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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) - 03:13, 27 May 2024
• 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);
  20 KB (3,497 words) - 15:37, 27 May 2024
• 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) - 17:46, 24 May 2024
• 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
• 2009 AIME I Problems
  ...s <math>3</math> distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest g ...s a complex number <math>z</math> with imaginary part <math>164</math> and a positive integer <math>n</math> such that
  7 KB (1,152 words) - 02:24, 23 July 2021
• 2009 AIME I Problems/Problem 4
  ...}</math> so that <math>\frac {AN}{AD} = \frac {17}{2009}</math>. Let <math>P</math> be the point of intersection of <math>\overline{AC}</math> and <math One of the ways to solve this problem is to make this parallelogram a straight line.
  7 KB (1,117 words) - 00:23, 9 January 2023
• 2009 AIME I Problems/Problem 11
  ...y = 2009</math>. Find the number of such distinct triangles whose area is a positive integer. == Solution 1 (Matrix's, Determinants) ==
  8 KB (1,319 words) - 15:01, 16 August 2020
• 2009 AIME I Problems/Problem 12
  ...cle having <math>\overline{CD}</math> as a diameter. Let <math>I</math> be a point outside <math>\triangle ABC</math> such that <math>\overline{AI}</mat ...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
  12 KB (1,970 words) - 22:53, 22 January 2024
• 2009 AIME I Problems/Problem 15
  ...and <math>c</math> is not divisible by the square of any prime. Find <math>a + b + c</math>. pair A,B,C,D,Ic,Ib,P;
  6 KB (1,048 words) - 19:35, 2 January 2023
• 2009 AIME II Problems/Problem 10
  ...and <math>r</math> is not divisible by the square of any prime. Find <math>p</math> + <math>q</math> + <math>r</math>. ...),NW,f); D(rightanglemark(C,B,A,20)); D(anglemark(D,A,E,35));D(anglemark(C,A,D,30));
  9 KB (1,415 words) - 13:56, 18 December 2022
• Radical axis
  ...n should peruse this material carefully, as it may contain the keys to one's future success. ...rtains to that situation. I hope after you read this text, you will become a better math student, armed with another tool to solve difficult problems. B
  12 KB (2,125 words) - 08:38, 23 May 2024
• 2009 AIME II Problems/Problem 15
  ...d <math>t</math> is not divisible by the square of any prime. Find <math>r+s+t</math>. ...c{3}{5}</math>, point <math>B</math> lies between <math>M</math> and <math>A</math> on the semicircular arc. We will first compute the length of <math>\
  11 KB (1,862 words) - 21:23, 23 May 2024
• 2009 IMO Problems/Problem 2
  ...BC</math> be a triangle with circumcentre <math>O</math>. The points <math>P</math> and <math>Q</math> are interior points of the sides <math>CA</math> dot("O", (50, 38), N);
  2 KB (280 words) - 01:16, 19 November 2023
• 2009 IMO Problems
  Let <math>n</math> be a positive integer and let <math>a_1,\ldots,a_k (k\ge2)</math> be distinct in ...BC</math> be a triangle with circumcentre <math>O</math>. The points <math>P</math> and <math>Q</math> are interior points of the sides <math>CA</math>
  3 KB (509 words) - 09:23, 10 September 2020
• 1992 AJHSME Problems
  <math>\text{(A)}\ -1 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 9 \qquad <math>\text{(A)}\ \dfrac{10}{8} \qquad \text{(B)}\ 1\dfrac{1}{4} \qquad \text{(C)}\ 1\dfra
  17 KB (2,346 words) - 13:36, 19 February 2020
• 2010 AMC 10B Problems
  \textbf{(A)}\ -20,000 <math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\
  12 KB (1,817 words) - 22:44, 22 December 2020
• Characteristic polynomial
  ...[operator]] refers to the [[polynomial]] whose roots are the [[eigenvalue]]s of the operator. It carries much information about the operator. ...<math>A</math> is defined as <math>P_A(t) = \det(tI - A)</math>, which is a <math>n</math>th degree polynomial in <math>t</math>. Here, <math>I</math>
  19 KB (3,412 words) - 14:57, 21 September 2022
• Mock AIME 2 2010 Answer Key
  ...an be easily proved through strong induction. Starting from 2010, which is a multiple of 15, we must first purge 1 lemming. We can then purge 4 lemmings ...atisfies <math>b_1+b_2+\ldots+b_{10}\equiv 0 \pmod{3}</math>, there exists a corresponding <math>(a_1, a_2, \ldots, a_{10})</math> such that <math>a_1+a
  36 KB (6,214 words) - 20:22, 13 July 2023
• 2010 AIME II Problems/Problem 14
  ...and <math>r</math> is not divisible by the square of any prime. Find <math>p+q+r</math>. Let <math>O</math> be the [[circumcenter]] of <math>ABC</math> and let the intersection
  10 KB (1,507 words) - 00:31, 19 November 2023
• 2010 AIME II Problems/Problem 9
  ...of the smaller hexagon to the area of <math>ABCDEF</math> be expressed as a fraction <math>\frac {m}{n}</math> where <math>m</math> and <math>n</math> pair A,B,C,D,E,F;
  5 KB (857 words) - 22:22, 27 August 2023
• 2010 AIME II Problems/Problem 11
  Define a <i>T-grid</i> to be a <math>3\times3</math> matrix which satisfies the following two properties: ...es are <math>1</math>'s, and the remaining four entries are <math>0</math>'s.</LI>
  6 KB (1,057 words) - 01:58, 8 January 2023
• 2010 USAMO Problems
  ...d <math>RS</math> is half the size of <math>\angle XOZ</math>, where <math>O</math> is the midpoint of segment <math>AB</math>. ...to make more than <math>\binom{n}{3}</math> such switches before reaching a position in which no further switches are possible.
  3 KB (525 words) - 13:44, 4 July 2013
• 2010 USAJMO Problems
  ...Find with proof the smallest <math>n</math> such that <math>P(n)</math> is a multiple of <math>2010</math>. ...d <math>RS</math> is half the size of <math>\angle XOZ</math>, where <math>O</math> is the midpoint of segment <math>AB</math>.
  3 KB (538 words) - 13:55, 16 June 2020
• 2010 USAMO Problems/Problem 1
  Let <math>AXYZB</math> be a convex pentagon inscribed in a semicircle of diameter <math>AB</math>. Denote by <math>P, Q, R, S</math> the feet of the perpendiculars from <math>Y</math> onto
  13 KB (2,271 words) - 14:08, 23 May 2024
• 1982 AHSME Problems/Problem 9
  pair A=origin, O=(10,0), B=(3,0), N=(0,5), C=(3,5), P=(5,0), D=(1,1), G=(9,1), F=(1,0); draw(G--A--D--cycle, linewidth(0.7));
  2 KB (315 words) - 12:30, 19 October 2020
• 1996 AJHSME Problems
  <math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad <math>\text{(A)}\ \text{Jose} \qquad \text{(B)}\ \text{Thuy} \qquad \text{(C)}\ \text{Kare
  13 KB (1,880 words) - 13:35, 19 February 2020
• 2011 AMC 12A Problems/Problem 13
  \textbf{(A)}\ 27 \qquad Let <math>O</math> be the incenter of <math>\triangle{ABC}</math>. Because <math>\overl
  4 KB (683 words) - 03:12, 23 January 2023
• 2011 AMC 12A Problems/Problem 15
  ...ircular base of a hemisphere of radius <math>2</math> rests on the base of a square pyramid of height <math>6</math>. The hemisphere is tangent to the o \textbf{(A)}\ 3\sqrt{2} \qquad
  2 KB (376 words) - 23:14, 5 January 2024
• 2011 AMC 12A Problems/Problem 25
  ...and <math>AC \geq AB</math>. Let <math>H</math>, <math>I</math>, and <math>O</math> be the orthocenter, incenter, and circumcenter of <math>\triangle AB \textbf{(A)}\ 60^{\circ} \qquad
  6 KB (1,046 words) - 13:05, 28 June 2022
• 2011 AMC 12A Problems/Problem 24
  ...of the largest possible circle that fits inside or on the boundary of such a quadrilateral? \textbf{(A)}\ \sqrt{15} \qquad
  4 KB (753 words) - 18:58, 2 June 2022
• 1958 AHSME Problems
  <math> \textbf{(A)}\ 5\qquad <math> \textbf{(A)}\ y - x\qquad \textbf{(B)}\ x - y\qquad \textbf{(C)}\ \frac {y - x}{xy}\qq
  25 KB (3,872 words) - 14:21, 20 February 2020
• 1993 AHSME Problems
  ...<math>c</math>, define <math>\boxed{a,b,c}</math> to mean <math>a^b-b^c+c^a</math>. Then <math>\boxed{1,-1,2}</math> equals <math>\text{(A)} \ -4 \qquad \text{(B)} \ -2 \qquad \text{(C)} \ 0 \qquad \text{(D)} \ 2 \
  20 KB (2,814 words) - 08:15, 27 June 2021
• 2021 AIME II Problems/Problem 5
  ...tau(s)</math> are congruent, is an interval <math>[a,b)</math>. Find <math>a^2+b^2</math>. We start by defining a triangle. The two small sides MUST add to a larger sum than the long side. We are given <math>4</math> and <math>10</ma
  14 KB (2,269 words) - 00:43, 2 January 2023
• 2011 AMC 12B Problems/Problem 20
  <math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \q ...> and <math>\triangle CEF</math> all intersect at <math>P</math>, so <math>P</math> is <math>X</math>.
  8 KB (1,200 words) - 19:31, 7 August 2023
• 2011 AIME I Problems/Problem 4
  ...h>AC=117</math> and <math>BC=120</math>. The angle bisector of angle <math>A</math> intersects <math> \overline{BC} </math> at point <math>L</math>, and ...CN}</math> such that they intersect line <math>{AB}</math> at points <math>P</math> and <math>Q</math>, respectively.
  6 KB (1,068 words) - 18:52, 2 August 2023
• 2011 AIME I Problems/Problem 14
  Let <math>A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8</math> be a regular octagon. Let <math>M_1</math>, <math>M_3</math>, <math>M_5</math>, pair A,B,C,D,E,F,G,H,M,N,O,O2,P,W,X,Y,Z;
  8 KB (1,344 words) - 18:39, 9 February 2023
• 2011 AIME II Problems/Problem 10
  ...14 intersect at point <math>P</math>. The distance between the [[midpoint]]s of the two chords is 12. The quantity <math>OP^2</math> can be represented The line through the midpoint of a chord of a circle and the center of that circle is perpendicular to that chord, so <ma
  11 KB (1,720 words) - 03:12, 18 December 2023
• 1999 AMC 8 Problems
  <math>\text{(A)} \div \qquad \text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \ ...e degree measure of the smaller angle formed by the hands of a clock at 10 o clock?
  17 KB (2,394 words) - 19:51, 8 May 2023
• 2011 USAJMO Problems/Problem 5
  ...h>PD</math> are tangent to <math>\omega</math>, (ii) <math>P</math>, <math>A</math>, <math>C</math> are collinear, and (iii) <math>\overline{DE} \parall ...<math>ABC</math>, from which it follows that <math>\overline{BE}</math> is a median of <math>\triangle ABC</math>. <math>\blacksquare</math>
  4 KB (717 words) - 17:00, 14 April 2024
• 2011 USAJMO Problems/Problem 3
  ...nate plane, let <math>\ell(P)</math> denote the line passing through <math>P</math> with slope <math>2a</math>. Consider the set of triangles with verti ...math> which are all distinct, and so there are no other roots. Using Vieta's, we get that
  15 KB (2,593 words) - 13:37, 29 January 2021
• 2011 AMC 10B Problems/Problem 17
  pair A=(-3cos(80),-3sin(80)); pair O=(0,0), E=(-3,0), B=(3,0);
  5 KB (699 words) - 04:53, 21 January 2023
• 2007 AMC 10B Problems
  Isabella's house has <math>3</math> bedrooms. Each bedroom is <math>12</math> feet lon <math>\textbf{(A) } 678 \qquad\textbf{(B) } 768 \qquad\textbf{(C) } 786 \qquad\textbf{(D) }
  15 KB (2,297 words) - 12:57, 19 February 2020
• Projective geometry
  A particularly striking result of projective geometry is the duality of point The first theorems of a projective character were established by the ancient Greeks, most notably P
  4 KB (720 words) - 15:03, 26 January 2015
• Block walking
  ...asier means of this problem from A to B by labeling closer distances. Let's label these points with letters. label("O", (1,3), NW);
  9 KB (1,228 words) - 21:47, 26 February 2022
• Mock Amc 12
  ...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 ...r more information on Mock AMCs, as well as tips on how to write them, and a complete list of all Mock AMC 10s and Mock AMC 12s.
