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• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> ...any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
  4 KB (692 words) - 22:33, 15 February 2021
• Fermat's Little Theorem
  ...] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to ...ibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>.
  16 KB (2,658 words) - 16:02, 8 May 2024
• Number theory/Advanced
  ...r theory include the [[Birch and Swinnerton-Dyer Conjecture]] and [[Fermat's Last Theorem]]. ...vertically upward as the point at infinity. We denote it by <math>\mathcal{O}</math>.
  5 KB (849 words) - 16:14, 18 May 2021
• 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 triangle <math>ABC</math> and intersects t .../math> is the Miquel Point of quadrilateral <math>ACNK</math>, so there is a spiral similarity centered at <math>M</math> that takes <math>KN</math> to
  3 KB (496 words) - 13:35, 18 January 2023
• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  ...ric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> that says: ..._1}+\cdots+\frac{1}{x_n}} \ge \sqrt[n_4]{\frac{x_1^{n_4}+\cdots+x_a^{n_4}}{a}}</cmath> where <math>n_1>1,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math
  5 KB (912 words) - 20:06, 14 March 2023
• Power of a Point Theorem
  ...e lengths of the [[line segment]]s formed when two [[line]]s [[intersect]] a [[circle]] and each other. ...of the lines is [[tangent line|tangent]] to the circle while the other is a [[secant line|secant]] (middle figure). In this case, we have <math> AB^2 =
  5 KB (827 words) - 17:30, 21 February 2024
• Mock AMC
  ...an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Prob == Tips for Writing a Mock AMC ==
  51 KB (6,175 words) - 20:58, 6 December 2023
• 2006 AIME I Problems/Problem 14
  A tripod has three legs each of length <math>5</math> feet. When the tripod i 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);
  6 KB (980 words) - 21:45, 31 March 2020
• 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
• 2006 AMC 12B Problems
  \text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } <math>\text {(A) } - 72 \qquad \text {(B) } - 27 \qquad \text {(C) } - 24 \qquad \text {(D)
  13 KB (2,058 words) - 12:36, 4 July 2023
• 2005 AMC 12A Problems
  {{AMC12 Problems|year=2005|ab=A}} (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D})
  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>? <math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2003 AMC 12B Problems
  \text {(A) } -1 \qquad \text {(B) } -\frac{2}{3} \qquad \text {(C) } \frac{2}{3} \qqu ...ks. A green pill costs 1 dollar more than a pink pill, and Al's pills cost a total of 546 dollars for the two weeks. How much does one green pill cost?
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2006 AMC 12B Problems/Problem 15
  ...s <math> C</math> and <math> D</math> are on the circle centered at <math> P</math>, such that <math> \overline{AD}</math> and <math> \overline{BC}</mat pair A, B, C, D;
  3 KB (458 words) - 16:40, 6 October 2019
• 2005 AIME II Problems/Problem 12
  ...r </math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math> ...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
• 2004 AIME II Problems/Problem 12
  ...k </math> and <math> p </math> are [[relatively prime]]. Find <math> k+m+n+p. </math> ...of the center circle be <math>r</math> and its center be denoted as <math>O</math>.
  3 KB (431 words) - 23:21, 4 July 2013
• 1983 AIME Problems
  ...th> and <math>z</math> all exceed <math>1</math> and let <math>w</math> be a positive number such that <math>\log_xw=24</math>, <math>\log_y w = 40</mat ...ue taken by <math>f(x)</math> for <math>x</math> in the [[interval]] <math>p \leq x\leq15</math>.
  7 KB (1,104 words) - 12:53, 6 July 2022
• 1992 AIME Problems
  A positive integer is called ascending if, in its decimal representation, the ...end of the weekend, her win ratio is greater than <math>.503</math>. What's the largest number of matches she could've won before the weekend began?
