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• Pythagorean Theorem
  ...n [[geometry]], and is one of the many tools in a good geometer's arsenal. A very large number of geometry problems can be solved by building right tria In these proofs, we will let <math>ABC </math> be any right triangle with a right angle at <math>{} C </math>.
  5 KB (886 words) - 13:51, 15 May 2024
• 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
• Number theory/Advanced
  ...r theory include the [[Birch and Swinnerton-Dyer Conjecture]] and [[Fermat's Last Theorem]]. An elliptic curve is the set of points <math>(x,y)</math> satisfying some two variable third degree equation. Using certain a
  5 KB (849 words) - 16:14, 18 May 2021
• Circle
  A '''circle''' is a geometric figure commonly used in Euclidean [[geometry]]. {{asy image|<asy>unitsize(2cm);draw(unitcircle,blue);</asy>|right|A basic circle.}}
  9 KB (1,581 words) - 18:59, 9 May 2024
• Ellipse
  An '''ellipse''' is a type of [[conic section]]. An ellipse is formed by cutting through a [[cone]] at an [[angle]].
  5 KB (892 words) - 21:52, 1 May 2021
• 2006 AIME I Problems/Problem 8
  ...> is a [[positive integer]], find the number of possible values for <math> K</math>. ...2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6);
  5 KB (730 words) - 15:05, 15 January 2024
• 2006 AMC 12A Problems
  {{AMC12 Problems|year=2006|ab=A}} Sandwiches at Joe's Fast Food cost <math>3</math> dollars each and sodas cost <math>2</math> do
  15 KB (2,223 words) - 13:43, 28 December 2020
• 2001 AMC 12 Problems
  The sum of two numbers is <math>S</math>. Suppose <math>3</math> is added to each number and then <math>\text{(A)}\ 2S + 3\qquad \text{(B)}\ 3S + 2\qquad \text{(C)}\ 3S + 6 \qquad\text{(D)
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2004 AMC 12B Problems
  <math>(\mathrm {A}) 3\qquad (\mathrm {B}) 6 \qquad (\mathrm {C}) 9 \qquad (\mathrm {D}) 12 \q In the expression <math>c\cdot a^b-d</math>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are 0, 1, 2, and
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at t \mathrm{(A)}\ 100 \qquad
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2005 AIME II Problems/Problem 12
  ...where <math> p,q, </math> and <math> r </math> are [[positive]] [[integer]]s and <math> r </math> is not divisible by the [[square]] of any [[prime]], f ...",G,(-1,1));label("\(O\)",O,(1,-1)); label("\(x\)",E/2+G/2,(0,1));label("\(y\)",G/2+F/2,(0,1)); label("\(450\)",(O+G)/2,(-1,1));
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME II Problems/Problem 14
  pair A = origin; pair C = rotate(15,A)*(A+dir(-50));
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2002 AIME I Problems
  ...plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right- ...gram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the r
  8 KB (1,374 words) - 21:09, 27 July 2023
• 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
• 1995 AIME Problems/Problem 9
  ...,</math> where <math>a</math> and <math>b</math> are integers. Find <math>a+b.</math> ...W); dot((2,0),ds); label("$C$",(2,0),SE); dot((1,0),ds); label("$M$",(1,0),S); dot((1,0.7),ds); label("$D$",(1,0.7),NE);
  7 KB (1,181 words) - 13:47, 3 February 2023
• 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> ...ngle with <math>\angle A=60^\circ</math>, and let <math>O</math> and <math>H</math> denote the circumcenter and orthocenter of <math>\triangle{ABC}</mat
  3 KB (600 words) - 16:42, 5 August 2023
• 2001 AIME I Problems/Problem 6
  .../math> and <math>n</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>m + n</math>. ...n the diagram below, the lowest <math>y</math>-coordinate at each of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> corresponds to t
  11 KB (1,729 words) - 20:50, 28 November 2023
• 2000 AIME II Problems/Problem 12
  ...<math>n</math> is not divisible by the square of any prime. Find <math>m+n+k</math>. ...split into <math>9-12-15</math> and <math>5-12-13</math> [[right triangle]]s):
  3 KB (532 words) - 13:14, 22 August 2020
• Dedekind domain
  A '''Dedekind domain''' is a [[integral domain]] <math>R</math> satisfying the following properties: * <math>R</math> is a [[noetherian]] [[ring]].