  18 KB (2,206 words) - 19:41, 24 December 2020
• 1984 AHSME Problems/Problem 18
  A point <math> (x, y) </math> is to be chosen in the [[coordinate plane]] so <math> \mathrm{(A) \ }\sqrt{2}-1 \qquad \mathrm{(B) \ }\frac{1}{2} \qquad \mathrm{(C) \ } 2-\
  3 KB (446 words) - 18:21, 4 June 2021
• 1967 AHSME Problems
  ...umber <math>5b9</math>. If <math>5b9</math> is divisible by 9, then <math>a+b</math> equals <math> \text{(A)}\ 2\qquad\text{(B)}\ 4\qquad\text{(C)}\ 6\qquad\text{(D)}\ 8\qquad\text{(E
  20 KB (3,108 words) - 14:14, 20 February 2020
• Simson line
  ...e]] ABC and a point P on its [[circumcircle]], the three closest points to P on lines AB, AC, and BC are [[collinear]]. Let a triangle <math>\triangle ABC</math> and a point <math>P</math> be given.
  4 KB (638 words) - 06:38, 26 April 2023
• 2003 AMC 8 Problems
  Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of <math>\mathrm{(A)}\ 12 \qquad\mathrm{(B)}\ 16 \qquad\mathrm{(C)}\ 20 \qquad\mathrm{(D)}\ 22
  16 KB (2,236 words) - 12:02, 19 February 2024
• 2009 AMC 8 Problems
  Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. The <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 11\qquad\
  18 KB (2,551 words) - 18:46, 27 February 2024
• 1996 AJHSME Problems/Problem 17
  Figure <math>OPQR</math> is a square. Point <math>O</math> is the origin, and point <math>Q</math> has coordinates (2,2). What pair O,P,Q,R,T;
  2 KB (257 words) - 11:20, 22 March 2015
• 1997 AHSME Problems
  If <math>\texttt{a}</math> and <math>\texttt{b}</math> are digits for which <math> \begin{array}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{array} </m
  17 KB (2,590 words) - 13:38, 19 February 2020
• 2010 AMC 8 Problems/Problem 23
  ...o of the combined areas of the two semicircles to the area of circle <math>O</math>? ...",(1.55,2.21),SE*lsf); label("$ R(-1,-1) $",(-2.72,-1.45),SE*lsf); label("$S(1,-1)$",(1.59,-1.49),SE*lsf);
  2 KB (298 words) - 00:29, 18 December 2023
• 1979 USAMO Problems/Problem 4
  ...<math>OA</math> and <math>R</math> on <math>OB</math> collinear with <math>P</math> so that <math>\frac{1}{PQ} + \frac{1}{PR}</math> is as large as poss ...</math> go to the circles <math>(O_1),(O_2)</math> passing through <math>P,O</math> and the line <math>QR</math> cuts <math>(O_1),(O_2)</math> again at
  3 KB (568 words) - 12:24, 11 March 2018
• 1996 AHSME Problems
  <math> \text{(A)}\ 4\qquad\text{(B)}\ 5\qquad\text{(C)}\ 6\qquad\text{(D)}\ 7\qquad\text{(E ...After <math>10</math> days of doing his chores daily, Walter has received a total of <math>36</math> dollars. On how many days did Walter do them excep
  15 KB (2,343 words) - 13:39, 19 February 2020
• 1996 AHSME Problems/Problem 25
  <math> \text{(A)}\ 72\qquad\text{(B)}\ 73\qquad\text{(C)}\ 74\qquad\text{(D)}\ 75\qquad\tex The first equation is a [[circle]], so we find its center and [[radius]] by [[completing the square
  9 KB (1,441 words) - 17:51, 22 October 2023
• 2003 AMC 8 Problems/Problem 18
  ...east one of her friends. How many classmates will not be invited to Sarah's party? pair a=(102,256), b=(68,131), c=(162,101), d=(134,150);
  2 KB (250 words) - 22:17, 5 January 2024
• 2009 PUMaC Problems
  '''Algebra A''' Given that <math>P(x)</math> is the least degree polynomial with rational coefficients such th
  25 KB (4,154 words) - 16:27, 2 September 2011
• 1985 AHSME Problems
  <math> \mathrm{(A)\ } 15 \qquad \mathrm{(B) \ }16 \qquad \mathrm{(C) \ } 17 \qquad \mathrm{( In an arcade game, the "monster" is the shaded sector of a circle of radius <math>1</math> cm, as shown in the figure. The missing pie
  17 KB (2,488 words) - 03:26, 20 March 2024
• 2010 AMC 8 Problems
  ...students in Mr. Newton's class, and <math>9</math> students in Mrs. Young's class taking the AMC 8 this year. How many mathematics students at Euclid M <math> \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qq
  18 KB (2,768 words) - 21:05, 9 January 2024
• 2011 AMC 8 Problems
  ...math> apples at a cost of <math> 50 </math> cents per apple. She paid with a 5-dollar bill. How much change did Margie receive? <math>\textbf{(A) }\ \textdollar 1.50 \qquad \textbf{(B) }\ \textdollar 2.00 \qquad \textbf{
  16 KB (2,371 words) - 17:34, 9 January 2024
• Barycentric coordinates
  ...Barycentric coordinates were discovered by Möbius in 1827 (Coxeter 1969, p. 217; Fauvel et al. 1993). The Central NC Math Group published a lecture concerning this topic at https://www.youtube.com/watch?v=KQim7-wrwL
  25 KB (5,106 words) - 11:00, 24 May 2024
• 2012 AMC 12A Problems
  {{AMC12 Problems|year=2012|ab=A}} A bug crawls along a number line, starting at <math>-2</math>. It crawls to <math>-6</math>, the
  14 KB (2,197 words) - 13:34, 12 August 2020
• 2012 AMC 12A Problems/Problem 16
  Circle <math>C_1</math> has its center <math>O</math> lying on circle <math>C_2</math>. The two circles meet at <math>X</ <math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf
  9 KB (1,496 words) - 02:40, 2 October 2022
• 2012 AMC 12A Problems/Problem 22
  ...igcup_{j=1}^{k}p_{j} </math>. The intersection of <math>P</math> and <math>S</math> consists of the union of all segments joining the midpoints of every <math> \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qqua
  5 KB (940 words) - 17:13, 4 April 2020
• 2015 AIME I Problems
  ...successive integers. Find the positive difference between integers <math>A</math> and <math>B</math>. ...uch that <math>16p+1</math> is the cube of a positive integer. Find <math>p</math>.
  10 KB (1,615 words) - 21:48, 13 January 2024
• 2012 AIME I Problems
  ...s with three not necessarily distinct digits, <math>abc</math>, with <math>a \neq 0</math> and <math>c \neq 0</math> such that both <math>abc</math> and ...h> miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are <math>n</math> miles from Dodge, and they have been trav
  10 KB (1,617 words) - 14:49, 2 June 2023
• 2012 AIME I Problems/Problem 7
  At each of the sixteen circles in the network below stands a student. A total of <math>3360</math> coins are distributed among the sixteen students pair A = (0,0);
  6 KB (1,058 words) - 01:49, 25 November 2023
• 2012 AIME I Problems/Problem 13
  ...and <math>d</math> is not divisible by the square of any prime. Find <math>a+b+c+d.</math> ...ath> A <math>60^\circ</math> counter-clockwise rotation about vertex <math>A</math> maps <math>X</math> to <math>X'</math> and <math>B</math> to <math>C
  11 KB (1,889 words) - 20:42, 25 January 2023
• Mock AIME II 2012 Problems
  ...ng a positive integer greater than <math>1</math> and <math>m</math> being a positive integer greater than 2, find the smallest possible value of <math> ...as the result obtained by repeatedly adding the digits of the number until a single digit remains. For example, the <math>\textit{digital root}</math> o
  7 KB (1,274 words) - 21:16, 8 March 2021
• Mock AIME I 2012 Problems
  ...and <math>q</math> and <math>r</math> are relatively prime. What is <math>p+q+r</math>? A permutation of the numbers <math>\{2</math>, <math>3</math>, <math>4</math>
  7 KB (1,309 words) - 11:13, 8 April 2012
• 2012 USAJMO Problems/Problem 1
  ...h>, <math>S</math> are concyclic (in other words, these four points lie on a circle). ...circle of triangle <math>PRS</math> is tangent to <math>AB</math> at <math>P</math>. Similarly, since <math>\angle CQR = \angle QSR</math>, the circumc
  4 KB (613 words) - 20:50, 19 December 2023
• 2015 IMO Problems/Problem 1
  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 (773 words) - 08:14, 19 July 2016
• 2002 Romanian NMO Problems
  Eight card players are seated around a table. One remarks that at some moment, any player and his two neighbours h Prove that any real number <math>0<x<1</math> can be written as a difference of two positive and less than <math>1</math> irrational numbers.
  10 KB (1,695 words) - 10:03, 10 May 2012
• 1974 AHSME Problems
  <math> \mathrm{(A)\ } 4x+3y=xy \qquad \mathrm{(B) \ }y=\frac{4x}{6-y} \qquad \mathrm{(C) \ } <math> \mathrm{(A)\ } -\frac{h}{3} \qquad \mathrm{(B) \ }\frac{h}{3} \qquad \mathrm{(C) \ }
  15 KB (2,151 words) - 14:04, 19 February 2020
• 1974 AHSME Problems/Problem 23
  ...gency. <math> PT''Q </math> is a third tangent with <math> T''' </math> as a point of tangency. If <math> TP=4 </math> and <math> T'Q=9 </math> then <ma pair O = (0,0); pair T = dir(90);
  2 KB (337 words) - 12:44, 5 July 2013
• 1974 AHSME Problems/Problem 25
  ...h> Q </math>. Lines <math> DP </math> and <math> CQ </math> meet at <math> O </math>. If the area of parallelogram <math> ABCD </math> is <math> k </mat label("A", (0,0), S);
  2 KB (232 words) - 12:44, 5 July 2013
• 1954 AHSME Problems
  <math>\textbf{(A)}\ y^2-5\sqrt{y^2-25} \qquad \textbf{(B)}\ -y^2 \qquad \textbf{(C)}\ y^2 \\ <math>\textbf{(A)}\ 4 \text{ and }1 \qquad \textbf{(B)}\ \text{only }1 \qquad \textbf{(C)}\
  23 KB (3,535 words) - 16:29, 24 April 2020
• 1955 AHSME Problems
  <math> \textbf{(A)}\ 3.75\times 10^{-7}\qquad\textbf{(B)}\ 3\frac{3}{4}\times 10^{-7}\qquad\t The smaller angle between the hands of a clock at <math>12:25</math> p.m. is:
  22 KB (3,509 words) - 21:29, 31 December 2023
• 1971 Canadian MO Problems/Problem 8
  ...h> is any point inside the pentagon. Perpendiculars are dropped from <math>P</math> to the sides, or the sides produced, of the pentagon. a) Prove that the sum of the lengths of these perpendiculars is constant.