  8 KB (1,117 words) - 05:32, 11 November 2023
• 2000 AIME II Problems
  A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola <math>x^2 - y^ ...h>m/n</math> be the probability that two randomly selected cards also form a pair, where <math>m</math> and <math>n</math> are relatively prime positive
  6 KB (947 words) - 21:11, 19 February 2019
• 2002 AIME II Problems
  Three [[vertex|vertices]] of a [[cube]] are <math>P=(7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=(11,3,9)</math>. What ...th>b - a</math> is the [[Perfect square|square]] of an integer. Find <math>a + b + c</math>.
  7 KB (1,177 words) - 15:42, 11 August 2023
• 1983 AIME Problems/Problem 4
  ...hat of <math>BC</math> is <math>2</math> cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B< pair O=(0,0),
  11 KB (1,741 words) - 22:40, 23 November 2023
• 1983 AIME Problems/Problem 15
  ...r arc <math>AB</math> is a rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <mat pair A=(-0.91,-0.41);
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1985 AIME Problems/Problem 12
  ...= \frac{n}{729}</math> be the probability that the bug is at vertex <math>A</math> when it has crawled exactly <math>7</math> meters. Find the value of ...when it has crawled exactly <math>k</math> meters. We wish to find <math>p=P(7).</math>
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1989 AIME Problems/Problem 10
  ...h>\beta</math>, <math>\gamma</math>, be the angles opposite them. If <math>a^2+b^2=1989c^2</math>, find ...[altitude]] <math>h</math> to <math>c</math>, to get two [[right triangle]]s.
  8 KB (1,401 words) - 21:41, 20 January 2024
• 1990 AIME Problems/Problem 14
  ...[[triangular pyramid]], all four of whose faces are [[isosceles triangle]]s. Find the volume of this pyramid. pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6);
  7 KB (1,086 words) - 08:16, 29 July 2023
• 1991 AIME Problems/Problem 12
  ...math>P^{}_{}</math>, <math>Q^{}_{}</math>, <math>R^{}_{}</math>, and <math>S^{}_{}</math> are interior points on sides <math>\overline{AB}</math>, <math ...)",S,W); label("\(15\)",B/2+P/2,N);label("\(20\)",B/2+Q/2,E);label("\(O\)",O,SW); </asy></center>
  8 KB (1,270 words) - 23:36, 27 August 2023
• 1991 AIME Problems/Problem 11
  ...ath>c^{}_{}</math> is not divisible by the square of any prime. Find <math>a+b+c^{}_{}</math>. label("$1$", (0.5,0), S);
  4 KB (740 words) - 19:33, 28 December 2022
• 1993 AIME Problems/Problem 13
  ...enny and Kenny can see each other again. If <math>t\,</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? ...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.
  8 KB (1,231 words) - 20:06, 26 November 2023
• 1995 AIME Problems/Problem 12
  // p = point on line triple lineintersectplan(triple p, triple d, triple q, triple n)
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1997 AIME Problems/Problem 15
  ...th>q</math> is not divisible by the square of any prime number. Find <math>p+q+r</math>. ...{2} - \frac {a\sqrt {3}}{2}\right) + \left(\frac {11\sqrt {3}}{2} + \frac {a}{2}\right)i = b + 10i</math>.
  4 KB (609 words) - 22:49, 17 July 2023
• 1999 AIME Problems/Problem 15
  ...The vertices of its midpoint triangle are the [[midpoint]]s of its sides. A triangular [[pyramid]] is formed by folding the triangle along the sides of ...;label("\(E\)",foot(B,A,C),SW);label("\(F\)",foot(C,A,B),NW);label("\(P\)",P,NW);label("\(Q\)",Q,NE);label("\(R\)",R,SE);</asy><asy>import three; defaul
  7 KB (1,169 words) - 15:28, 13 May 2024
• 2014 USAJMO Problems
  ...frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc. </cmath> ...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
• 2002 AIME II Problems/Problem 15
  ...me, and <math>a</math> and <math>c</math> are relatively prime. Find <math>a + b + c</math>. ...<math>b</math> respectively. We know that the point <math>(9,6)</math> is a point on both circles, so we have that
  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 ...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
• 2002 AIME II Problems/Problem 12
  ...<math>c</math> are positive integers. Find <math>\left(p+q+r+s\right)\left(a+b+c\right)</math>. ...y=\frac{2x}{3}</math>. We can find the number of such paths using a Pascal's Triangle type method below, computing the number of paths to each point tha
  7 KB (1,127 words) - 13:34, 19 June 2022
• 2000 AIME II Problems/Problem 10
  ...al]] <math>ABCD</math>, [[tangent]] to <math>\overline{AB}</math> at <math>P</math> and to <math>\overline{CD}</math> at <math>Q</math>. Given that <mat ...e vertices of the quadrilateral, four pairs of congruent [[right triangle]]s are formed.