  9 KB (1,648 words) - 16:36, 14 October 2017
• Congruent (geometry)
  '''Congruency''' is a property of multiple geometric figures. Two [[geometry | geometric]] [[figure]]s are congruent if one of them can be turned and/or flipped and placed exactl
  10 KB (1,655 words) - 21:43, 24 March 2022
• 1970 Canadian MO Problems
  Find all number triples <math>(x,y,z)</math> such that when any of these numbers is added to the product of th ...prove that <math>a+h\ge b+k</math>. Find under what conditions <math>a+h=b+k</math>.
  4 KB (604 words) - 04:32, 8 October 2014
• 2007 iTest Problems
  A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</ <math>\mathrm{(A)}\, 4</math>
  30 KB (4,794 words) - 23:00, 8 May 2024
• 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
• 2003 AMC 10A Problems/Problem 22
  ...gle <math>ABCD</math>, we have <math>AB=8</math>, <math>BC=9</math>, <math>H</math> is on <math>BC</math> with <math>BH=6</math>, <math>E</math> is on < pair D=(0,0), Ep=(4,0), A=(9,0), B=(9,8), H=(3,8), C=(0,8), G=(-6,20), F=(-6,0);
  9 KB (1,446 words) - 22:48, 8 May 2024
• 1987 IMO Problems/Problem 4
  ...h> must be even. If <math>k = 2h</math>, then we may take <math>f(n) = n + h</math>. ...m + k(s + 2)</math>. But <math>f(m + k) = n + k</math>, so <math>n = m + k(s + 1) > n</math>. Contradiction.
  5 KB (923 words) - 19:51, 21 January 2024
• 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
• 2007 AIME I Problems/Problem 9
  .../math> and <math>q</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>p+q</math>. ...th>x = AD = AF</math> and <math>y = EB = BG</math>. We know that <math>x + y + 2r = 34</math>.
  11 KB (1,851 words) - 12:31, 21 December 2021
• 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 7
  ...th> a+b=1000 </math> and neither <math> a </math> nor <math> b </math> has a zero digit. ...(notice that the only time both <math>a</math> and <math>b</math> can have a 0 in the tens digit is when they are divisible by 100, which falls into the
  7 KB (1,114 words) - 03:41, 12 September 2021
• 2000 PMWC Problems
  ...r <math>1</math> and itself). What is the smallest possible value of <math>y</math>? Given that <math>A^4=75600\times B</math>. If <math>A</math> and <math>B</math> are positive integers, find the smallest value of
  11 KB (1,713 words) - 22:47, 13 July 2023
• De Longchamps point
  label("H",(24,0),(1,1)); ...hamps <br />point (<math>L</math>) is the the <br />[[orthocenter]] (<math>H</math>) reflected <br /> through the [[circumcenter]]<br /> (<math>C</math>
  10 KB (1,780 words) - 09:23, 17 November 2022
• 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
• 1959 AHSME Problems
  Each edge of a cube is increased by <math>50</math>%. The percent of increase of the surfa <math>\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300
  22 KB (3,345 words) - 20:12, 15 February 2023
• 1966 AHSME Problems
  ...constant, and <math>y = 3</math> when <math>x = 2</math>, then, when <math>y = 12</math>, <math>x</math> equals: <math>\text{(A)} \ \frac 18 \qquad \text{(B)} \ \frac 73 \qquad \text{(C)} \ \frac78 \qqua
  19 KB (3,159 words) - 22:10, 11 March 2024
• Isogonal conjugate
  ...nal conjugates''' are pairs of [[point]]s in the [[plane]] with respect to a certain [[triangle]]. ...ll</math> and a point <math>O</math> lying on <math>\ell</math> be given. A pair of lines symmetric with respect to <math>\ell</math> and containing t
  54 KB (9,416 words) - 08:40, 18 April 2024
• Mock AIME 5 Pre 2005 Problems
  Function <math>g(y)</math> is given such way that for all <math>y</math>, <cmath>6g(1 + (1/y)) + 12g(y + 1) = \log_{10} y</cmath>
  6 KB (909 words) - 07:27, 12 October 2022
• 2008 AIME II Problems/Problem 14