  2 KB (363 words) - 22:38, 11 March 2016
• 1998 USAMO Problems/Problem 6
  ...er <math>k</math> (as a function of <math>n</math>) such that there exists a convex <math>n</math>-gon <math>A_{1}A_{2}\dots A_{n}</math> for which exac pair A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U;
  5 KB (871 words) - 18:59, 10 May 2023
• 1999 USAMO Problems
  (a) every square that does not contain a checker shares a side with one that does; ...ven squares, such that every two consecutive squares of the sequence share a side.
  3 KB (465 words) - 13:35, 4 July 2013
• 2008 AMC 8 Problems/Problem 25
  ...hown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Which of the following is closest to the per pair O=origin;
  1 KB (227 words) - 20:38, 2 January 2023
• 2008 AMC 8 Problems
  <math>\textbf{(A)}\ 12 \qquad <math>\textbf{(A)}\ 8671 \qquad
  14 KB (2,035 words) - 15:23, 26 January 2024
• 2012 AMC 8 Problems
  ...many pounds of meat does she need to make <math> 24 </math> hamburgers for a neighborhood picnic? <math> \textbf{(A)}\hspace{.05in}6\qquad\textbf{(B)}\hspace{.05in}6\frac{2}3\qquad\textbf{(C)
  13 KB (1,835 words) - 08:51, 8 March 2024
• 2013 AMC 12B Problems/Problem 24
  Let <math>ABC</math> be a triangle where <math>M</math> is the midpoint of <math>\overline{AC}</math> <math>\textbf{(A)}\ \frac{10-6\sqrt{2}}{7} \qquad \textbf{(B)}\ \frac{2}{9} \qquad \textbf{
  11 KB (1,876 words) - 00:08, 12 October 2023
• 1982 USAMO Problems/Problem 5
  ...</math>. Prove that the sum of their radii is equal to the radius of <math>S</math>. ...an be flattened to the plane <math>P'</math> through <math>A</math>, <math>O</math>, <math>O_1</math> and <math>O_2</math>.
  3 KB (524 words) - 16:08, 11 July 2018
• 2013 AIME I Problems
  ...les twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling? (a) the number <math>n</math> is divisible by <math>5,</math>
  9 KB (1,580 words) - 13:07, 24 February 2024
• 1991 AHSME Problems/Problem 10
  ...ius <math>15</math>. How many different chords of the circle contain <math>P</math> and have integer lengths? (A) 11 (B) 12 (C) 13 (D) 14 (E) 29
  2 KB (387 words) - 14:27, 23 June 2021
• 2013 AIME II Problems
  ...\text{B}</math>, and <math>\text{C}</math> are digits. Find <math>100\text{A}+10\text{B}+\text{C}</math>. Positive integers <math>a</math> and <math>b</math> satisfy the condition
  8 KB (1,402 words) - 12:17, 13 March 2020
• 2013 AIME II Problems/Problem 8
  ...where <math>p</math> and <math>q</math> are positive integers. Find <math>p+q</math>. real a;
  8 KB (1,357 words) - 09:23, 11 March 2024
• 2013 USAJMO Problems/Problem 5
  ...</math>. Point <math>Z</math> is the foot of the perpendicular from <math>P</math> to line <math>XY</math>. Point <math>C</math> lies on <math>\omega< ...ing the y's in the equation we find the intersection of these lines, <math>P</math>, is <math>\left(\frac{u-v}{u+v}\right), \frac{2uv}{u+v})</math>. Als
  7 KB (1,250 words) - 18:05, 1 October 2021
• Rotation
  A rotation of a planar figure is a transformation that preserves area and angles, but not orientation. The res ...le <math>ABC</math> <math>60^{\circ}</math> clockwise around a point <math>O</math>, also known as the '''center of rotation'''.
  3 KB (432 words) - 23:22, 13 January 2021
• CSS: Pseudo-class and Pseudo-element
  A pseudo-class and a pseudo-element are extensions to an element in a CSS document that configure how the document looks in specific situations. A pseudo-class is used to manipulate an element in a such way that cannot be done with simple selectors and/or access to the doc
  15 KB (2,197 words) - 22:49, 24 July 2023
• Mock AIME 6 2006-2007 Problems
  ...egers of the form <math>2^r\cdot3^s</math>, where <math>r</math> and <math>s</math> are nonnegative integers that do not exceed <math>4</math>. Find th Draw in the diagonals of a regular octagon. What is the sum of all distinct angle measures, in degree
  7 KB (1,173 words) - 21:04, 7 December 2018
• 2014 AIME II Problems/Problem 11
  ...me positive integers, and <math>b</math> is a positive integer. Find <math>a+b+c</math>. .... Let <math>O</math> be the foot of altitude <math>RO</math> and set <math>O</math> as the origin. Now we notice special right triangles! In particular,
  6 KB (1,059 words) - 18:24, 20 January 2024
• 2014 AIME II Problems/Problem 10
  ...ac{1}{z+w}=\frac{1}{z}+\frac{1}{w}</math>. Then the area enclosed by <math>P</math> can be written in the form <math>n\sqrt{3}</math>, where <math>n</ma ...{i \theta_w}</math> and likewise for <math>z</math>. Consider circle <math>O</math> with the origin as the center and radius 2014 on the complex plane.
  6 KB (1,045 words) - 13:08, 21 January 2024
• 2014 AIME II Problems/Problem 14
  ...h>N</math> is the midpoint of the segment <math>HM</math>, and point <math>P</math> is on ray <math>AD</math> such that <math>PN\perp{BC}</math>. Then < ...--C),N), H = MP("H",foot(A,B,C),N), N = MP("N",0.5(H+M),NE), P = MP("P",IP(A--D,L(N,N-(1,1),0,10)));
  11 KB (1,442 words) - 19:28, 21 October 2023
• ASIA TEAM Problems
  ...1</math> for the first time in <math>r</math> jumps. In particular, <math>o(1)=0</math> because Kelvin is already at lily pad <math>1</math>. ...first fly, he places it on <math>a_1</math>. After that, when he catches a fly, he places it on <math>a_i</math> for which <math>i</math> is the least
  10 KB (1,710 words) - 23:23, 10 January 2020
• 1966 AHSME Problems/Problem 38
  ...nd <math>AB</math>, respectively, intersect in point <math>O</math>. <math>P</math> is the midpoint of side <math>AC</math>, and <math>MP</math> interse <math>\text{(A) } 16n \quad \text{(B) } 18n \quad \text{(C) } 21n \quad \text{(D) } 24n \q
  2 KB (415 words) - 20:21, 3 April 2023
• 1951 AHSME Problems/Problem 46
  ...er a semicircle, the bisector of angle <math>OCD</math> cuts the circle in a point that always: <math> \textbf{(A)}\ \text{bisects the arc }AB\qquad\textbf{(B)}\ \text{trisects the arc }AB\
  2 KB (283 words) - 17:26, 1 January 2014
• 2013 AMC 8 Problems
  <math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \ ...lf-pound packages for just \$3 per package." What is the regular price for a full pound of fish, in dollars? (Assume that there are no deals for bulk)
  15 KB (2,162 words) - 20:05, 8 May 2023
• 1968 AHSME Problems
  ...ing from an increase in <math>\pi</math> units in the diameter. Then <math>P</math> equals: <math>\text{(A) } \frac{1}{\pi}\quad\text{(B) } \pi\quad\text{(C) } \frac{\pi^2}{2}\quad\t
  16 KB (2,571 words) - 14:13, 20 February 2020
• 1994 AHSME Problems/Problem 17
  An <math>8</math> by <math>2\sqrt{2}</math> rectangle has the same center as a circle of radius <math>2</math>. The area of the region common to both the <math> \textbf{(A)}\ 2\pi \qquad\textbf{(B)}\ 2\pi+2 \qquad\textbf{(C)}\ 4\pi-4 \qquad\textbf
  2 KB (287 words) - 03:07, 28 May 2021
• 1983 IMO Problems/Problem 2
  Let <math>A</math> be one of the two distinct points of intersection of two unequal cop ...with centers <math>O_1</math> and <math>O_2</math> respectively. Let <math>S</math> be such point on line <math>O_1O_2</math> so that tangents on <math>
  7 KB (1,267 words) - 23:35, 29 January 2021
• 2014 USAJMO Problems/Problem 2
  ...teral, acute triangle with <math>\angle A=60^{\circ}</math>, and let <math>O</math> and <math>H</math> denote the circumcenter and orthocenter of <math> (a) Prove that line <math>OH</math> intersects both segments <math>AB</math> a
  7 KB (1,197 words) - 19:40, 28 February 2023
• 1960 AHSME Problems
  If <math>2</math> is a solution (root) of <math>x^3+hx+10=0</math>, then <math>h</math> equals: <math>\textbf{(A)}10\qquad
  21 KB (3,242 words) - 21:27, 30 December 2020
• 2014 AMC 10B Problems/Problem 23
  A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is <math>\text{(A) } \dfrac32 \quad \text{(B) } \dfrac{1+\sqrt5}2 \quad \text{(C) } \sqrt3 \q
  6 KB (1,060 words) - 19:15, 11 August 2023
• 2014 AMC 12B Problems/Problem 19
  A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is real s=sqrt(r);
  4 KB (703 words) - 16:24, 9 September 2022
• 2014 AIME I Problems/Problem 13
  ...rline{EG}</math> and <math>\overline{FH}</math> intersect at a point <math>P</math>, and the areas of the quadrilaterals <math>AEPH, BFPE, CGPF,</math> pair A = (0,sqrt(850));
  9 KB (1,404 words) - 21:07, 13 October 2023
• 1993 AHSME Problems/Problem 25
  MP("P",(1,0),E); ...g the sides of a <math>120^{\circ}</math> angle, and let <math>P</math> be a fixed point inside the angle
  2 KB (375 words) - 19:13, 29 July 2019
• 1973 AHSME Problems
  A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length <math> \textbf{(A)}\ 3\sqrt3\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 6\sqrt3\qquad\textbf{(D
  18 KB (2,788 words) - 13:55, 20 February 2020
• 1969 AHSME Problems
  <math>\frac{a}{b},a \ne b,b \ne 0</math>, the value of the fraction is changed to <math>\frac{c <math>\text{(A) } \frac{1}{c-d}\quad
  16 KB (2,662 words) - 14:12, 20 February 2020
• 1970 AHSME Problems
  <math>\textbf{(A) }\sqrt{2}+\sqrt{3}\qquad A square and a circle have equal perimeters. The ratio of the area of the circle to the ar
  15 KB (2,366 words) - 07:52, 26 December 2023
• 1988 AHSME Problems
  <math>\text{(A)}\ \sqrt{20} \qquad Triangles <math>ABC</math> and <math>XYZ</math> are similar, with <math>A</math> corresponding to <math>X</math> and <math>B</math> to <math>Y</math>
  17 KB (2,535 words) - 13:45, 19 February 2020
• 1986 AHSME Problems
  <math>\textbf{(A)}\ 2y \qquad <math>\textbf{(A)}\ y = \frac{1}{3} x + 8 \qquad
  17 KB (2,512 words) - 18:30, 12 October 2023
• 1982 AHSME Problems
  <math>\textbf{(A)} \ 2 \qquad If a number eight times as large as <math>x</math> is increased by two, then one
  17 KB (2,500 words) - 19:05, 11 September 2023
• 1979 AHSME Problems
  label("$A$",(-2,1),W); label("$G$",(0,-1),S);
  17 KB (2,664 words) - 01:34, 19 March 2022
• 1977 AHSME Problems
  <math>\text{(A)}\ x \qquad <math>\textbf{(A)}\ \text{ equiangular}\qquad
  15 KB (2,412 words) - 05:09, 27 November 2020
• 1976 AHSME Problems
  <math>\textbf{(A) }-2\qquad For how many real numbers <math>x</math> is <math>\sqrt{-(x+1)^2}</math> a real number?