  2 KB (399 words) - 17:37, 2 January 2024
• Radius
  The '''radius''' of a [[circle]] is the distance from the center to any point on the circle. Ide The radius of a circle is often denoted using R or r.
  929 bytes (156 words) - 22:49, 5 January 2023
• 2006 AMC 10B Problems
  <math> \mathrm{(A) \ } -2006\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 0\qquad \mathrm{( <math> \mathrm{(A) \ } -72\qquad \mathrm{(B) \ } -27\qquad \mathrm{(C) \ } -24\qquad \mathrm{
  14 KB (2,059 words) - 01:17, 30 January 2024
• 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 ...ee below an attempted solution to understand why this problem doesn't have a solution:
  3 KB (541 words) - 17:32, 22 November 2023
• 1977 Canadian MO Problems/Problem 2
  ...on the circumference of the circle such that the angle <math>OPA</math> is a maximum. ...> 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
  ...^{7}</math>. If <math>m</math> is the number of elements in <math>\mathcal{S}</math>, compute <math>\star(m)</math>. ...\sqrt{7}</math>, <math>CA=1</math>, and <math>AB=3</math>. If <math>\angle A=\frac{\pi}{n}</math> where <math>n</math> is an integer, find the remainder
  8 KB (1,355 words) - 14:54, 21 August 2020
• 1977 Canadian MO Problems
  ...uation <math>4f(a)=f(b)</math> has no solutions in positive integers <math>a</math> and <math>b.</math> ...on the circumference of the circle such that the angle <math>OPA</math> is a maximum.
  3 KB (560 words) - 19:23, 10 March 2015
• 2006 Romanian NMO Problems/Grade 9/Problem 3
  ...ircle]] of [[radius]] <math>r</math>, for which there is a [[point]] <math>P</math> on <math>CD</math> such that <math>CB=BP=PA=AB</math>. (a) Prove that there are points <math>A,B,C,D,P</math> which fulfill the above conditions.
  6 KB (1,080 words) - 19:28, 21 September 2014
• 2005 Canadian MO Problems/Problem 3
  Let <math>S</math> be a set of <math>n\ge 3</math> points in the interior of a circle. ...d <math>c</math> is closer to <math>C</math> than any other point in <math>S</math>.
  2 KB (460 words) - 13:35, 9 June 2011
• P versus NP
  ...me, or whether there are problems in <math>NP</math> that are not in <math>P</math>. Since all modern computers (with the exception of a few quantum computers) are deterministic, non-deterministic algorithms are
  6 KB (1,104 words) - 15:11, 25 October 2017
• Orthocenter
  The '''orthocenter''' of a [[triangle]] is the point of intersection of its [[altitude|altitudes]]. I ...existence is a trivial consequence of the trigonometric version of [[Ceva's Theorem]]; however, the following proof, due to [[Leonhard Euler]], is much
  5 KB (829 words) - 13:11, 20 February 2024
• 2007 iTest Problems
  ...p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin <math>\mathrm{(A)}\, 4</math>
  30 KB (4,794 words) - 23:00, 8 May 2024
• 2005 AMC 10A Problems
  {{AMC10 Problems|year=2005|ab=A}} <math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \
  14 KB (2,026 words) - 11:45, 12 July 2021
• Euler line
  ...fact of Euclidean [[geometry]]. Certain fixed orders and distance [[ratio]]s hold among these points. In particular, <math>\overline{OGNH}</math> and < ...s examine what else this transformation, which we denote as <math>\mathcal{S}</math>, will do.