  ...math>. Let <math>\rho</math> be the maximum possible value of <math>\frac {a}{b}</math> for which the system of equations a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
  8 KB (1,387 words) - 11:56, 29 January 2024
• 2003 AMC 12A Problems/Problem 17
  ...h>M</math> intersects a circle with radius <math>4</math> and center <math>A</math> at points <math>P</math> and <math>D</math>. What is the distance fr pair A,B,C,D,M,P;
  6 KB (1,026 words) - 22:35, 29 March 2023
• Sylow Theorems
  ...eory of [[finite]] [[group]]s. They give a partial converse to [[Lagrange's Theorem]], and are one of the most important results in the field. They ar :'''Theorem.''' Every finite group contains a [[Sylow p-subgroup |Sylow <math>p</math>-subgroup]].
  11 KB (2,071 words) - 12:25, 9 April 2019
• KGS math club
  A group of people on [http://www.gokgs.com Kiseido Go Server] Mathematics roo ...our, or of a less-than-or-equal rank. What is the greatest possible length a parade can reach?
  22 KB (3,358 words) - 15:17, 18 July 2017
• 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
• 2021 GMC 10B
  <math>\textbf{(A)} ~99 \qquad\textbf{(B)} ~100 \qquad\textbf{(C)} ~101 \qquad\textbf{(D)} ~1 The radius of a circle that has an area of <math>\frac{\pi}{\sqrt{2}}</math> is <math>r</ma
  11 KB (1,695 words) - 14:33, 7 March 2022
• 2009 AMC 12A Problems
  {{AMC12 Problems|year=2009|ab=A}} ...rs and <math>m</math> minutes, with <math>0 < m < 60</math>, what is <math>h + m</math>?
  13 KB (2,105 words) - 13:13, 12 August 2020
• 2001 AMC 12 Problems/Problem 24
  \text{(A) }54^\circ ...The comparable ratio is that <math>BH-DH=t</math>. If we involve our <math>k</math>, we get:
  8 KB (1,316 words) - 22:48, 7 March 2024
• 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
  10 KB (1,797 words) - 02:05, 24 October 2023
• Constructible number
  ...can be constructed with a [[straight edge]] and [[compass]] starting with a segment of length <math>1</math>. ...h> is constructible as a real number if and only if it is constructible as a complex number, i.e., our two definitions coincide in this case.
  8 KB (1,305 words) - 08:39, 21 August 2009
• 2010 AMC 12A Problems
  {{AMC12 Problems|year=2010|ab=A}} <math>\textbf{(A)}\ -4020 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}
  12 KB (1,817 words) - 15:00, 12 August 2020
• 2010 AMC 10A Problems
  {{AMC10 Problems|year=2010|ab=A}} Mary’s top book shelf holds five books with the following widths, in centimeters:
  13 KB (1,902 words) - 11:20, 5 March 2023
• 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
• Proofs without words
  The following demonstrate [[proof]]s of various identities and theorems using pictures, inspired from [http://us Minkowski's inequality
  55 KB (7,986 words) - 17:04, 20 December 2018
• Pell equation
  ...st square factor <math>d^2 | D</math> into <math>y</math> by setting <math>y' = dy</math>. Note that if <math>D = d^2</math> is a perfect square, then this problem can be solved using [[difference of squar
  6 KB (1,076 words) - 10:20, 29 October 2023
• 2010 AIME I Problems/Problem 11
  ...et of points in the [[coordinate plane]] that satisfy both <math>|8 - x| + y \le 10</math> and <math>3y - x \ge 15</math>. When <math>\mathcal{R}</math> pair A = (8,10), B = (4.5,6.5), C= (9.75,8.25), F=foot(A,B,C), G=2*F-A; fill(A--B--C--cycle,rgb(0.9,0.9,0.9));
  4 KB (636 words) - 16:46, 25 November 2023
• 2010 AIME II Problems
  ...math> denote the distance from <math>P</math> to the closest side of <math>S</math>. The probability that <math>\frac{1}{5}\le d(P)\le\frac{1}{3}</math> ...t positive integer <math>n</math> such that <math>2^n</math> divides <math>K</math>.