  17 KB (2,835 words) - 14:36, 8 September 2021
• 1975 AHSME Problems
  <math>\textbf{(A)}\ 3/4 \qquad <math>\textbf{(A)}\ \text{all }m\qquad
  16 KB (2,512 words) - 04:48, 27 November 2021
• 2007 iTest Problems/Problem 19
  ...rechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!" ...son a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.
  3 KB (397 words) - 01:18, 29 October 2023
• 2007 iTest Problems/Problem 18
  ...ic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers. If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?
  2 KB (393 words) - 17:01, 10 June 2018
• 2007 iTest Problems/Problem 21
  ...1's in a row and randomly writes + or - between each pair of consecutive 1's. <math>\text{(A) }0\qquad
  2 KB (247 words) - 03:34, 14 June 2018
• 2007 iTest Problems/Problem 22
  <math>\text{(A) } 0 \quad \text{(O) } 14 \quad
  929 bytes (137 words) - 22:05, 10 January 2019
• 2007 iTest Problems/Problem 25
  Ted's favorite number is equal to Find the remainder when Ted’s favorite number is divided by 25.
  2 KB (270 words) - 14:35, 29 July 2018
• 2007 iTest Problems/Problem 23
  <math>\text{(A) } 0\quad \text{(O) } 3-2i\quad</math>
  1 KB (202 words) - 16:48, 24 November 2018
• 2007 iTest Problems/Problem 24
  ...such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube. <math>\text{(A) }0 \quad
  2 KB (340 words) - 19:49, 30 June 2018
• 1971 AHSME Problems
  <math>\textbf{(A) }9\qquad <math>\textbf{(A) }fb^2\qquad
  18 KB (2,703 words) - 20:50, 11 September 2023
• 2007 IMO Problems/Problem 4
  ...at <math>R</math>, the perpendicular bisector of <math>BC</math> at <math>P</math>, and the perpendicular bisector of <math>AC</math> at <math>Q</math> real c=8.1,a=5*(c+sqrt(c^2-64))/6,b=5*(c-sqrt(c^2-64))/6;
  8 KB (1,480 words) - 14:52, 5 August 2022
• 2014 IMO Problems/Problem 3
  ...of the perpendicular from <math>A</math> to <math>BD</math>. Points <math>S</math> and <math>T</math> lie on sides <math>AB</math> and <math>AD</math>, pair A = (0,1), C=(0,-1);
  6 KB (1,071 words) - 03:58, 8 September 2018
• 1964 AHSME Problems
  <math>\textbf{(A)}\ \log_{10}50 \qquad <math>\textbf{(A)}\ \text{a parabola} \qquad
  19 KB (2,907 words) - 14:16, 20 February 2020
• 1963 AHSME Problems
  <math>\textbf{(A)}\ (0,0)\qquad <math>\textbf{(A)}\ -14 \qquad
  20 KB (3,122 words) - 14:17, 20 February 2020
• 1957 AHSME Problems
  ...lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is: <math>\textbf{(A)}\ 9\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 5\qqua
  26 KB (3,950 words) - 21:09, 31 August 2020
• 1956 AHSME Problems
  <math>\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 36 <math> \textbf{(A)}\ \text{broke even}\qquad
  20 KB (3,039 words) - 22:44, 12 February 2021
• 1986 AHSME Problems/Problem 28
  ...P, AQ</math> and <math>AR</math> are the perpendiculars dropped from <math>A</math> onto <math>CD, CB</math> extended and <math>DE</math> extended, respectively. Let <math>O</math> be the center of the pentagon. If <math>OP = 1</math>, then <math>AO
  4 KB (702 words) - 17:13, 17 April 2020
• 2015 AMC 10A Problems
  {{AMC10 Problems|year=2015|ab=A}} <math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf
  12 KB (1,887 words) - 18:52, 14 May 2024
• 2015 AIME II Problems
  ...ect to take Latin. The probability that a randomly chosen Latin student is a sophomore is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</ma ...where <math>p</math> and <math>q</math> are positive integers. Find <math>p + q</math>.
  8 KB (1,326 words) - 19:15, 13 January 2024
• 2015 AIME II Problems/Problem 11
  ...e{OB}</math> intersects lines <math>AB</math> and <math>BC</math> at <math>P</math> and <math>Q</math>, respectively. Also <math>AB=5</math>, <math>BC=4 pair A,B,C,O, Q, P, M, N;
  9 KB (1,249 words) - 21:59, 26 November 2023
• 2015 AIME II Problems/Problem 15
  ...A line <math>\ell</math> through <math>A</math> intersects <math>\mathcal{P}</math> again at <math>D</math> and intersects <math>\mathcal{Q}</math> aga D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5));
  31 KB (5,086 words) - 19:15, 20 December 2023
• 2015 USAMO Problems/Problem 2
  ...n segment <math>\overline{PQ}</math>, show that <math>M</math> moves along a circle. <cmath>A=(1,0) P=(1-a,b), Q=(1-a,-b)</cmath>,
  5 KB (902 words) - 09:58, 20 August 2021
• 2015 USAJMO Problems/Problem 3
  ...n segment <math>\overline{PQ}</math>, show that <math>M</math> moves along a circle. pair A=(1,0);
  4 KB (760 words) - 16:45, 29 April 2020
• Aops font
  The AoPS font is a font that can be used to make certain special symbols in the forum. You enc | <span class="aops-font">a</span>
  6 KB (924 words) - 21:31, 22 February 2024
• 1982 AHSME Problems/Problem 14
  ...us <math>15</math> and the line <math>AG</math> is tangent to circle <math>P</math> at <math>G</math>. If <math>AG</math> intersects circle <math>N</mat ..."$D$", D, dir(0)); label("$P$", P, S); label("$N$", N, S); label("$O$", O, S); label("$E$", E, dir(120)); label("$F$", F, NE); label("$G$", G, dir(100))
  1 KB (248 words) - 22:35, 16 August 2022
• 2017 AIME I Problems
  ...points are designated on <math>\triangle ABC</math>: the 3 vertices <math>A</math>, <math>B</math>, and <math>C</math>; <math>3</math> other points on ...der is always the positive integer <math>s \neq r</math>. Find <math>m+n+r+s</math>.
  7 KB (1,163 words) - 16:43, 2 June 2022
• 2015 AMC 8 Problems
  ...feet long and <math>9</math> feet wide? (There are <math>3</math> feet in a yard.) <math>\textbf{(A) }12\qquad\textbf{(B) }36\qquad\textbf{(C) }108\qquad\textbf{(D) }324\qquad
  16 KB (2,322 words) - 14:04, 2 February 2024
• 2019 AMC 12A Problems
  {{AMC12 Problems|year=2019|ab=A}} ...us <math>4</math> inches is <math>N</math> percent larger than the area of a pizza with radius <math>3</math> inches. What is the integer closest to <ma
  14 KB (2,180 words) - 22:25, 25 April 2024
• 2016 AMC 10B Problems
  ...is the value of <math>\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}</math> when <math>a= \tfrac{1}{2}</math>? <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\
  12 KB (1,868 words) - 17:50, 30 December 2023
• 2016 AIME I Problems/Problem 6
  real c=8.1,a=5*(c+sqrt(c^2-64))/6,b=5*(c-sqrt(c^2-64))/6; pair A=(0,0),B=(c,0),D=(c/2,-sqrt(25-(c/2)^2));
  14 KB (2,397 words) - 20:04, 27 August 2023
• 2016 AIME II Problems/Problem 14
  ...). There is a point <math>O</math> whose distance from each of <math>A,B,C,P,</math> and <math>Q</math> is <math>d</math>. Find <math>d</math>. ...ngle ABC</math>, and <math>L</math> be the foot of the altitude from <math>A</math> to <math>BC</math>. We must have <math>\tan(\angle KLP+ \angle QLK)=
  15 KB (2,560 words) - 01:44, 1 July 2023
• 2016 AIME II Problems
  ...ex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats <math>5</math> of his peanuts, Betty eats ...ath> and <math>b</math> are relatively prime positive integers. Find <math>a+b</math>.
  8 KB (1,312 words) - 21:16, 3 March 2021
• 2006 iTest Problems
  <math>\mathrm{(A)}\, 8</math> <math>\mathrm{(A)}\, 15\quad\mathrm{(B)}\, \frac{40}{3}</math>
  31 KB (4,811 words) - 00:02, 4 November 2023
• 1954 AHSME Problems/Problem 37
  ...ath>, <math>PQ</math> extended to <math>D</math> and <math>\angle n</math> a right angle, then: path anglemark2(pair A, pair B, pair C, real t=8, bool flip=false)
  3 KB (560 words) - 10:11, 3 May 2020
• 2016 USAMO Problems
  ...ldots, 99\}</math>. Find the smallest possible number of elements in <math>S</math>. ...ABC</math> be an acute triangle, and let <math>I_B, I_C,</math> and <math>O</math> denote its <math>B</math>-excenter, <math>C</math>-excenter, and cir
  4 KB (608 words) - 13:49, 22 November 2023
• 2016 USAJMO Problems
  ...on the arc <math>\stackrel{\frown}{BC}</math> that does not contain <math>A</math>, and let <math>I_B</math> and <math>I_C</math> denote the incenters ...the circumcircle of triangle <math>\triangle PI_BI_C</math> passes through a fixed point.
  3 KB (414 words) - 16:43, 5 August 2023
• 2016 USAJMO Problems/Problem 5
  ...th>A</math> to line <math>\overleftrightarrow{BC}</math>, and points <math>P</math> and <math>Q</math> are the feet of the perpendiculars from <math>H</ Given that <cmath>AH^2=2\cdot AO^2,</cmath>prove that the points <math>O,P,</math> and <math>Q</math> are collinear.