  59 KB (10,203 words) - 04:47, 30 August 2023
• 1985 IMO Problems/Problem 1
  A [[circle]] has center on the side <math>AB</math> of the [[cyclic quadrilat Let <math>O</math> be the center of the circle mentioned in the problem. Let <math>T</
  4 KB (684 words) - 07:28, 3 October 2021
• 2020 AMC 10A Problems
  {{AMC10 Problems|year=2020|ab=A}} <math>\textbf{(A)}\ {-}\frac{2}{3}\qquad\textbf{(B)}\ \frac{7}{36}\qquad\textbf{(C)}\ \frac{
  13 KB (1,968 words) - 18:32, 29 February 2024
• Asymptote: Useful commands and their Output
  For each of the following, we have put a blue dot at the origin in order to indicate relative location of the output In addition, a comment after a line such as <tt>//math - extension</tt> indicates that the command (in thi
  6 KB (871 words) - 21:14, 12 June 2023
• Mock AIME 1 Pre 2005 Problems
  ...tive integers with three distinct digits. Compute the remainder when <math>S</math> is divided by <math>1000</math>. ...h that <math>AB = BC = 1, CD = 2,</math> and <math>DE = 9</math>. If <math>P</math> can be any point in space, what is the smallest possible value of <m
  6 KB (1,100 words) - 22:35, 9 January 2016
• Mock AIME 2 Pre 2005/Problems
  <math>x</math> is a real number with the property that <math>x + \frac{1}{x} = 3</math>. Let < ...l, and a black ball. These balls are randomly drawn out of the box one at a time (without replacement) until two of the same color have been removed.
  6 KB (990 words) - 15:23, 11 November 2009
• Mock AIME 3 Pre 2005 Problems
  A function <math>f(x)</math> is defined for all real numbers <math>x</math>. ...> for which <math>f(x) = 2004</math>. Compute the integer nearest to <math>S</math>.
  7 KB (1,135 words) - 23:53, 24 March 2019
• Mock AIME 3 Pre 2005 Problems/Problem 5
  ...>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 the ...te the number of <math>n</math>-letter words ending in a vowel followed by a constant (<tt>VC</tt> - the only other combination, two vowels, is impossib
  5 KB (795 words) - 16:03, 17 October 2021
• 2007 AIME II Problems/Problem 3
  [[Square]] <math>ABCD</math> has side length <math>13</math>, and [[point]]s <math>E</math> and <math>F</math> are exterior to the square such that <mat pair A, B, C, D, E, F;
  6 KB (933 words) - 00:05, 8 July 2023
• 2007 AIME II Problems/Problem 15
  .../math>, and <math>\omega</math> is [[externally tangent]] to <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math>. If the sides pair A,B,C,X,Y,Z,P,Q,R;
  11 KB (2,099 words) - 17:51, 4 January 2024
• 2006 AIME II Problems/Problem 12
  ...| divisible]] by the [[perfect square | square]] of any prime, find <math> p+q+r. </math> size(250); pointpen = black; pathpen = black + linewidth(0.65); pen s = fontsize(8);
  6 KB (1,033 words) - 02:36, 19 March 2022
• 1998 PMWC Problems
  Find a pair of consecutive Triangular Numbers and the difference between a pair of consecutive Square Numbers whose difference are both <math>11</math ...<math>OB = OA=3</math> cm. <math>\angle BOA</math> is a right angle. <math>A</math> and <math>B</math> are two points on the circumference of circle of
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2001 PMWC Problems
  void wdot(pair p) fill(circle(p,.025),white);
  4 KB (641 words) - 21:24, 21 April 2014
• 2007 AMC 12B Problems
  Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 fee <math> \mathrm{(A) \ } 678\qquad \mathrm{(B) \ } 768\qquad \mathrm{(C) \ } 786\qquad \mathrm{
  12 KB (1,814 words) - 12:58, 19 February 2020
• Asymptote: Useful functions
  The bundled <code>markers</code> package provides a useful function for marking angles. The exact same function exists in the < arrowbar ''arrow''=None, pen ''p''=currentpen,<br/>
  4 KB (646 words) - 21:18, 26 March 2024
• Excircle
  ...is a [[circle]] [[Tangent line|tangent]] to the extensions of two sides of a [[triangle]] and the third side. pair excenter(pair A, pair B, pair C){
  5 KB (843 words) - 03:02, 1 July 2020
• 2001 USAMO Problems/Problem 2
  ...t <math>AD_2</math> at two points, the closer of which to the vertex <math>A</math> is denoted by <math>Q</math>. Prove that <math>AQ = D_2P</math>. ...math>. It follows that <math>2CD_3 = AB + BC - AC</math>, and <math>CD_3 = s-b = BD_1 = CD_2</math>, so <math>D_3 \equiv D_2</math>.)