  8 KB (1,246 words) - 21:58, 10 August 2020
• 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
• 1997 AJHSME Problems
  <math>\text{(A)}\ 0.0026 \qquad \text{(B)}\ 0.0197 \qquad \text{(C)}\ 0.1997 \qquad \text{ Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is
  12 KB (1,702 words) - 12:35, 6 November 2022
• 2011 AMC 12A Problems
  {{AMC12 Problems|year=2011|ab=A}} A cell phone plan costs <math>\$20</math> dollars each month, plus <math>5</m
  13 KB (1,994 words) - 13:52, 3 July 2021
• 2011 AMC 12B Problems
  <math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \ Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>,
  13 KB (1,978 words) - 16:28, 12 July 2020
• 2011 AIME I Problems
  ...ath> and <math>n</math> are relatively prime positive integers, find <math>k + m + n</math>. ...ive <math>x</math>-axis, and point <math>B</math> is on the positive <math>y</math>-axis. The point <math>P</math> with coordinates <math>(-14,27)</mat
  10 KB (1,634 words) - 22:21, 28 December 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 I Problems/Problem 8
  ...itive integer that is not divisible by the square of any prime. Find <math>k+m+n</math>. pair[] A, B, C, U, V, W, X, Y, Z;
  6 KB (1,077 words) - 21:47, 12 April 2022
• 1999 AMC 8 Problems
  <math>\text{(A)} \div \qquad \text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \ What is the 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 3
  For a point <math>P = (a, a^2)</math> in the coordinate plane, let <math>\ell(P)</math> denote the line ...by the real part - of all <math>w_n^3</math> is constant, say it is <math>k</math>. Then for <math>n=1,2,3</math>,
  15 KB (2,593 words) - 13:37, 29 January 2021
• Block walking
  ...lines, what is the number of ways you can get from <math>X</math> to <math>Y</math>? label("Y", (3,0), SE);
  9 KB (1,228 words) - 21:47, 26 February 2022
• 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
• 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
• 2010 PUMaC Problems
  == Division A == ...math> such that <math>4^x = x^4</math>. Find the nearest integer to <math>S</math>.
  22 KB (3,694 words) - 23:58, 3 June 2022
• 2016 AMC 10A Problems/Problem 19
  ...of <math>r,s,</math> and <math>t</math> is <math>1.</math> What is <math>r+s+t</math>? <math>\textbf{(A) } 7 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 15
  8 KB (1,386 words) - 15:10, 8 October 2023
• 2009 PUMaC Problems
  '''Algebra A''' ...>f(h+k)+f(hk)=f(h)f(k)+1</math>, for all integers <math>h</math> and <math>k</math>.