  10 KB (1,733 words) - 19:15, 14 June 2020
• 2016 USAMO Problems/Problem 3
  ...ABC</math> be an acute triangle, and let <math>I_B, I_C,</math> and <math>O</math> denote its <math>B</math>-excenter, <math>C</math>-excenter, and cir Lines <math>I_B F</math> and <math>I_C E</math> meet at <math>P.</math> Prove that <math>\overline{PO}</math> and <math>\overline{YZ}</math
  6 KB (998 words) - 21:36, 17 October 2022
• 2016 USAMO Problems/Problem 5
  ...Q\in\overline{AC},</math> and <math>N, P\in\overline{BC}.</math> Let <math>S</math> be the intersection of lines <math>MN</math> and <math>PQ.</math> De ...t <math>\overline{OI}</math> is parallel to <math>\ell,</math> where <math>O</math> is the circumcenter of triangle <math>ABC,</math> and <math>I</math>
  8 KB (1,449 words) - 00:09, 12 October 2023
• 1976 AHSME Problems/Problem 18
  O=(0,0), c1=circle(O, r1);
  2 KB (361 words) - 08:05, 9 April 2023
• Mock AIME 4 Pre 2005/Problems
  ...> 3</math> and <math>a_{2004} = 7.</math> <math>b_1, b_2, \ldots</math> is a sequence of real numbers in which <math>b_n</math> is the geometric mean of ...nt in its interior such that <math>ABP</math> is equilateral. There exists a unique pair <math>\{m, n\}</math> of relatively prime positive integers suc
  7 KB (1,094 words) - 15:39, 24 March 2019
• Mock AIME 3 2010 Problems
  ...can distribute 1 piece of candy among 243 hungry schoolchildren sitting in a row -3. We have a triangle abc with sides ab, bc, ca with lengths α, β, and γ, respectivel
  14 KB (2,904 words) - 18:24, 16 May 2017
• 2017 AMC 12A Problems/Problem 24
  Quadrilateral <math>ABCD</math> is inscribed in circle <math>O</math> and has side lengths <math>AB=3, BC=2, CD=6</math>, and <math>DA=8</ ...math>\overline{AC}</math>. Let <math>G</math> be the point on circle <math>O</math> other than <math>C</math> that lies on line <math>CX</math>. What is
  8 KB (1,322 words) - 03:24, 7 October 2023
• 2017 AMC 12A Problems/Problem 16
  ...the largest semicircle. What is the radius of the circle centered at <math>P</math>? pair P = (-1,0)+(2+6/7)*dir(36.86989);
  13 KB (1,982 words) - 17:12, 20 December 2022
• 2017 AMC 12A Problems
  {{AMC12 Problems|year=2017|ab=A}} <math>\textbf{(A)}\ 8\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qqua
  15 KB (2,418 words) - 16:58, 7 November 2022
• 2017 AIME I Problems/Problem 4
  A pyramid has a triangular base with side lengths <math>20</math>, <math>20</math>, and <ma ...<math>O</math>. Note that <math>O</math> is equidistant from each of <math>A</math>, <math>B</math>, and <math>C</math>. Then,
  7 KB (1,177 words) - 15:55, 5 January 2024
• 2017 AIME I Problems/Problem 8
  ...measures of <math>\angle POQ</math> and <math>\angle POR</math> are <math>a</math> and <math>b</math> respectively, and <math>\angle OQP</math> and <ma ...and <math>\angle ORP</math> are right angles, we realize that we can draw a semicircle with diameter <math>\overline{OP}</math> and points <math>Q</mat
  9 KB (1,539 words) - 15:47, 17 February 2024
• 2017 AIME I Problems/Problem 15
  ...<math>p</math> is not divisible by the square of any prime. Find <math>m+n+p.</math> pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0);
  22 KB (3,622 words) - 17:11, 6 January 2024
• 2017 AIME II Problems/Problem 10
  ...of <math>\overline{CM}</math> and <math>\overline{DN}</math>. Point <math>P</math> lies on the quadrilateral <math>BCON</math>, and <math>\overline{BP} pair A,B,C,D,M,n,O,P;
  4 KB (718 words) - 00:21, 27 January 2023
• 2017 AIME II Problems/Problem 3
  ...ath> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. pair A,B,C,D,X,Z,P;
  9 KB (1,411 words) - 22:18, 29 January 2024
• Circular Inversion
  ...e transformed point on the ray extending from <math>O</math> through <math>A</math>. pair O = (0, 0);
  16 KB (2,516 words) - 23:48, 15 January 2024
• 2017 USAJMO Problems/Problem 3
  ...Let <math>ABC</math> be an equilateral triangle and let <math>P</math> be a point on its circumcircle. Let lines <math>PA</math> and <math>BC</math> in ..., P = (0, -sqrt(3)), D = (0, 0), E1 = (6, -3sqrt(3)), F = (-6, -3sqrt(3)), O = (0, sqrt(3));
  10 KB (1,829 words) - 03:13, 16 March 2022
• 2017 USAJMO Problems/Problem 5
  Let <math>O</math> and <math>H</math> be the circumcenter and the orthocenter of an acu ...N'</math> is the midpoint of minor arc <math>\overarc{BC}</math>, so <math>A, D, N'</math> are collinear in that order. It suffices to show that <math>\
  5 KB (805 words) - 18:31, 21 September 2021
• 2017 UNCO Math Contest II Problems
  A prime is an integer strictly greater than one that is evenly divisible by n pair A=24*dir(40),B=15*dir(220);
  7 KB (1,192 words) - 15:14, 20 August 2020
• 1957 AHSME Problems/Problem 18
  ...other. <math>AM</math> is any chord intersecting <math>CD</math> at <math>P</math>. pair O = origin;
  1 KB (210 words) - 18:43, 20 October 2018
• 2017 JBMO Problems
  ...h> \Gamma</math> at point <math>X\neq E</math>. Prove that the point <math>A,M</math> and <math>X</math> are collinear . ...>P</math> can be seen .Find the number of distinct such colorings of <math>P</math> (two colorings are considered distinct if at least one of sides is c
  2 KB (437 words) - 16:32, 16 September 2017
• MIE 2015
  (a) <math>(G\cup F)-(F-H)</math> (a) <math>-2</math>
  7 KB (1,127 words) - 18:23, 11 January 2018
• 2018 AMC 12B Problems
  Kate bakes a <math>20</math>-inch by <math>18</math>-inch pan of cornbread. The cornbrea <math>\textbf{(A) } 90 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 180 \qquad \textbf{(D)
  14 KB (2,118 words) - 15:36, 28 October 2021
• 2018 AMC 12B Problems/Problem 21
  ...th> and <math>I</math> denote the circumcenter and incenter, respectively. A circle with center <math>M</math> is tangent to the legs <math>AC</math> an <math>\textbf{(A)}\ \frac52\qquad\textbf{(B)}\ \frac{11}{4}\qquad\textbf{(C)}\ 3\qquad\textb
  3 KB (540 words) - 04:12, 30 August 2022
• 2018 AMC 10B Problems/Problem 24
  Let <math>ABCDEF</math> be a regular hexagon with side length <math>1</math>. Denote by <math>X</math>, <math>\textbf{(A)}\ \frac {3}{8}\sqrt{3} \qquad \textbf{(B)}\ \frac {7}{16}\sqrt{3} \qquad \
  17 KB (2,562 words) - 03:38, 6 September 2022
• 2016 JBMO Problems/Problem 3
  Find all triplets of integers <math>(a,b,c)</math> such that the number <cmath>N = \frac{(a-b)(b-c)(c-a)}{2} + 2</cmath>
  4 KB (639 words) - 15:14, 28 November 2023
• 2018 AIME I Problems/Problem 10
  ...mber of paths with <math>15</math> steps that begin and end at point <math>A</math>. Find the remainder when <math>n</math> is divided by <math>1000.</m real s = 4;
  11 KB (1,934 words) - 12:18, 29 March 2024
• 2018 AIME II Problems/Problem 14
  ...ction of <math>\overline{AX}</math> with <math>\omega</math>. Points <math>P</math> and <math>Q</math> lie on <math>\overline{AB}</math> and <math>\over ...air I=incenter(A,B,C); P=extension(A,B,Y,Y+dir(90)*(Y-I)); Q=extension(A,C,P,Y);
  9 KB (1,518 words) - 14:32, 28 January 2024
• 2018 USAJMO Problems/Problem 3
  ...ver lines <math>BA</math> and <math>BC</math>, respectively, and let <math>P</math> be the intersection of lines <math>BD</math> and <math>EF</math>. Su label("$O$", (0.03388760411534265,0.08889671794036069), NE * 0.5);
  8 KB (920 words) - 20:30, 4 November 2020
• 1977 AHSME Problems/Problem 19
  ...vex quadrilateral <math>ABCD</math>, and let <math>P,Q,R</math>, and <math>S</math> be the centers of the circles <math>\textbf{(A) }PQRS\text{ is a parallelogram}\\
  2 KB (305 words) - 08:38, 28 June 2018
• 2003 Indonesia MO Problems
  ...ath>a^9 - a</math> is divisible by <math>6</math> for every integers <math>a</math>. ...>, respectively. <math>PR</math> and <math>QS</math> intersects at <math>O</math>. Prove that <math>PO = OR</math> and <math>QO = OS</math>.
  3 KB (497 words) - 02:20, 19 May 2024
• 2003 Indonesia MO Problems/Problem 2
  ...>, respectively. <math>PR</math> and <math>QS</math> intersects at <math>O</math>. Prove that <math>PO = OR</math> and <math>QO = OS</math>. pair A=(0,0), B=(60,0), C=(40,80), D=(10,50), P=(30,0), Q=(50,40), R=(25,65), SX=(5,25);
  3 KB (583 words) - 17:52, 22 January 2024
• 1973 AHSME Problems/Problem 35
  ...hords <math>MP</math>, <math>PQ</math>, and <math>NR</math> are each <math>s</math> units long and chord <math>MN</math> is <math>d</math> units long. label("1", (5,0), S);
  7 KB (1,181 words) - 20:41, 17 April 2022
• 2006 iTest Problems/Problem 18
  <math>\text{(A) Ptolemy; Archimedes}\qquad \text{(O) Goldbach; Cauchy}\qquad</math>
  2 KB (252 words) - 12:04, 27 November 2018
• 2006 iTest Problems/Problem 8
  ...th center <math>O</math>. Perpendicular lines are drawn from <math>P</math> to perpendicular diameters, <math>AB</math> and <math>CD</math> <math>\mathrm{(A)}\,4\quad\mathrm{(B)}\,8\quad\mathrm{(C)}\,6\sqrt{3}\quad\mathrm{(D)}\,4\sq
  2 KB (307 words) - 20:50, 29 November 2018
• 2019 AMC 10A Problems
  {{AMC10 Problems|year=2019|ab=A}} <math>\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qqu
  13 KB (2,024 words) - 16:07, 22 April 2024
• 2019 AIME II Problems/Problem 15
  In acute triangle <math>ABC</math> points <math>P</math> and <math>Q</math> are the feet of the perpendiculars from <math>C</ pen s = linewidth(0.8)+fontsize(8);
  7 KB (1,115 words) - 03:11, 7 January 2024
• 2019 AMC 12A Problems/Problem 22
  ...with <math>\text{gcd}(m,n) = \text{gcd}(p,q) = 1</math>. What is <math>m+n+p+q</math>? <math>\textbf{(A) } 42 \qquad \textbf{(B) }86 \qquad \textbf{(C) } 92 \qquad \textbf{(D) } 1
  3 KB (536 words) - 17:03, 30 July 2022
• 2019 AMC 10A Problems/Problem 20
  ...>1,2,\dots,9</math> are randomly placed into the <math>9</math> squares of a <math>3 \times 3</math> grid. Each square gets one number, and each of the <math>\textbf{(A) }\frac{1}{21}\qquad\textbf{(B) }\frac{1}{14}\qquad\textbf{(C) }\frac{5}{63
  8 KB (1,347 words) - 10:39, 18 April 2024
• 2019 AIME I Problems/Problem 6
  ...verline{KN}</math> intersects diagonal <math>\overline{KM}</math> at <math>O</math> with <math>KO = 8</math>. Find <math>MO</math>. ...math>\angle MKN=\alpha</math> and <math>\angle LNK=\beta</math>. Let <math>P</math> be the project of <math>L</math> onto line <math>NK</math>. Note <ma
  11 KB (1,717 words) - 20:11, 19 January 2024
• 2019 AIME II Problems/Problem 1
  pair A = (0,0); draw(A--B--C--cycle);
  11 KB (1,794 words) - 15:32, 14 January 2024
• 2019 AIME I Problems/Problem 15
  ...<math>\omega_1</math> and <math>\omega_2</math> intersect at points <math>P</math> and <math>Q</math>. Line <math>PQ</math> intersects <math>\omega</ma pair O, A, B, P, O1, O2, Q, X, Y;
  13 KB (2,252 words) - 11:32, 1 February 2024
• Mock AIME 2 Pre 2005 Problems
  <math>x</math> is a real number with the property that <math>x+\tfrac1x = 3</math>. Let <math>S ...ll, 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. T
  6 KB (1,052 words) - 13:52, 9 June 2020
• What is the definition of Pure Mathematics?
  ...is welcomed to edit, BUT ONLY IN GOOD WAYS! The AoPS Secret Governemtn has a backup of this page. ...ng it. For example, can a butterfly that flaps its wings in Brazil trigger a tornado in Texas?
