B(1,1)</math> and <math>C(0,1)</math>. Let <math>u=x^2-y^2</mat
  16 KB (2,662 words) - 14:12, 20 February 2020
• 1970 AHSME Problems/Problem 24
  Let <math>ABCDEF</math> be our regular hexagon, with centre <math>O</math> - and join <math>AO, BO, CO, DO, EO, </math> and <math>FO</math>. No
  1 KB (166 words) - 02:32, 29 June 2018
• 1970 AHSME Problems/Problem 29
  It is now between 10:00 and 11:00 o'clock, and six minutes from now, the minute hand of a watch will be exactly
  589 bytes (80 words) - 00:32, 13 January 2022
• 1952 AHSME Problems/Problem 49
  ...frac{2\sqrt{3}}{\sqrt{7}}.</math> <math>A</math> is vertically above <math>O</math> by <math>2\sqrt{3}</math> units. The scale factor is thus <math>\fra
  7 KB (1,136 words) - 10:01, 23 December 2023
• 1952 AHSME Problems/Problem 29
  ...tersection of <math>CH</math> and <math>AB</math> be <math>N</math>, <math>O</math> be the center of the circle, <math>ON = a</math> and <math>CN = x</m
  1 KB (191 words) - 13:20, 15 January 2018
• 1970 AHSME Problems
  It is now between <math>10:00</math> and <math>11:00</math> o'clock, and six minutes from now, the minute hand of the
  15 KB (2,366 words) - 07:52, 26 December 2023
• 1988 AHSME Problems
  ...CD</math>, and <math>BC</math> is tangent to the circle with center <math>O</math> and diameter <math>AD</math>. pair O=origin, A=(-1/sqrt(2),1/sqrt(2)), B=(-1/sqrt(2),-1), C=(1/sqrt(2),-1), D=(1
  17 KB (2,535 words) - 13:45, 19 February 2020
• 1986 AHSME Problems
  respectively. Let <math>O</math> be the center of the pentagon. If <math>OP = 1</math>, then <math>AO pair O=origin, A=2*dir(90), B=2*dir(18), C=2*dir(306), D=2*dir(234), E=2*dir(162),
  17 KB (2,512 words) - 18:30, 12 October 2023
• 1958 AHSME Problems/Problem 21
  ...math> \overline{DE}</math> are equal chords of a circle with center <math> O</math>. Arc <math> AB</math> is a quarter-circle. Then the ratio of the are MP("O",(0,0),N);MP("C",(-10,0),W);MP("D",(10,0),E);;MP("E",(0,10),N);
  1 KB (166 words) - 01:06, 1 January 2024
• 1958 AHSME Problems/Problem 42
  In a circle with center <math> O</math>, chord <math> \overline{AB}</math> equals chord <math> \overline{AC}
  1 KB (204 words) - 23:46, 19 April 2024
• 1958 AHSME Problems/Problem 48
  Diameter <math> \overline{AB}</math> of a circle with center <math> O</math> is <math> 10</math> units. <math> C</math> is a point <math> 4</math
  2 KB (332 words) - 00:49, 1 January 2024
• 1982 AHSME Problems
  ...M=intersectionpoint(A--B, O--O+40*dir(180)), N=intersectionpoint(A--C, O--O+40*dir(0)); draw(B--M--O--B--C--O--N--C^^N--A--M);
  17 KB (2,500 words) - 19:05, 11 September 2023
• 1979 AHSME Problems
  pair A=(-2,0),B,C=(-1,0),D=(1,0),EE,O=(0,0); draw(arc(O,1, 0, 180));
  17 KB (2,664 words) - 01:34, 19 March 2022
• 2001 IMO Problems
  ...\triangle ABC</math> issuing from the vertex <math>A</math>, and let <math>O</math> be the [[circumcenter]] of triangle <math>\triangle ABC</math>. Assu
  2 KB (343 words) - 21:52, 15 April 2021
• 1977 AHSME Problems
  & & & & & C & O & C & & & & &\\ & & & & C & O & N & O & C & & & &\\
  15 KB (2,412 words) - 05:09, 27 November 2020
• 1976 AHSME Problems
  O=(0,0), c1=circle(O, r1);
  17 KB (2,835 words) - 14:36, 8 September 2021
• 1975 AHSME Problems
  ...math>BC</math>; and <math>AN</math> and <math>CM</math> intersect at <math>O</math>. The ratio of the area of <math>AOCD</math> to the area of <math>ABC label("O", (1.2, .8));
  16 KB (2,512 words) - 04:48, 27 November 2021
• 2007 iTest Problems/Problem 19
  \textbf{(O) }\dfrac32\qquad
  3 KB (397 words) - 01:18, 29 October 2023
• 2007 iTest Problems/Problem 18
  \text{(O) It could be } 4 \text{ or } -4 \qquad
  2 KB (393 words) - 17:01, 10 June 2018
• 1969 IMO Problems/Problem 4
  ...t angle triangle <math>\triangle ABC</math> centered at the midpoint <math>O</math> of its hypotenuse <math>c = AB</math>. Let <math>R, S, T</math> be t
  5 KB (904 words) - 13:42, 29 January 2021
• 2007 iTest Problems/Problem 17
  \text{(O) }\frac{5}{6}\qquad
  2 KB (266 words) - 21:30, 4 February 2023
• 2007 iTest Problems/Problem 16
  \text{(O) }520\qquad
  3 KB (420 words) - 20:28, 17 June 2018
• 2007 iTest Problems/Problem 21
  \text{(O) }\frac{1}{2^{14}}\qquad
  2 KB (247 words) - 03:34, 14 June 2018
• 2007 iTest Problems/Problem 22
  \text{(O) } 14 \quad
  929 bytes (137 words) - 22:05, 10 January 2019
• 2007 iTest Problems/Problem 25
  \text{(O) } 14\qquad
  2 KB (270 words) - 14:35, 29 July 2018
• 2013 Canadian MO Problems
  Let <math>O</math> denote the circumcentre of an acute-angled triangle <math>ABC</math>
  2 KB (381 words) - 13:45, 8 October 2014
• 1973 Canadian MO Problems/Problem 6
  ...</math> are fixed points on a given circle not collinear with center <math>O</math> of the circle, and if <math>XY</math> is a variable diameter, find t
  445 bytes (76 words) - 17:51, 8 October 2014
• 2007 iTest Problems/Problem 23
  \text{(O) } 3-2i\quad</math>
  1 KB (202 words) - 16:48, 24 November 2018
• 2007 iTest Problems/Problem 24
  \text{(O) }14 \quad
  2 KB (340 words) - 19:49, 30 June 2018
• 1971 AHSME Problems
  ...inute hand passes the hour hand at the usual dial position(<math>12</math> o'clock, etc.)
