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• ICTM Math Contest
  ...consists of five members who work together on 20 questions for 25 minutes. A school may field as many five member teams as it wishes. The individual com A school team will qualify by taking first place in a region with four or more schools competing.
  8 KB (1,182 words) - 14:26, 3 April 2024
• Ptolemy's theorem
  ...orem''' gives a relationship between the side lengths and the diagonals of a [[cyclic quadrilateral]]; it is the [[equality condition | equality case]] Given a [[cyclic quadrilateral]] <math>ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
  7 KB (1,198 words) - 20:39, 9 March 2024
• LaTeX:Symbols
  This article provides a short list of commonly used LaTeX symbols. .../math> on the web, (technically an AJAX library simulating it.)) maintains a [http://docs.mathjax.org/en/latest/tex.html#supported-latex-commands list o
  16 KB (2,324 words) - 16:50, 19 February 2024
• 2004 AMC 12A Problems
  {{AMC12 Problems|year=2004|ab=A}} <math>\text{(A) } 0.0029 \qquad \text{(B) } 0.029 \qquad \text{(C) } 0.29 \qquad \text{(D)
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2006 AMC 10A Problems/Problem 17
  pair A,B,C,D,E,F,G,H,W,X,Y,Z; A=(0,2); B=(1,2); C=(2,2); D=(3,2);
  6 KB (1,066 words) - 00:21, 2 February 2023
• 2005 AIME II Problems/Problem 12
  ...> AB </math> with <math> AE<BF </math> and <math> E </math> between <math> A </math> and <math> F, m\angle EOF =45^\circ, </math> and <math> EF=400. </m ...D--A);draw(E--O--F);draw(G--O); dot(A^^B^^C^^D^^E^^F^^G^^O); label("\(A\)",A,(-1,1));label("\(B\)",B,(1,1));label("\(C\)",C,(1,-1));label("\(D\)",D,(-1,
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME II Problems/Problem 15
  ...h> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally [[tangent (geometry)|tangent]] to <math> w_2 </ma pair A = (-5, 12), B = (5, 12), C = (0, 0);
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2004 AIME II Problems/Problem 7
  <math> ABCD </math> is a rectangular sheet of paper that has been folded so that corner <math> B </m pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect
  9 KB (1,501 words) - 05:34, 30 October 2023
• 1992 AIME Problems
  A positive integer is called ascending if, in its decimal representation, the ...she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math>0.500</math>. During the weekend, s
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1983 AIME Problems/Problem 11
  The solid shown has a square base of side length <math>s</math>. The upper edge is parallel to th triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);
  5 KB (865 words) - 21:11, 6 February 2023
• 1984 AIME Problems/Problem 3
  A [[point]] <math>P</math> is chosen in the interior of <math>\triangle ABC</ pair A=(0,0),B=(12,0),C=(4,5);
  4 KB (726 words) - 13:39, 13 August 2023
• 1985 AIME Problems/Problem 4
  A small [[square (geometry) | square]] is constructed inside a square of [[area]] 1 by dividing each side of the unit square into <math>n< ...qrt{2n^2 - 2n + 1}</math>, so the parallelogram has height <math>h = \frac{A}{l} = \frac{1}{\sqrt{2n^2 - 2n + 1}}</math>. But the height of the paralle
  3 KB (484 words) - 21:40, 2 March 2020
• 1991 AIME Problems/Problem 12
  ...--B--C--D--cycle);draw(P--R..Q--S); draw(P--Q--R--S--cycle); label("\(A\)",A,NW);label("\(B\)",B,NE);label("\(C\)",C,SE);label("\(D\)",D,SW); label("\(P ...bus; <math>PQ = \sqrt{15^2 + 20^2} = 25</math>. Since the [[diagonal]]s of a rhombus are [[perpendicular bisector]]s, we have that <math>OP = 15, OQ = 2
  8 KB (1,270 words) - 23:36, 27 August 2023