  35 KB (5,882 words) - 18:08, 28 June 2021
• 2006 iTest Problems/Problem 20
  <math>\textbf{(A) }2\qquad \textbf{(O) }23\qquad
  4 KB (334 words) - 12:49, 15 January 2020
• 2006 iTest Problems/Problem 19
  <math>\textbf{(A) }7\qquad \textbf{(O) }21\qquad
  4 KB (332 words) - 00:10, 27 January 2023
• 2020 AIME I Problems/Problem 6
  A flat board has a circular hole with radius <math>1</math> and a circular hole with radius <math>2</math> such that the distance between the pair A, B, C, D, O, P, H, L, X, Y;
  5 KB (819 words) - 07:54, 15 December 2023
• 2020 AIME I Problems/Problem 8
  ...walked the previous day. The bug gets arbitrarily close to the point <math>P</math>. Then <math>OP^2=\tfrac{m}{n}</math>, where <math>m</math> and <math pair O, A, B, C, D, F, G, H, I, P, X;
  9 KB (1,380 words) - 16:12, 2 January 2024
• 2020 AIME I Problems/Problem 13
  ...positive integer not divisible by the square of any prime. Find <math>m+n+p.</math> label("$A$", (-6.8002301023571095,8.267690318323321), NE * labelscalefactor);
  35 KB (5,215 words) - 23:08, 29 October 2023
• 2020 AIME I Problems/Problem 15
  The following is a power of a point solution to this menace of a problem: pen p=fontsize(10)+gray+0.4;
  16 KB (2,678 words) - 22:45, 27 November 2023
• 2002 Pan African MO Problems/Problem 5
  ...e circle through <math> E </math> and <math> F </math> intersect at <math> P </math>. Show that <math> P </math> lies on the altitude through the vertex <math> C </math>.
  7 KB (1,158 words) - 13:47, 27 May 2024
• Mock AMC 10B Problems
  <math>\textbf{(A)}\ 10\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 25\qqu ...x Boy Scouts that will be split up into two groups of three Boy Scouts for a boating trip. How many ways are there to split up the six boys if the two g
  13 KB (2,059 words) - 02:59, 21 January 2021
• 2002 Pan African MO Problems/Problem 2
  ...ively such that <math>AC=BD</math>. Show that there is a fixed point <math>P</math> through which the perpendicular bisector of <math>CD</math> always p ...tion Postulate, <math>OC = OD</math>. This, <math>\triangle OCD</math> is a 45-45-90 triangle.
  4 KB (710 words) - 14:01, 23 December 2019
• 2016 IMO Problems
  ...>CF</math>. Let <math>X</math> be the point such that <math>AMXE</math> is a parallelogram. Prove that <math>BD,FX</math> and <math>ME</math> are concur ...filled with one of the letters <math>I,M</math> and <math>O</math> in such a way that:
  4 KB (755 words) - 09:48, 20 December 2023
• 2020 AMC 10A Problems/Problem 18
  ...ath>\{0,1,2,3\}.</math> For how many such quadruples is it true that <math>a\cdot d-b\cdot c</math> is odd? (For example, <math>(0,3,1,1)</math> is one <math>\textbf{(A) } 48 \qquad \textbf{(B) } 64 \qquad \textbf{(C) } 96 \qquad \textbf{(D) }
  6 KB (1,044 words) - 13:50, 4 April 2024
• 2020 AMC 12A Problems/Problem 14
  <math>\textbf{(A) } \frac{\sqrt{2}}{4} \qquad \textbf{(B) } \frac{\sqrt{2}}{2} \qquad \textb <math>ACEG</math> is a square. WLOG <math>AB = 1,</math> then using Law of Cosines, <math>AC^2 = [
  11 KB (1,654 words) - 02:01, 17 September 2023
• 2020 AMC 12B Problems/Problem 10
  ...<math>\overline{AM}</math> intersects <math>\omega</math> at a point <math>P</math> different from <math>M.</math> What is <math>AP?</math> <math>\textbf{(A) } \frac{\sqrt5}{12} \qquad \textbf{(B) } \frac{\sqrt5}{10} \qquad \textbf{
  9 KB (1,328 words) - 16:14, 11 September 2023
• 2020 AMC 12B Problems/Problem 12
  ...th> be a chord in the circle that intersects <math>\overline{AB}</math> at a point <math>E</math> such that <math>BE=2\sqrt5</math> and <math>\angle AEC <math>\textbf{(A)}\ 96 \qquad\textbf{(B)}\ 98 \qquad\textbf{(C)}\ 44\sqrt5 \qquad\textbf{(D
  10 KB (1,599 words) - 10:05, 6 October 2023
• 2021 AMC 10A Problems
  {{AMC10 Problems|year=2021|ab=A}} What is the value of<cmath>(2^2-2)-(3^2-3)+(4^2-4)?</cmath><math>\textbf{(A)} ~1 \qquad\textbf{(B)} ~2 \qquad\textbf{(C)} ~5 \qquad\textbf{(D)} ~8 \qqu
  15 KB (2,302 words) - 23:41, 14 April 2024
• 2020 AIME I Problems
  ...\overline{AC},</math> and point <math>E</math> lies strictly between <math>A</math> and <math>B</math> on side <math>\overline{AB}</math> such that <mat ...h>\log_8(2x),\log_4x,</math> and <math>\log_2x,</math> in that order, form a geometric progression with positive common ratio. The number <math>x</math>
  7 KB (1,257 words) - 17:51, 5 February 2022
• 2024 AIME II Problems
  ...n a garden spade. In addition, each of the <math>900</math> residents owns a bag of candy hearts. There are <math>437</math> residents who own exactly t A list of positive integers has the following properties:
  8 KB (1,236 words) - 23:11, 12 March 2024
• 2007 Indonesia MO Problems/Problem 7
  ...<math> S</math> at <math> C</math> and <math> D</math> intersect at <math> P</math>. Points <math> Q</math> and <math> R</math> are the intersections of (a) Prove that <math> P</math>, <math> Q</math>, and <math> R</math> are collinear.
  4 KB (729 words) - 15:08, 26 March 2020
• 2022 AMC 10A Problems
  {{AMC10 Problems|year=2022|ab=A}} <math>\textbf{(A)}\ \frac{31}{10}\qquad\textbf{(B)}\ \frac{49}{15}\qquad\textbf{(C)}\ \frac{
  18 KB (2,662 words) - 02:08, 9 March 2024
• AMC 12C 2020 Problems
  A plane flies at a speed of <math>590</math> miles/hour. How many miles in two hours ...<math>4</math> red marbles. If each marble is pulled out <math>1</math> at a time, what is the probability that the <math>6th</math> marble pulled out r
  8 KB (1,223 words) - 15:02, 27 November 2022
• AoPS Schoolhouse
  ...by students. For example, in [[Mathcounts Week/Countdown Round|The World’s Largest Countdown Round]], there were 17 classroom assistants. ...f a moderator feels a question requires further discussion, they will open a mini window with you to more deeply discuss the question.
  11 KB (1,269 words) - 19:28, 28 March 2024
• 2020 AIME II Problems/Problem 15
  Let <math>O</math> be the circumcenter of <math>\triangle ABC</math>; say <math>OT</mat ...cyclic, so <math>\angle XTY=180^{\circ}-A</math>, so <math>\cos XTY=-\cos A</math>, and the cosine law in <math>\triangle TXY</math> gives <cmath>1143-
  7 KB (1,221 words) - 16:46, 29 January 2023
• 2020 FMC 10A Problems
  ...ain stops halfway between the bank and the store to pick up some food from a local restaurant. Then, he decides he instead wants to go to the park and h <math>\textbf{(A)}\ \dfrac{9}{16}\qquad\textbf{(B)}\ \dfrac{3}{16}\qquad\textbf{(C)}\ \dfrac
  14 KB (2,267 words) - 12:49, 9 June 2020
• 2020 USAMO Problems
  ...ngle inscribed in a circle <math>\omega</math> with center <math>O</math>. A variable point <math>X</math> is chosen on minor arc <math>AB</math> of <ma ...th> grid of square unit cells is drawn on each of its six faces. A beam is a <math>1 \times 1 \times 2020</math> rectangular prism. Several beams are pl
  4 KB (614 words) - 13:47, 22 November 2023
• 2020 USOJMO Problems/Problem 5
  ..._j</math>. But all pairs are distinct and <math>a_i=a_j</math>, so this is a contradiction. ...math>. Similarly <math>b_j=1</math>, but all pairs are distinct so we have a contradiction.
  8 KB (1,486 words) - 19:16, 6 October 2023
• 1989 IMO Problems/Problem 3
  .../math> and <math> k</math> be positive integers and let <math> S</math> be a set of <math> n</math> points in the plane such that i.) no three points of <math> S</math> are collinear, and
  3 KB (580 words) - 11:56, 30 January 2021
• 2021 AIME I Problems/Problem 6
  ...rline{AG}</math> is a diagonal through the center of the cube. Point <math>P</math> satisfies <math>BP=60\sqrt{10}</math>, <math>CP=60\sqrt{5}</math>, < ...>, point <math>A</math> have coordinates <math>(0, 0, 0)</math>, and <math>s</math> be the side length. Then we have the equations
  4 KB (575 words) - 00:50, 12 January 2024
• 2021 AIME I Problems/Problem 13
  ...se line <math>AB</math> intersects <math>\omega</math> at two points <math>P</math> and <math>Q</math> such that the measure of minor arc <math>\widehat <font color=red><b>Claim.</b></font color> <math>DPEQ</math> is a harmonic quadrilateral.
  17 KB (2,852 words) - 03:59, 7 February 2024
• 2021 AMC 12A Problems
  {{AMC12 Problems|year=2021|ab=A}} <math>\textbf{(A) }0 \qquad \textbf{(B) }50 \qquad \textbf{(C) }52 \qquad \textbf{(D) }54 \q
  15 KB (2,383 words) - 09:49, 25 June 2023
• 2022 AIME I Problems/Problem 11
  ...{BC}</math> intersects diagonal <math>\overline{AC}</math> at points <math>P</math> and <math>Q</math> with <math>AP < AQ,</math> as shown. Suppose that pair A,B,C,D,P,Q;
  16 KB (2,517 words) - 20:22, 31 January 2024
• 2022 AIME I Problems/Problem 14
  ...h> into two polygons of equal perimeter. Let <math>\triangle ABC</math> be a triangle where <math>BC = 219</math> and <math>AB</math> and <math>AC</math ...mple angle chasing reveals the condition is now equivalent to <math>\angle A=120^\circ</math>.
  16 KB (2,730 words) - 02:56, 4 January 2023
• 2022 AIME I Problems/Problem 15
  ...th>\sqrt{2z}</math>, and <math>l</math>, with altitude from <math>l</math>'s equal to <math>\sqrt{xz}</math>. <math>l = \sqrt{2x - xz} + \sqrt{2z - xz}< The circumcircle's central angle to a side is <math>2 \arcsin(l/2)</math>, so the 3 triangles' <math>l=1, \sqrt{2
  15 KB (2,208 words) - 01:25, 1 February 2024
• 2022 AIME II Problems
  ...made up <math>\frac5{12}</math> of the crowd of people at a concert. After a bus carrying <math>50</math> more people arrived, adults made up <math>\fra ...ath> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
  8 KB (1,397 words) - 09:18, 15 August 2022
• 2020 CIME I Problems
  A knight begins on the point <math>(0,0)</math> in the coordinate plane. From ...es tax of <math>7.5\%</math> is applied on all items. If the total cost of a purchase, after tax, is an integer number of cents, find the minimum possib
  7 KB (1,188 words) - 18:00, 31 August 2020
• 2021 Fall AMC 10B Problems
  <math>(\textbf{A})\: 10{,}000\qquad(\textbf{B}) \: 10{,}010\qquad(\textbf{C}) \: 10{,}110\qq pen s = linewidth(0.8)+fontsize(8);
  16 KB (2,450 words) - 00:13, 12 November 2023
• 2021 Fall AMC 10B Problems/Problem 25
  ...length <math>1,</math> and a rectangle <math>R</math> are inscribed inside a larger square as shown. The sum of all possible values for the area of <mat label("$3$",(11/2,9/2),S);
  16 KB (2,274 words) - 09:02, 10 December 2022
• 2021 AIME II Problems
  ...etic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as <math>777</math> o ...oint <math>D</math> lies on the same side of line <math>BC</math> as <math>A</math> such that <math>\overline{BD} \perp \overline{BC}</math>. The line <
  8 KB (1,429 words) - 14:31, 26 February 2024
• 2021 Fall AMC 10A Problems/Problem 17
  ...l that will not be parallel to the ground. The heights of pillars at <math>A</math>, <math>B</math>, and <math>C</math> are <math>12</math>, <math>9</ma <math>\textbf{(A) }9 \qquad\textbf{(B) } 6\sqrt{3} \qquad\textbf{(C) } 8\sqrt{3} \qquad\text
  10 KB (1,705 words) - 00:13, 17 July 2023
• 1978 IMO Problems/Problem 4
  ...angle is also tangent to the sides <math>AB, AC</math> in the points <math>P,</math> respectively <math>Q.</math> Prove that the midpoint of <math>PQ</m ...{s - a}{b} \cdot \frac{2R}{h}</math> Denote <math>h' = AL</math> the <math>A</math>-altitude of the triangle <math>\triangle APQ</math>. Then
  3 KB (541 words) - 17:04, 29 January 2021
• 2020 IMO Problems/Problem 6
  Prove that there exists a positive constant <math>c</math> such that the following statement is true: ...> separating <math>S</math> such that the distance from any point of <math>S</math> to <math>\ell</math> is at least <math>cn^{- \frac{1}{3}}</math>.