  18 KB (2,703 words) - 20:50, 11 September 2023
• 2007 IMO Problems/Problem 4
  pair O = circumcenter(A,B,C); dot(O);
  8 KB (1,480 words) - 14:52, 5 August 2022
• 2014 IMO Problems/Problem 3
  pair O = circumcenter(H,S,T); path circle2 = Circle(O, length(O-H));
  6 KB (1,071 words) - 03:58, 8 September 2018
• 1964 AHSME Problems
  ...ts are drawn connecting these points to each other and to the origin <math>O</math>.
  19 KB (2,907 words) - 14:16, 20 February 2020
• 1963 AHSME Problems
  ...d <math>\triangle ABC</math> is inscribed in a circle with center at <math>O</math>; <math>\stackrel \frown {AB} = 120</math> and <math>\stackrel \frown
  20 KB (3,122 words) - 14:17, 20 February 2020
• 1957 AHSME Problems
  Circle <math>O</math> has diameters <math>AB</math> and <math>CD</math> perpendicular to e pair O = origin;
  26 KB (3,950 words) - 21:09, 31 August 2020
• 1956 AHSME Problems
  The points <math>A,B,C</math> are on a circle <math>O</math>. The tangent line at <math>A</math> and the secant <math>BC</math> i Triangle <math>PAB</math> is formed by three tangents to circle <math>O</math> and <math>\angle APB = 40^{\circ}</math>; then <math>\angle AOB</mat
  20 KB (3,039 words) - 22:44, 12 February 2021
• 2014 UMO Problems
  pair O=(-sqrt(3)/2,-1/2),E=(sqrt(3),0),NW=(-sqrt(3)/2,3/2),NE=(sqrt(3)/2,3/2),SE=( ...+2E+NE)--(O+2E+NE+SE)--(O+2E+2NE+SE)--(O+2E+2NE+SE+NW)--(O+3E+2NE+SE+NW)--(O+3E+3NE+SE+NW),black+linewidth(2));
  6 KB (982 words) - 15:11, 22 October 2018
• 2014 UMO Problems/Problem 6
  pair O=(-sqrt(3)/2,-1/2),E=(sqrt(3),0),NW=(-sqrt(3)/2,3/2),NE=(sqrt(3)/2,3/2),SE=( ...+2E+NE)--(O+2E+NE+SE)--(O+2E+2NE+SE)--(O+2E+2NE+SE+NW)--(O+3E+2NE+SE+NW)--(O+3E+3NE+SE+NW),black+linewidth(2));
  3 KB (628 words) - 19:22, 20 May 2021
• 2007 UNCO Math Contest II Problems/Problem 9
  Let <math>OM</math> be the inscribed circle's radius, with <math>O</math> being its center and <math>M</math> being the midpoint of <math>AB</
  3 KB (412 words) - 18:49, 29 January 2018
• 1988 AHSME Problems/Problem 27
  ...CD</math>, and <math>BC</math> is tangent to the circle with center <math>O</math> and diameter <math>AD</math>. pair O=origin, A=(-1/sqrt(2),1/sqrt(2)), B=(-1/sqrt(2),-1), C=(1/sqrt(2),-1), D=(1
  1 KB (261 words) - 22:22, 13 August 2023
• 1986 AHSME Problems/Problem 28
  respectively. Let <math>O</math> be the center of the pentagon. If <math>OP = 1</math>, then <math>AO pair O=origin, A=2*dir(90), B=2*dir(18), C=2*dir(306), D=2*dir(234), E=2*dir(162),
  4 KB (702 words) - 17:13, 17 April 2020
• 2014 AMC 8 Problems
  The circumference of the circle with center <math>O</math> is divided into 12 equal arcs, marked the letters <math>A</math> thr pair O=origin;
  13 KB (1,957 words) - 12:08, 13 January 2024

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