• 1992 AIME Problems/Problem 14
  ...AC^{}_{}</math>, and <math>AB^{}_{}</math>, respectively. Given that <math>AA'</math>, <math>BB'</math>, and <math>CC'</math> are concurrent at the point ...and <math>ABC</math> having the same base, <cmath>\frac{AO}{OA'}+1=\frac{AA'}{OA'}=\frac{[ABC]}{[BOC]}=\frac{K_A+K_B+K_C}{K_A}.</cmath> Therefore, we
  4 KB (667 words) - 01:26, 16 August 2023
• 1996 AIME Problems/Problem 15
  ...=(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); D(MP("A",A,N)--MP("B",B)--MP("C",C)--MP("D",D,N)--cycle); D(B--D); D(A--C); D(MP("O",O,SE));
  5 KB (710 words) - 21:04, 14 September 2020
• 1997 AIME Problems/Problem 8
  ...umn must have 2 1's and 2 -1's. Let's consider the first column. There are a total of <math>6</math> ways to arrange 2 1's and 2 -1's. Let's consider th ...e that first there are <math>\binom{4}{2}</math> values of the fixed <math>A</math>. We then have the following cases:
  4 KB (638 words) - 16:41, 22 January 2024
• 1998 AIME Problems/Problem 8
  ...[negative]] term encounted. What positive integer <math>x</math> produces a sequence of maximum length? ...ont> || <math>1000 - x</math> || <math>2x - 1000</math><font color="white">a</font> || <math>2000 - 3x</math> || <math>5x - 3000</math> || <math>5000 -
  2 KB (354 words) - 19:37, 24 September 2023
• 1999 AIME Problems/Problem 14
  pair A=(0,0),B=(13,0),C=IP(circle(A,15),circle(B,14)); D(MP("A",A)--MP("B",B)--MP("C",C,N)--cycle);
  7 KB (1,184 words) - 13:25, 22 December 2022
• 2001 AIME I Problems/Problem 7
  ...), I=incenter(A,B,C), D=IP((0,I.y)--(20,I.y),A--B), E=IP((0,I.y)--(20,I.y),A--C); D(MP("A",A,N)--MP("B",B)--MP("C",C)--cycle); D(MP("I",I,NE)); D(MP("E",E,NE)--MP("D",D
  9 KB (1,540 words) - 08:31, 1 December 2022
• 2002 AIME I Problems/Problem 15
  ...CD</math> is a square with <math>AB = 12;</math> face <math>ABFG</math> is a trapezoid with <math>\overline{AB}</math> parallel to <math>\overline{GF},< triple A=(-6,-6,0), B = (-6,6,0), C = (6,6,0), D = (6,-6,0), E = (2,0,12), H=(-6+2*s
  7 KB (1,181 words) - 20:32, 8 January 2024
• 2002 AIME I Problems/Problem 10
  In the diagram below, angle <math>ABC</math> is a right angle. Point <math>D</math> is on <math>\overline{BC}</math>, and <ma ...> is <math>10/3</math> that of triangle <math>AEG</math>, since they share a common side and angle, so the area of triangle <math>AGF</math> is <math>10
  4 KB (643 words) - 22:44, 8 August 2023
• 2003 AIME II Problems/Problem 11
  Triangle <math>ABC</math> is a right triangle with <math>AC = 7,</math> <math>BC = 24,</math> and right an ...>, we get <math>DM=\frac{5\sqrt{11}} {2}</math>. Since <math>ABC</math> is a right triangle, <math>M</math> is the circumcenter and thus, <math>CM=\frac
  5 KB (772 words) - 19:47, 1 August 2023
• 2002 AIME II Problems/Problem 14
  ...is <math>152</math>, and the angle <math>PAM</math> is a [[right angle]]. A [[circle]] of [[radius]] <math>19</math> with center <math>O</math> on <mat ...MPA</math> are similar. Also note that <math>AM = BM</math> by [[power of a point]]. Using the fact that the ratio of corresponding sides in similar tr
  4 KB (658 words) - 19:15, 19 December 2021
• 2001 AIME II Problems/Problem 13
  pair A=(0,0), D=(x,0), B=IP(CR(A,8),CR(D,10)), E=(-3x/5,0), C=IP(CR(E,16),CR(D,64/5)); D(MP("A",A)--MP("B",B,NW)--MP("C",C,NW)--MP("D",D)--cycle); D(B--D); D(A--MP("E",E)--B,d);
  4 KB (743 words) - 03:32, 23 January 2023
• 2006 USAMO Problems/Problem 6
  ...th>, <math>SBF</math>, <math>TCF</math>, and <math>TDE</math> pass through a common point. ...imilarity, and <math>X</math> is the center of spiral similarity for <math>A,E,B,</math> and <math>F</math>.