  3 KB (574 words) - 11:30, 14 May 2021
• 2021 AMC 12A Problems/Problem 17
  ...s <math>\overline{AC}</math> and <math>\overline{BD}</math>, and let <math>P</math> be the midpoint of <math>\overline{BD}</math>. Given that <math>OP=1 <math>\textbf{(A) }65 \qquad \textbf{(B) }132 \qquad \textbf{(C) }157 \qquad \textbf{(D) }19
  10 KB (1,548 words) - 00:08, 12 February 2024
• 2021 AMC 12A Problems/Problem 23
  ...if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops? <math>\textbf{(A)} ~\frac{9}{16}\qquad\textbf{(B)} ~\frac{5}{8}\qquad\textbf{(C)} ~\frac{3}{
  17 KB (2,801 words) - 07:29, 4 November 2022
• 2021 AMC 12A Problems/Problem 24
  ...a positive integer not divisible by the square of any prime. What is <math>a+b+c</math>? <math>\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }1
  16 KB (2,539 words) - 07:30, 30 December 2023
• 2020 INMO Problems/Problem 1
  .../math> again in <math>X</math> and <math>Y</math>, respectively. Let <math>P</math> and <math>Q</math> be the circumcentres of triangles <math>ACD</math Let <math>O, S, T</math> be the circumcenters of triangles <math>BCD</math>, <math>XAY</ma
  2 KB (385 words) - 12:56, 5 November 2020
• 1979 IMO Problems/Problem 3
  ...h> in the plane such that the two points are always equidistant from <math>P.</math> ...Let A' be the other intersection of the 2 circles different from the point A. Since the angles <math>\angle BO_1A = \angle CO_2A</math> are equal, the i
  8 KB (1,485 words) - 22:55, 29 January 2021
• 2021 AMC 12B Problems/Problem 8
  Three equally spaced parallel lines intersect a circle, creating three chords of lengths <math>38,38,</math> and <math>34</ <math>\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textb
  6 KB (878 words) - 00:15, 19 July 2023
• 2021 AMC 12B Problems/Problem 24
  ...tively, onto the line <math>BD;</math> and points <math>R</math> and <math>S</math> are the projections of <math>B</math> and <math>D,</math> respective ...ot("$D$",D,W); dot("$P$",P,SE); dot("$Q$",Q,NE); dot("$R$",R,N); dot("$S$",S,dir(270)); </asy>
  11 KB (1,646 words) - 21:14, 28 May 2024
• 2021 AIME II Problems/Problem 10
  ...the sphere with radius <math>13</math> is tangent to plane <math>\mathcal{P}</math> is <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</mat triple O1, O2, O3, T1, T2, T3, A, L1, L2;
  13 KB (2,124 words) - 01:00, 14 January 2023
• 2021 AIME II Problems/Problem 14
  ...f the line tangent to the circumcircle of <math>\Delta ABC</math> at <math>A</math> and the line perpendicular to <math>GO</math> at <math>G</math>. Let pair A, B, C, O, G, X, Y;
  10 KB (1,526 words) - 16:50, 25 December 2022
• 2022 AIME I Problems
  ...h the two points <math>(16,54)</math> and <math>(20,53).</math> Find <math>P(0) + Q(0).</math> ...rline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},</math> where <math>a,</math> <math>b,</math> and <math>c</math> are (not necessarily distinct) d
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• G285 2021 MC10A
  <math>\textbf{(A)}\ -2\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 1\qquad ...h> with equal chances. What is the probability that the number selected is a perfect square, given <math>n=4</math>?
  8 KB (1,385 words) - 12:55, 23 June 2021
• 2022 AIME II Problems/Problem 15
  ...h>O_1O_2 = 15</math>, <math>CD = 16</math>, and <math>ABO_1CDO_2</math> is a convex hexagon. Find the area of this hexagon. circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B);
  14 KB (2,217 words) - 00:28, 29 June 2023
• G285 2021 Summer Problem Set
  <math>\textbf{(A)}\ 42\qquad\textbf{(B)}\ 43\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 45\qqu \end{cases}</cmath> If <math>y</math> is a positive integer, find the sum of all values of <math>x</math> such that <m
  9 KB (1,577 words) - 23:28, 28 June 2021
• 2021 Fall AMC 12B Problems/Problem 24
  ...sects the circumcircle of <math>\triangle{ABC}</math> in point <math>E \ne A</math>. The circumcircle of <math>\triangle{BED}</math> intersects the line <math>\textbf{(A) } 28 \qquad \textbf{(B) } 20\sqrt{2} \qquad \textbf{(C) } 30 \qquad \textb
  11 KB (1,733 words) - 11:11, 23 November 2023
• 2021 IMO Problems/Problem 4
  ...Gamma</math> be a circle with centre <math>I</math>, and <math>ABCD</math> a convex quadrilateral such that each of The extension of <math>BA</math> beyond <math>A</math> meets <math>\Omega</math> at <math>X</math>, and the extension of <m
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• 2021 Fall AMC 12B Problems/Problem 22
  ...cle centered at <math>P</math> is tangent to line <math>AC</math> at <math>A</math> and passes through <math>B</math>. What is <math>OP</math>? <math>\textbf{(A)}\ \frac{23}{8} \qquad\textbf{(B)}\ \frac{29}{10} \qquad\textbf{(C)}\ \fr
  6 KB (943 words) - 00:41, 6 August 2023
• 2021 Fall AMC 10B Problems/Problem 11
  A regular hexagon of side length <math>1</math> is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the re
  6 KB (1,021 words) - 19:40, 1 November 2023
• 2021 Fall AMC 10B Problems/Problem 15
  In square <math>ABCD</math>, points <math>P</math> and <math>Q</math> lie on <math>\overline{AD}</math> and <math>\over pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0),
  11 KB (1,733 words) - 23:06, 13 May 2024
• 2021 Fall AMC 10B Problems/Problem 17
  ...<math>\ell</math> is <math>5x - y = 0</math>, and the coordinates of <math>P''</math> are <math>(4,1)</math>. What is the equation of line <math>m?</mat <math>(\textbf{A})\: 5x+2y=0\qquad(\textbf{B}) \: 3x+2y=0\qquad(\textbf{C}) \: x-3y=0\qquad(
  8 KB (1,331 words) - 22:44, 16 December 2023
• 2021 Fall AMC 10B Problems/Problem 18
  ...<math>c</math> is not divisible by the square of any prime. What is <math>a+b+c?</math> pair O,A1,B1,C1,A2,B2,C2,A3,B3,C3,A4,B4,C4;
  18 KB (3,011 words) - 22:05, 26 September 2023
• 2021 Fall AMC 12A Problems
  {{AMC12 Problems|year=2021 Fall|ab=A}} <math>\textbf{(A) } 7 \qquad\textbf{(B) } 21 \qquad\textbf{(C) } 49 \qquad\textbf{(D) } 64 \
  15 KB (2,452 words) - 19:37, 7 June 2023
• 2021 Fall AMC 12A Problems/Problem 13
  <math>\textbf{(A)} \ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)} \ \frac{1+\sqrt{7}}{2} \qquad pair O;
  16 KB (2,526 words) - 00:53, 6 May 2023
• 2021 Fall AMC 12A Problems/Problem 21
  ...e on diagonal <math>\overline{AC}</math> with <math>X</math> between <math>A</math> and <math>Y</math>, as shown in the figure. Suppose <math>\angle AXD pair O, pair M, pair B, real size=5,
  5 KB (756 words) - 03:40, 23 January 2023
• 2021 Fall AMC 12B Problems
  <math>\textbf{(A)}\: 10{,}000\qquad\textbf{(B)} \: 10{,}010\qquad\textbf{(C)} \: 10{,}110\qq pen s = linewidth(0.8)+fontsize(8);
  14 KB (2,191 words) - 19:57, 12 November 2023
• 2023 AIME I Problems/Problem 5
  Let <math>P</math> be a point on the circle circumscribing square <math>ABCD</math> that satisfies ==Solution 1 (Ptolemy's Theorem)==
  19 KB (3,107 words) - 23:31, 17 January 2024
• AoPSWiki:Sandbox
  ...bulletin board post. Adapted and edited by <b>aquadragon.</b></i></small></p> * If your code is not working check to make sure you have a semicolon after each line, no extra parentheses or brackets, and no open pa
  28 KB (4,808 words) - 19:43, 3 June 2022
• Kimberling’s point X(24)
  == Kimberling's point X(24) == ...2 = \triangle UVW</math> be Orthic Triangle of <math>T_1</math>. Let <math>O</math> and <math>H</math> be the circumcenter and orthocenter of <math>T_0.
  10 KB (1,751 words) - 15:34, 25 November 2022
• 2023 AMC 8 Problems
  <math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 18 A square piece of paper is folded twice into four equal quarters, as shown be
  29 KB (4,048 words) - 17:25, 21 February 2024
• 2022 AMC 10A Problems/Problem 23
  ...with <math>BC < AD</math> and <math>AB = CD.</math> There is a point <math>P</math> in the plane such that <math>PA=1, PB=2, PC=3,</math> and <math>PD=4 <math>\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\q
  12 KB (1,806 words) - 12:52, 26 March 2024
• 2022 AMC 10A Problems/Problem 13
  ...pendicular to <math>\overline{AP}</math> intersects the line through <math>A</math> parallel to <math>\overline{BC}</math> at point <math>D.</math> Supp <math>\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 11
  7 KB (1,170 words) - 08:04, 19 May 2024
• 2022 AMC 10B Problems/Problem 22
  ...)^{2}+y^{2}=3</math>. What is the sum of the areas of all circles in <math>S</math>? <math>\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~1
  7 KB (1,202 words) - 01:15, 10 June 2023
• 2022 AMC 12B Problems/Problem 25
  Four regular hexagons surround a square with side length 1, each one sharing an edge with the square, ...ath>m \sqrt{n} + p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are integers and <math>n</math> is not divisible by the square of an
  20 KB (2,980 words) - 18:17, 2 January 2024
• 2022 AMC 12B Problems/Problem 19
  ...sitive integer not divisible by the square of any prime. What is <math>m+n+p?</math> <math>\textbf{(A) }44 \qquad \textbf{(B) }48 \qquad \textbf{(C) }52 \qquad \textbf{(D) }56 \
  5 KB (710 words) - 03:53, 12 November 2023
• Kimberling’s point X(22)
  ...{2}{J^2–3} (\vec H – \vec O), J = \frac {|OH|}{R},</math> where <math>O</math> - circumcenter, <math>H</math> - orthocenter, <math>R</math> - circu ...f <math>BC, AC,</math> and <math>AB,</math> respectively. The points <math>A, G, A_1,</math> and <math>A_0</math> are collinear. Similarly the points <m
  6 KB (1,002 words) - 09:24, 25 November 2022
• Kimberling’s point X(21)
  ...th> and <math>\triangle ABC</math> are concurrent at Schiffler point <math>S = X(21), \frac {OS}{SG} = \frac {3R}{2r}</math>. ...ler line <math>OG</math> of <math>\triangle ABC</math> at such point <math>S,</math> that <math>\frac {OS}{SG} = \frac {3R}{2r}</math>.
