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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 Lemma: 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 Lemma 1: 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

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