  25 KB (4,154 words) - 16:27, 2 September 2011
• 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
  ...triangle <math> \Delta{A_1}{A_2}{A_3} </math>. These masses then determine a point <math> P </math>, which is the geometric centroid of the three masses The Central NC Math Group published a lecture concerning this topic at https://www.youtube.com/watch?v=KQim7-wrwL
  25 KB (5,067 words) - 22:15, 31 March 2024
• Mathtime Version 1 Issue 2
  ...e the positive integer <math>n</math> such that <math>n^2+20n+40</math> is a perfect square. (Proposed by djmathman) ...umber of ordered integer pairs <math>(a,b)</math> such that <math>a^2+b^2|(a+b)^2</math>. (Proposed by djmathman)
  15 KB (2,444 words) - 21:46, 1 January 2012
• 1989 AHSME Problems
  <math> \textrm{(A)}\ -7\qquad\textrm{(B)}\ -2\qquad\textrm{(C)}\ 0\qquad\textrm{(D)}\ 1\qquad <math> \textrm{(A)}\ \frac{1}5\qquad\textrm{(B)}\ \frac{1}4\qquad\textrm{(C)}\ \frac{2}7\qqua
  15 KB (2,247 words) - 13:44, 19 February 2020
• 2012 AMC 10A Problems/Problem 15
  pair A=(0,0), B=(1,0); pair C=(0.8,-0.4); draw(A--(2,0)); draw((0,-1)--(2,-1)); draw((0,-2)--(1,-2));
  6 KB (935 words) - 23:41, 13 September 2023
• 2012 AMC 12B Problems/Problem 17
  <math> \textbf{(A)}\ 6\qquad\textbf{(B) }\frac{31}5\qquad\textbf{(C) }\frac{32}5\qquad\textbf ...; pair DA = extension(D,A,(0,-2),(1,-2)); draw(A--AB--B--BC--C--CD--D--DA--A, dotted); dot(AB^^BC^^CD^^DA);</asy>
  12 KB (2,183 words) - 21:05, 23 December 2023
• 2015 AIME I Problems
  ...successive integers. Find the positive difference between integers <math>A</math> and <math>B</math>. ...a prime number <math>p</math> such 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 ...math>, and in general, the <math>k</math>th term is increased by the <math>k</math>th odd positive integer. The terms of the new sequence add to <math>8
  10 KB (1,617 words) - 14:49, 2 June 2023
• 1974 AHSME Problems
  ...ath> y\not=0 </math> or <math> 6 </math>, then <math> \frac{2}{x}+\frac{3}{y}=\frac{1}{2} </math> is equivalent to ...\mathrm{(B) \ }y=\frac{4x}{6-y} \qquad \mathrm{(C) \ } \frac{x}{2}+\frac{y}{3}=2 \qquad </math>
  15 KB (2,151 words) - 14:04, 19 February 2020
• 1954 AHSME Problems
  The square of <math>5-\sqrt{y^2-25}</math> is: ...\textbf{(C)}\ y^2 \\ \textbf{(D)}\ (5-y)^2\qquad\textbf{(E)}\ y^2-10\sqrt{y^2-25} </math>
  23 KB (3,535 words) - 16:29, 24 April 2020
• 1998 USAMO Problems/Problem 6
  ...th>n</math>-gon <math>A_{1}A_{2}\dots A_{n}</math> for which exactly <math>k</math> of the quadrilaterals <math>A_{i}A_{i+1}A_{i+2}A_{i+3}</math> have a 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
• 2013 AMC 10A Problems/Problem 23
  ...<math>AB = 86</math>, and <math>AC=97</math>. A circle with center <math>A</math> and radius <math>AB</math> intersects <math>\overline{BC}</math> at <math> \textbf{(A)}\ 11\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 33\qquad\textbf{(D)}\ 61\qqu
  5 KB (846 words) - 23:02, 21 August 2023
• 1991 AHSME Problems
  ...math>, <math>b</math>, and <math>c</math> we define <math>f(a,b,c)=\frac{c+a}{c-b}</math>, then <math>f(1,-2,-3)</math> is <math> \textbf{(A) } -2 \qquad \textbf{(B) } -\frac{2}{5} \qquad \textbf{(C) } -\frac{1}{4} \
  16 KB (2,451 words) - 04:27, 6 September 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 10
  ...and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c+d</math>. pair A, B, K, L;
  5 KB (823 words) - 17:57, 29 December 2023
• 2013 AIME II Problems/Problem 13