  11 KB (2,091 words) - 08:35, 16 November 2017
• 2007 AMC 10A Problems
  {{AMC10 Problems|year=2007|ab=A}} ...ath>25\%</math> discount. Pam buys 5 tickets using a coupon that gives her a <math>30\%</math> discount. How many more dollars does Pam pay than Susan?
  13 KB (2,058 words) - 17:54, 29 March 2024
• 1995 AHSME Problems
  ...7, 83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will <math> \mathrm{(A) \ \text{remain the same} } \qquad \mathrm{(B) \ \text{increase by 1} } \qq
  17 KB (2,387 words) - 22:44, 26 May 2021
• 1951 AHSME Problems
  <math> \textbf{(A) \ } \frac {100(M - N)}{M} \qquad \textbf{(B) \ } \frac {100(M - N)}{N} \qq A rectangular field is half as wide as it is long and is completely enclosed
  23 KB (3,641 words) - 22:23, 3 November 2023
• 1966 AHSME Problems
  <math>\text{(A)} \ \frac 18 \qquad \text{(B)} \ \frac 73 \qquad \text{(C)} \ \frac78 \qqua When the base of a triangle is increased 10% and the altitude to this base is decreased 10%, t
  19 KB (3,159 words) - 22:10, 11 March 2024
• 1998 AHSME Problems/Problem 26
  ...\circ}</math>, angles <math>B</math> and <math>D</math> are [[right angle]]s, <math>AB = 13</math>, and <math>AD = 46</math>. Then <math>AC=</math> <math>\mathrm{(A)}\ 60
  3 KB (391 words) - 14:30, 5 July 2013
• 2004 AMC 12B Problems/Problem 19
  ...d cone has horizontal bases with radii <math>18</math> and <math>2</math>. A sphere is tangent to the top, bottom, and lateral surface of the truncated <math>\mathrm{(A)}\ 6
  3 KB (520 words) - 19:12, 20 November 2023
• Isogonal conjugate
  ...nal conjugates''' are pairs of [[point]]s in the [[plane]] with respect to a certain [[triangle]]. ...nt <math>O</math> be called isogonals with respect to the pair <math>(\ell,O).</math>
  54 KB (9,416 words) - 08:40, 18 April 2024
• 2008 AMC 10A Problems
  {{AMC10 Problems|year=2008|ab=A}} ...}\ {\small\text{AM}}</math> the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
  14 KB (2,138 words) - 15:08, 18 February 2023
• 2008 AMC 12A Problems
  {{AMC12 Problems|year=2008|ab=A}} ...}\ {\small\text{AM}}</math> the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
  13 KB (2,025 words) - 13:56, 2 February 2021
• 2008 AMC 12A Problems/Problem 13
  ...rcle]] centered at <math>O</math>, and <math>\angle AOB = 60^\circ</math>. A second circle is internally [[tangent]] to the first and tangent to both <m <math>\mathrm{(A)}\ \frac{1}{16}\qquad\mathrm{(B)}\ \frac{1}{9}\qquad\mathrm{(C)}\ \frac{1}{
  4 KB (630 words) - 20:32, 4 June 2021
• 2008 AMC 12A Problems/Problem 20
