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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

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