  ...3[EDB]=3[AEB]=\frac{3}{8}[ABC]</math>, because of base-ratios. Using Heron's Formula on <math>\triangle EDB</math>, as it is simplest, we see that <math After drawing the figure, we suppose <math>BD=a</math>, so that <math>CD=3a</math>, <math>AC=4a</math>, and <math>AE=ED=b</
  13 KB (2,116 words) - 23:24, 21 March 2024
• ASIA TEAM Problems
  Let the order of a lily pad number <math>x</math>, denoted <math>o(x)</math>, be the minimum n ...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
• 2013 AMC 8 Problems/Problem 24
  <math> \textbf{(A)}\hspace{.05in}\frac{1}{4}\qquad\textbf{(B)}\hspace{.05in}\frac{7}{24}\qqua pair A,B,C,D,E,F,G,H,I,J;
  6 KB (1,076 words) - 22:15, 20 December 2023
• 1968 AHSME Problems
  Let <math>P</math> units be the increase in circumference of a circle resulting from an increase in <math>\pi</math> units in the diameter <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
• 2014 AMC 10A Problems
  {{AMC10 Problems|year=2014|ab=A}} <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}
  15 KB (2,190 words) - 15:21, 22 December 2020
• 2014 AMC 10A Problems/Problem 16
  ...line{CD}</math>, and <math>\overline{AD}</math>, respectively. Point <math>H</math> is the midpoint of <math>\overline{GE}</math>. What is the area of t pair A = (0,2);
  9 KB (1,411 words) - 19:51, 25 July 2023
• 1960 AHSME Problems
  ...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 12A Problems
  {{AMC12 Problems|year=2014|ab=A}} <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}
  12 KB (1,863 words) - 19:04, 11 April 2024
• 2014 AMC 12B Problems
  ...d have the same number of pennies and nickels. In cents, how much are Leah's coins worth? <math> \textbf{(A)}\ 33\qquad\textbf{(B)}\ 35\qquad\textbf{(C)}\ 37\qquad\textbf{(D)}\ 39\qqu
  13 KB (2,066 words) - 14:08, 1 November 2022
• 2014 AMC 12B Problems/Problem 21
  In the figure, <math> ABCD </math> is a square of side length <math> 1 </math>. The rectangles <math> JKHG </math> ...sqrt(3),0), F=(2-sqrt(3),1), G=(1,sqrt(3)/2), H=(2.5-sqrt(3),1), J=(.5,0), K=(2-sqrt(3),1-sqrt(3)/2);
  8 KB (1,307 words) - 02:07, 1 January 2023
• 2014 AIME II Problems
  ...s Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is pai ...has none of the three risk factors given that he does not have risk factor A is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are re
  8 KB (1,410 words) - 00:04, 29 December 2021
• 2014 AIME I Problems
  ...each of the ends of the lace must extend at least 200 mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters. ...nd urn contains <math>16</math> green balls and <math>N</math> blue balls. A single ball is drawn at random from each urn. The probability that both bal
  9 KB (1,472 words) - 13:59, 30 November 2021
• 2014 AIME I Problems/Problem 6
  ...ectively, and each graph has two positive integer x-intercepts. Find <math>h</math>. ...e that because the two parabolas have to have positive x-intercepts, <math>h\ge32</math>.
  7 KB (1,158 words) - 20:50, 8 December 2021
• 1992 AHSME Problems
  <math>\text{(A) } 2P\quad <math>\text{(A) } 40\%\quad
  16 KB (2,548 words) - 13:40, 19 February 2020
• 1988 AHSME Problems
  <math>\text{(A)}\ \sqrt{20} \qquad ...<math>A</math> corresponding to <math>X</math> and <math>B</math> to <math>Y</math>. If <math>AB=3, BC=4</math>, and <math>XY=5</math>, then <math>YZ</m
  17 KB (2,535 words) - 13:45, 19 February 2020
• 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

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