  <math>\mathrm{(A)}\ \frac{1}{28}\left(10-\sqrt{2}\right)\qquad\mathrm{(B)}\ \frac{3}{56}\lef pair C=(0,0),A=(0,3),B=(4,0),D=(4-2.28571,1.71429);
  6 KB (951 words) - 16:31, 2 August 2019
• 2008 AMC 12B Problems
  A basketball player made <math>5</math> baskets during a game. Each basket was worth either <math>2</math> or <math>3</math> points. <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \
  14 KB (2,199 words) - 13:43, 28 August 2020
• 2008 AIME I Problems/Problem 14
  ...a</math>. Point <math>P</math> is the foot of the perpendicular from <math>A</math> to line <math>CT</math>. Suppose <math>\overline{AB} = 18</math>, an ...= O - (9,0), A= O + (9,0), C=A+(18,0), T = 9 * expi(-1.2309594), P = foot(A,C,T);
  8 KB (1,333 words) - 00:18, 1 February 2024
• Mock AIME 5 Pre 2005 Problems
  &\text {ii. all ten digits, a through j are all distinct.}\\ &\text {iii.} a + b + c + d + e + f + g + h + i + j = 45\end{align*}</cmath>
  6 KB (909 words) - 07:27, 12 October 2022
• 2008 AIME II Problems/Problem 9
  ...<math>(p,q)</math>, find the greatest integer less than or equal to <math>|p| + |q|</math>. ...th>(x', y')</math> is the position of the particle after a move from <math>P</math>, then we have two equations for <math>x'</math> and <math>y'</math>:
  5 KB (725 words) - 22:37, 28 January 2024
• 2008 AIME II Problems/Problem 11
  ...rline{BC}</math>. Circle <math>Q</math> is externally [[tangent]] to <math>P</math> and is tangent to <math>\overline{AB}</math> and <math>\overline{BC} ...-28,0),C=(28,0),X=C-(64/3,0),Y=B+(4*r/3,0),P=X+(0,16),Q=Y+(0,r),M=foot(Q,X,P);
  6 KB (1,065 words) - 20:12, 9 August 2022
• 2005 USAMO Problems/Problem 3
  ...th>. Prove that points <math>B_1, C_1,P</math>, and <math>Q</math> lie on a circle. pair A=(2,5), B=(-1,0), C=(5,0);
  6 KB (1,117 words) - 01:17, 11 October 2021
• 2008 USAMO Problems/Problem 2
  ...rove that points <math>A</math>, <math>N</math>, <math>F</math>, and <math>P</math> all lie on one circle. ...le OFA = 90^{\circ}</math>, as this would immediately prove that <math>A,P,O,F,N</math> are concyclic.
  20 KB (3,565 words) - 11:54, 1 May 2024
• Millennium Problems
  ...set of seven problems for which the [[Clay Mathematics Institute]] offered a US \$7 million prize fund (\$1 million per problem) to celebrate the new mi ...Conjecture]], the [[Hodge Conjecture]], the [[Navier-Stokes Equations]], [[P versus NP]], the [[Poincaré Conjecture]], the [[Riemann Hypothesis]], and
  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
• 2008 iTest Problems
  ...s as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fue When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo.