  5 KB (986 words) - 22:46, 18 May 2015
• Dedekind domain
  A '''Dedekind domain''' is a [[integral domain]] <math>R</math> satisfying the following properties: * <math>R</math> is a [[noetherian]] [[ring]].
  9 KB (1,648 words) - 16:36, 14 October 2017
• 1959 IMO Problems/Problem 6
  ...</math> (with <math>AB </math> [[parallel]] to <math>DC </math>) in which a [[circle]] can be inscribed, and with [[vertex | vertices]] <math>B </math> ...that we must have both lines <math>AB </math> (which we shall denote <math>a </math>) and <math>DC </math> (which we shall denote <math>c </math>) paral
  2 KB (383 words) - 05:58, 11 February 2024
• 2006 Romanian NMO Problems/Grade 8/Problem 1
  We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove t ...o non-coplanar circles in space that intersect at two points, there exists a point P such that P is equidistant from any point on one circle and any poi
  3 KB (509 words) - 23:22, 15 August 2012
• 2006 Romanian NMO Problems/Grade 9/Problem 2
  ...ath>\triangle BDC</math>. Now assume that P' is the point such that DP' is a median of <math>\triangle EDF</math> (it is on <math>EF</math>). It is on D
  3 KB (488 words) - 14:05, 15 December 2022
• Equilateral triangle
  pair A=(0,50),B=(43.301,-25),C=(-43.301,-25); draw(A--B--C--A);
  1 KB (186 words) - 19:57, 15 September 2022
• Similarity (geometry)
  ...similar in every aspect except possibly size or orientation. For example, a globe and the surface of the earth are, in theory, similar. ...id motion]]s). We say two objects are similar if they are congruent up to a [[dilation]].
  2 KB (261 words) - 20:42, 25 November 2023
• Euler line
  ...] <math>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain fixed orders and distan ...followed by a homothety with scale factor <math>2</math> centered at <math>A</math> brings <math>\triangle ABC \to \triangle O_AO_BO_C</math>. Let us ex
  59 KB (10,203 words) - 04:47, 30 August 2023
• 2003 AMC 10A Problems/Problem 22
  pair D=(0,0), Ep=(4,0), A=(9,0), B=(9,8), H=(3,8), C=(0,8), G=(-6,20), F=(-6,0); draw(D--A--B--C--D--F--G--Ep);
  9 KB (1,446 words) - 22:48, 8 May 2024
• 2006 Canadian MO Problems/Problem 2
  Let <math>ABC</math> be an [[acute angle]]d [[triangle]]. Inscribe a [[rectangle]] <math>DEFG</math> in this triangle so that <math>D</math> is ...}{AB}</math> and let <math>H</math> be the foot of the altitude from <math>A</math> to <math>BC</math>. Then by [[similarity]], <math>\frac{AE}{AC} = \
  2 KB (416 words) - 20:00, 21 September 2014
• Mock AIME 3 Pre 2005 Problems/Problem 11
  ...th> is an acute triangle with perimeter <math>60</math>. <math>D</math> is a point on <math>\overline{BC}</math>. The circumcircles of triangles <math>A ...ath>B</math> to <math>E</math> and <math>F</math> to <math>D</math>, so by a fundamental theorem of spiral similarity <math>\triangle BDE\sim\triangle F
  2 KB (325 words) - 15:32, 22 March 2015
• Circumradius
  ...very triangle is [[cyclic]], every triangle has a circumscribed circle, or a [[circumcircle]]. ==Formula for a Triangle==
  4 KB (729 words) - 16:52, 19 February 2024
• 2006 AIME II Problems/Problem 6
  ...d <math> b</math> is not divisible by the square of any prime. Find <math> a+b+c. </math> pair B = (0, 0), C = (1, 0), D = (1, 1), A = (0, 1);
  7 KB (1,067 words) - 12:23, 8 April 2024
• 1961 IMO Problems
  \quad x + y + z \!\!\! &= a \; \, \\ ...</math> and <math>b </math> are constants. Give the conditions that <math>a </math> and <math>b </math> must satisfy so that <math>x, y, z </math> (the