  3 KB (576 words) - 16:41, 23 November 2022
• Complete Quadrilateral
  Let four lines made four triangles of a complete quadrilateral. In the diagram these are <math>\triangle ABC, \tria *[[Miquel's point]]
  8 KB (1,460 words) - 16:16, 5 May 2023
• 2017 EGMO Problems
  ...</math>.Prove that the points <math>M,N,A</math> and <math>C</math> lie on a circle. ...tive integers <math>\mathbb{Z}_{>0}</math> with <math>k</math> colours and a function <math>f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}</math> with the followi
  4 KB (738 words) - 14:03, 24 December 2022
• 2022 EGMO Problems
  ...and <math>CP</math>. Prove that <math>T</math>, <math>H</math>, and <math>S</math> are collinear. ...N} \rightarrow \mathbb{N}</math> such that for any positive integers <math>a</math> and <math>b</math>, the following two conditions hold:
  3 KB (626 words) - 14:13, 24 December 2022
• Gossard perspector
  ...at the Euler line of a given triangle together with two of its sides forms a triangle whose Euler line is in parallel with the third side of the given t ...d by the Euler line and the sides, taken by two, of a given triangle, form a triangle which is perspective with the given triangle and has the same Eule
  18 KB (3,046 words) - 06:44, 19 January 2023
• 2024 AMC 8 Problems
  <math>\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\t <math>\textbf{(A) } 6.4\qquad\textbf{(B) } 6.504\qquad\textbf{(C) } 6.54\qquad\textbf{(D) }
  22 KB (2,999 words) - 19:21, 30 May 2024
• 2023 AIME I Problems/Problem 8
  ...P</math> on the incircle of the rhombus such that the distances from <math>P</math> to the lines <math>DA,AB,</math> and <math>BC</math> are <math>9,</m pair A, B, C, D, O, P, R, S, T;
  17 KB (2,612 words) - 14:54, 3 July 2023
• 2023 AIME I Problems/Problem 13
  Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths <math>\sqrt{21}</math> and <math>\sqrt ...vely prime positive integers. Find <math>m + n</math>. A parallelepiped is a solid with six parallelogram faces
  13 KB (2,130 words) - 01:52, 31 January 2024
• 2023 AIME II Problems/Problem 12
  ...h> such that <math>M</math> is on <math>\overline{AP}.</math> There exists a unique point <math>Q</math> on segment <math>\overline{AM}</math> such that ...th>M</math> is the midpoint of <math>BC</math>, following from the Stewart's theorem, <math>AM = 2 \sqrt{37}</math>.
  12 KB (1,900 words) - 18:14, 28 January 2024
• 2023 AIME II Problems/Problem 14
  ...athcal{P}</math> at a height of <math>7</math> meters above <math>\mathcal{P}.</math> The volume of water is <math>\frac{m}{n}</math> cubic meters, wher pair A = origin, B = (6,3), X = rotate(40)*B, Y = rotate(70)*X, C = X+Y, Z = X+B,
  10 KB (1,574 words) - 16:42, 8 March 2024
• 2018 TSTST Problems
  ...{\mathbb Z}[x] \to {\mathbb Z}</math> such that for any polynomials <math>p,q \in {\mathbb Z}[x]</math>, <math>\theta(p+1) = \theta(p)+1</math>, and
  5 KB (930 words) - 08:38, 22 February 2023
• 2013 TSTST Problems
  ...ath>\ell_a</math> passes through the feet of the perpendiculars from <math>A</math> to <math>DB</math> and <math>DC</math>. Line <math>m_a</math> passes ...<math>b = a_k</math>. Find the maximum possible number of forced terms in a regular sequence with <math>1000</math> terms.
  5 KB (873 words) - 08:45, 24 February 2023
• 2022 SSMO Speed Round Problems
  ...a fair coin until it lands on tails. Bobby wins if the coin lands on heads a positive even number of times in the sequence of tosses. Then the probabili A bag is big enough to hold exactly 6 large pencils, 12 medium pencils, or 30
  5 KB (824 words) - 21:35, 31 May 2023
• 2022 SSMO Accuracy Round Problems
  ...distinct element in the range) <math>f:S\rightarrow S,</math> where <math>S=\{1, 2, 3, 4, 5\}</math>. What is the average of <math>f(1) + f(2) + f(3)</ ...base of the cone. We rotate the cone <math>360^{\circ}</math> about <math>P</math> such that the plane that the base of the cone lies on stays the same
  5 KB (830 words) - 13:04, 14 December 2023
• 2023 IMO Problems/Problem 2
  ...\neq B</math>. Prove that the line tangent to <math>\omega</math> at <math>P</math> meets line <math>BS</math> on the internal angle bisector of <math>\ ...e the point diametrically opposite to a point <math>S</math> through <math>S' \implies AS'</math> is the internal angle bisector of <math>\angle BAC</ma
  7 KB (1,318 words) - 06:38, 1 May 2024
• 2024 AIME I Problems/Problem 14
  ...<math>n</math> is not divisible by the square of any prime. Find <math>m+n+p</math>. ...(B~(4,5,0)\), \(C~(0,5,8)\), and \(D~(4,0,8)\). Then the plane \(BCD\) has a normal
  13 KB (2,042 words) - 09:34, 3 April 2024
• 2006 iTest Problems/Ultimate Question
  ...n of the [[2006 iTest]], where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substituted Points <math>A</math> and <math>B</math> lie on a circle centered at <math>O</math> such that <math>\angle AOB</math> is right. Points <math>C</math> a
  4 KB (612 words) - 11:13, 4 April 2024
• 2023 AMC 12A Problems/Problem 2
  ...ghs <math>\frac{1}{4}</math> of a pound. What is the weight, in pounds, of a large pizza? <math>\textbf{(A) }1\frac{4}{5}\qquad\textbf{(B) }2\qquad\textbf{(C) }2\frac{2}{5}\qquad\tex
  3 KB (509 words) - 00:52, 7 May 2024
• 2023 AMC 10B Problems/Problem 7
  <math>\text{(A)}\ 24^{\circ} \qquad \text{(B)}\ 35^{\circ} \qquad \text{(C)}\ 30^{\circ} \ First, let's call the center of both squares <math>I</math>. Then, <math>\angle{AIE} = 2
  4 KB (577 words) - 02:09, 27 November 2023
• 2023 AMC 12B Problems/Problem 25
  ...tices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? <math>\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qqu
  19 KB (2,967 words) - 16:56, 24 February 2024
• 2000˙AIME˙II˙Problems/Problem˙3
  ...;L'AS;JLOASLKHNKC,ZM AS;/LAS;KOASJDNK JLASJASLK ;LAS/O[POIUPAIO'O[I9UPOPO';P;/;';.,;LXKOZPCJ DLK.;LSZJCILKZXJCZCGUSGDSKAHAGASUAS5214D 's\]
  814 bytes (128 words) - 10:36, 11 November 2023
• 2023 AMC 12B Problems/Problem 19
  <math>\textbf{(A) } \frac{2}{3} \qquad\textbf{(B) } \frac{3}{10} \qquad\textbf{(C) } \frac{1 ...s and Bars does not provide an exact probability. However, it does provide a good estimate for the approximate answer as on average, the number of arran
  15 KB (2,521 words) - 18:51, 26 May 2024
• Mock AIME 6 2006-2007 Problems/Problem 13
  ...n internal tangent intersects the common external tangents at points <math>A</math> and <math>B</math>. Given that the radius of the larger circle is < ...h> where <math>A_1</math> is the area of <math>\Delta PBA</math> and <math>s</math> is half the perimeter of <math>\Delta PBA</math>
  2 KB (280 words) - 03:36, 26 November 2023
• Bisector
  Let a triangle <math>\triangle ABC, BC = a, AC = b, AB = c</math> be given. he segments <math>BB'</math> and <math>A'C'</math> meet at point <math>D.</math> Find <cmath>\frac {BI}{BB'}, \frac
  10 KB (1,668 words) - 09:47, 26 March 2024
• 1997 OIM Problems/Problem 5
  ...nt <math>O</math>. The lines <math>AE</math> and <math>AO</math> intersect a second time the circumference circumscribes triangle <math>ABC</math> at po ...math>, the intersection of <math>BC</math> with <math>OM</math>; and <math>S</math> the intersection of <math>HR</math> with <math>OP</math>.
  1 KB (176 words) - 15:34, 13 December 2023
• 2023 SSMO Accuracy Round Problems
  ...nd a <math>2</math> second pause between paragraphs. If it takes him <math>S</math> minutes to read the whole Hamburger Proclamation, find <math>\left\l Suppose that <math>a, b, c</math> are real numbers such
  6 KB (891 words) - 03:41, 23 December 2023
• 2024 AIME I Problems/Problem 5
  ...<math>D,E,C,F</math> are collinear. Also, <math>A,D,H,G</math> all lie on a circle. If <math>BC=16</math>,<math>AB=107</math>,<math>FG=17</math>, and < pair A = (0,0);pair B = (70,0);pair C = (70,16);pair D = (0,16);pair E = (3,16);pa
  7 KB (1,144 words) - 17:43, 22 April 2024
• DVI exam
  ...State University named after M.V. Lomonosov. The first four problems have a standard level. ...le of its upper base touches each of the faces which contains vertex <math>S.</math>
  32 KB (5,375 words) - 11:43, 5 May 2024
• 2024 AIME II Problems/Problem 8
  ...utside of <math>S</math>, it is externally tangent to <math>S</math> along a circle with radius <math>r_o</math>. The difference <math>r_i-r_o</math> ca First, let's consider a section <math>\mathcal{P} </math> of the solids, along the axis.
  5 KB (880 words) - 16:18, 23 April 2024
• Problems Collection
  This is a page where you can share the problems you made (try not to use past exams). <cmath>(p^2+1)(q^2+1)-((pq)^2-pq+1)</cmath>
  43 KB (7,006 words) - 14:24, 19 February 2024
• 2024 AIME II Problems/Problem 10
  Let <math>\triangle ABC</math> have circumcenter <math>O</math> and incenter <math>I</math> with <math>\overline{IA}\perp\overline{O Start off by (of course) drawing a diagram! Let <math>I</math> and <math>O</math> be the incenter and circumcenters of triangle <math>ABC</math>, resp
  17 KB (2,854 words) - 15:18, 23 April 2024
• 2024 AIME II Problems/Problem 12
  ...q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p+q\). ...critical condition that must be satisfied by our parameters is that <math>a^2+b^2=1</math>, since the length of <math>\overline{PQ}=1</math>.
  15 KB (2,558 words) - 21:32, 30 May 2024
• 2024 USAJMO Problems/Problem 1
  .../math> such that <math>CR = DS = 2</math>. Prove that <math>PQRS</math> is a cyclic quadrilateral. ....5</math>, <math>DF=CF=4</math>, and <math>SF=RF=2</math>. Also, let <math>O</math> be the circumcenter of <math>ABCD</math>.
  12 KB (1,874 words) - 15:42, 30 April 2024
• Sharygin Olympiads, the best
  ...y, popularizer of science. He wrote many textbooks on geometry and created a number of beautiful problems. He headed the mathematics section of the Russ A point <math>P</math> moves along a circle <math>\Omega.</math> Let <math>A</math> and <math>B</math> be fixed points of <math>\Omega,</math> and <math
  29 KB (4,997 words) - 18:06, 16 May 2024