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2005 AMC 8 Problems
  Connie multiplies a number by 2 and gets 60 as her answer. However, she should <math> \textbf{(A)}\ 7.5\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 120\q
  13 KB (1,821 words) - 22:18, 5 December 2023
• Spiral similarity
  ...milarity is a plane transformation composed of a rotation of the plane and a dilation of the plane having the common center. The order in which the com Any two directly similar figures are related either by a translation or by a spiral similarity (directly similar figures are similar and have the same o
  28 KB (4,863 words) - 00:29, 16 December 2023
• Van Aubel's Theorem
  ...; yielding centers <math>P_{AB}, P_{BC}, P_{CD}, P_{DA}</math>. Van Aubel's Theorem states that the two line segments connecting opposite centers are p pair A, B, C, D, O, P, Q, R, SS;
  2 KB (410 words) - 14:01, 4 March 2023
• 2001 USAMO Problems/Problem 4
  Let <math>P</math> be a point in the plane of triangle <math>ABC</math> such that the segments <mat Set <math>A(0,0)</math>, <math>B(1,0)</math>, <math>C(x,y)</math>, <math>P(p,q)</math>.
  8 KB (1,470 words) - 22:24, 18 June 2022
• 2004 AMC 10B Problems
  ...math> seats. Rows <math>12</math> through <math>22</math> are reserved for a youth club. How many seats are reserved for this club? <math> \mathrm{(A) \ } 297 \qquad \mathrm{(B) \ } 330\qquad \mathrm{(C) \ } 363\qquad \mathrm
  13 KB (1,988 words) - 20:19, 15 May 2024
• 2000 AMC 10 Problems
  ...= 2001</math>. What is the largest possible value of the sum <math>I + M + O</math>? <math>\textbf{(A)}\ 23 \qquad\textbf{(B)}\ 55 \qquad\textbf{(C)}\ 99 \qquad\textbf{(D)}\ 111
  14 KB (2,035 words) - 21:57, 2 May 2024
• 1976 USAMO Problems/Problem 2
  ...>AX</math> and <math>BY</math>. You may assume that <math>AB</math> is not a diameter. ...(pi/3), X1=r*expi(-pi/12), Y=r*expi(4*pi/3), Y1=r*expi(11*pi/12), O=(0,0), P, P1;
  5 KB (848 words) - 23:41, 6 July 2020
• 1976 USAMO Problems
  ...or white. Prove that with <i>any</i> such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board such as * (b) Exhibit a black-white coloring of a <math>4\times 6</math> board in which the four corner squares of every rect
  3 KB (510 words) - 19:01, 3 July 2013
• 1986 AJHSME Problems
  <math>\text{(A)}\ \frac{366}{31\times 24} \qquad \text{(B)}\ \frac{366\times 31}{24}\qquad <math>\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{2}{5} \qquad \text{(C)}\ 1 \qquad \
  14 KB (2,054 words) - 15:41, 8 August 2020
• 2004 AMC 10B Problems/Problem 22
  A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those cir <math> \mathrm{(A) \ } \frac{3\sqrt{5}}{2} \qquad \mathrm{(B) \ } \frac{7}{2} \qquad \mathrm{
  5 KB (700 words) - 13:46, 6 April 2024
• 2004 AMC 10B Problems/Problem 25
  ...ath> at points <math>A</math> and <math>B</math>, where <math>AB</math> is a diameter of the smaller circle. What is the area of the region, shaded in t \mathrm{(A) \ } \frac{5}{3} \pi - 3\sqrt 2
  4 KB (703 words) - 22:37, 2 November 2022
• 2001 AMC 12 Problems/Problem 17
  ...s selected at random from the interior of the pentagon with vertices <math>A = (0,2)</math>, <math>B = (4,0)</math>, <math>C = (2 \pi + 1, 0)</math>, <m \text{(A) }\frac {1}{5}
  5 KB (792 words) - 15:23, 30 November 2021
• 2009 AMC 10B Problems
  ...nt muffin or a <math>75</math>-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? <math>\text{(A) } 1\qquad\text{(B) } 2\qquad\text{(C) } 3\qquad\text{(D) } 4\qquad\text{(E
  15 KB (2,262 words) - 00:53, 18 June 2021
• 2009 AMC 10B Problems/Problem 22
  ...inches of cake and <math>s</math> square inches of icing. What is <math>c+s</math>? pair P=foot((1,-1),(1,1),(-1,0));
  12 KB (1,868 words) - 03:36, 30 September 2023

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