  3 KB (425 words) - 21:18, 20 August 2020
• 1961 IMO Problems/Problem 6
  ...o not form a triangle.) What is the locus of point <math>G</math> as <math>A', B', C'</math> range independently over the plane <math>\epsilon</math>? ...\prime B^\prime C^\prime</math>. It is obvious that the centroid of <math>A^\prime B^\prime C^\prime</math> can be any point on plane <math>\epsilon</m
  2 KB (301 words) - 23:29, 18 July 2016
• 1962 IMO Problems
  (a) Its decimal representation has 6 as the last digit. ...> begin their motion at the same instant from the starting positions <math>A</math> and <math>B'</math>, respectively. Determine and draw the locus of t
  3 KB (431 words) - 21:17, 20 August 2020
• 2004 AMC 12A Problems/Problem 12
  ...intersect at <math>C = (2,8)</math>. What is the length of <math>\overline{A'B'}</math>? <math>\text {(A)} 2 \qquad \text {(B)} 2\sqrt2 \qquad \text {(C)} 3 \qquad \text {(D)} 2 +
  1 KB (181 words) - 20:17, 3 July 2013
• 2004 AMC 12A Problems/Problem 8
  pair A, B, C, D, E; A=(0, 0);
  5 KB (692 words) - 21:15, 5 April 2024
• De Longchamps point
  The '''de Longchamps point''' of a [[triangle]] is the [[reflection (mathematics)|reflection]] of the triangle The de Longchamps point of a triangle is the radical center of the power circles of the triangle. Prove
  10 KB (1,780 words) - 09:23, 17 November 2022
• 1991 IMO Problems/Problem 1
  ...isectors of the angles <math>A,B,C</math> meet the opposite sides in <math>A^\prime,B^\prime,C^\prime</math> respectively. Prove that <math>\frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}</math>
  2 KB (311 words) - 09:40, 19 November 2023
• 1990 USAMO Problems/Problem 5
  ...\, Q \,</math>. Prove that the points <math>\, M, N, P, Q \,</math> lie on a common circle. ...of the altitude from <math>A</math>: <math>A</math>, <math>H</math>, <math>A'</math> are concurrent, where <math>H</math> is the orthocentre.
  3 KB (604 words) - 20:52, 24 October 2018
• Isogonal conjugate
  ...nal conjugates''' are pairs of [[point]]s in the [[plane]] with respect to a certain [[triangle]]. ...ll</math> and a point <math>O</math> lying on <math>\ell</math> be given. A pair of lines symmetric with respect to <math>\ell</math> and containing t
  54 KB (9,416 words) - 08:40, 18 April 2024
• 2007 AMC 12B Problems/Problem 21
  ...hese base-<math>3</math> representations are palindromes? (A palindrome is a number that reads the same forward and backward.) <math>\mathrm {(A)}\ 100 \qquad \mathrm {(B)}\ 101 \qquad \mathrm {(C)}\ 102 \qquad \mathrm {
  2 KB (267 words) - 16:35, 15 February 2021
• 2008 AMC 12B Problems/Problem 25
  ...B=11, BC=5, CD=19,</math> and <math>DA=7</math>. Bisectors of <math>\angle A</math> and <math>\angle D</math> meet at <math>P</math>, and bisectors of < <math>\textbf{(A)}\ 28\sqrt{3}\qquad \textbf{(B)}\ 30\sqrt{3}\qquad \textbf{(C)}\ 32\sqrt{3}
  12 KB (2,015 words) - 20:54, 9 October 2022
• Mock AIME 1 Pre 2005 Problems/Problem 8
  ...th>P</math>, dividing <math>P</math> into a [[frustum]] <math>F</math> and a smaller pyramid <math>P'</math>. Let <math>X</math> denote the center of th Thus, the top of the frustum is a rectangle <math>A'B'C'D'</math> with <math>A'B' = 6</math> and <math>B'C' = 8</math>.
  3 KB (446 words) - 00:18, 10 February 2020
• 2008 AIME I Problems
  Of the students attending a school party, <math>60\%</math> of the students are girls, and <math>40\%</ ...e without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. Th
  9 KB (1,536 words) - 00:46, 26 August 2023

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