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• Computer science books
  ...roduction to Algorithms] by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein. ...p://olympiads.win.tue.nl/ioi/study/books.html Reading List] (someone have a linky?)
  2 KB (251 words) - 00:45, 17 November 2023
• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> if, for 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
• Principle of Inclusion-Exclusion
  ...ized method/formula to find the number of [[element]]s in the [[union]] of a given group of [[set]]s, the size of each set, and the size of all possible ...counted once and only once. In particular, memorizing a formula for PIE is a bad idea for problem solving.
  9 KB (1,703 words) - 07:25, 24 March 2024
• Rearrangement Inequality
  ...ss than or equal to exactly <math>{i}</math> of the other members of <math>A</math>, then <math>{b_k}</math> is also greater than or equal to exactly <m ...re more pennies than nickels, more nickels than dimes, and so on. This is a simple application of the rearrangement inequality. It is also an applicat
  5 KB (804 words) - 13:54, 26 January 2023
• Incenter/excenter lemma
  ...ircumcenter. Then, the theorem states that <math>L</math> is the center of a circle through <math>I</math>, <math>B</math>, <math>I_A</math>, and <math> ...second, <math>I_A</math> is the reflection of <math>I</math> across <math>L</math>. Both of these follow easily from the main proof.
  2 KB (291 words) - 16:31, 18 May 2021
• Base numbers
  ...each number in 2746 is actually just a placeholder which shows how many of a certain power of 10 there are. The first digit to the left of the decimal ...digits 0-9. Usually, the base, or '''radix''', of a number is denoted as a subscript written at the right end of the number (e.g. in our example above
  4 KB (547 words) - 17:23, 30 December 2020
• Henstock-Kurzweil integral
  ...ble generalizations of the [[Integral|Riemann integral]], but it also uses a strikingly simple and elegant idea. It was developed independently by [[Ral Let <math>f:[a,b]\rightarrow\mathbb{R}</math>
  2 KB (401 words) - 09:46, 31 January 2018
• Fundamental Theorem of Calculus
  The '''Fundamental Theorem of Calculus''' establishes a link between the two central operations of [[calculus]]: [[derivative|diffe ...imes <math>t=1</math> and <math>t=2</math> geometrically, as an area under a curve.
  11 KB (2,082 words) - 15:23, 2 January 2022
• Sequence
  A '''sequence''' is an ordered list of terms. Sequences may be either [[fini A sequence of real numbers is simply a function <math>f : \mathbb{N} \rightarrow \mathbb{R}</math>. For instance,
  2 KB (413 words) - 21:18, 13 November 2022
• Area
  ...size of the region that a two-[[dimension]]al figure occupies. The size of a region in higher dimensions is referred to as [[volume]]. It is often possible to find the area of a region bounded by parts of [[circle]]s and [[line segment]]s through elemen
  6 KB (1,181 words) - 22:37, 22 January 2023
• Riemann Hypothesis
  The '''Riemann Hypothesis''' is a famous [[conjecture]] in [[analytic number theory]] that states that all no * All nontrivial zeros of all [[L-series]] have real part one half where an L-series is of the form <math>\sum_{n=1}^\infty \frac{a_n}{n^s}</math>. This
  2 KB (425 words) - 12:01, 20 October 2016
• Note that with two sequences <math>\mathbf{a}</math> and <math>\mathbf{b}</math>, and <math>\lambda_a = \lambda_b = 1/2< ...nces of nonnegative reals, and let <math>\{ \lambda_i \}_{i=1}^n</math> be a sequence of nonnegative reals such that <math>\sum \lambda = 1</math>. The
  4 KB (774 words) - 12:12, 29 October 2016
• 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
• Algebraic topology
  ...of [[topology]] using methods from [[abstract algebra]]. In general, given a [[topological space]], we can associate various algebraic objects, such as ...<math>\pi_1(X)=L/\sim</math>. That is, we equate any two elements of <math>L</math> which are equivalent under <math>\sim</math>.
  3 KB (479 words) - 15:35, 1 December 2015
• Riemann zeta function
  The '''Riemann zeta function''' is a function very important in [[number theory]]. In particular, the [[Riemann Hypothesis]] is a conjecture
  9 KB (1,547 words) - 03:04, 13 January 2021
• Modular arithmetic/Introduction
  ...is article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily ...ple, except let's replace the <math>12</math> at the top of the clock with a <math>0</math>.
  15 KB (2,396 words) - 20:24, 21 February 2024
• Limit
  ...of mathematics. It took several centuries to articulate the definition of a limit and to make it rigorous. ...ue to which a function grows close when its argument is near (but not at!) a particular value. For example,
  7 KB (1,325 words) - 13:51, 1 June 2015
• 2006 AIME I Problems
  In quadrilateral <math> ABCD , \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <mat ...th> and let <math> S </math> be the sum of the elements of <math> \mathcal{A}. </math> Find the number of possible values of <math> S. </math>
  7 KB (1,173 words) - 03:31, 4 January 2023
• 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 AIME I Problems/Problem 10
  ...are positive integers whose [[greatest common divisor]] is 1. Find <math> a^2+b^2+c^2. </math> ...h>, which can be easily solved to be <math>6x = 2y + 5</math>. Thus, <math>a^2 + b^2 + c^2 = \boxed{065}</math>.
  4 KB (731 words) - 17:59, 4 January 2022
• 2003 AMC 12A Problems
  {{AMC12 Problems|year=2003|ab=A}} <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \
  13 KB (1,955 words) - 21:06, 19 August 2023
• 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 21
  ...06</math>. An ellipse with area <math>2006\pi</math> passes through <math>A</math> and <math>C</math> and has foci at <math>B</math> and <math>D</math> \mathrm{(A)}\ \frac {16\sqrt {2006}}{\pi}
  2 KB (339 words) - 13:15, 12 July 2015
• Root (polynomial)
  ...<math>a</math> in the [[domain]] of the function such that <math>f(a) = (x-a) = 0</math>. ...ath> with all <math>c_j \in \mathbb C</math> and <math>c_n \neq 0</math>, a general degree-<math>n</math> polynomial. The degree of <math>P(x)</math> i
  8 KB (1,427 words) - 21:37, 13 March 2022
• 2022 USAJMO Problems/Problem 1
  ...exist integers <math>a,d,r</math> with <math>m \nmid d</math> and <math>m|a+(n-1)d-gr^{n-1}</math> for all integers <math>n>1</math>. <cmath>m | a+nd-gr^n \; (1),</cmath>
  4 KB (792 words) - 00:29, 13 April 2024
• 2005 AIME I Problems/Problem 13
  A particle moves in the [[Cartesian plane]] according to the following rules: ...> the particle may only move to <math> (a+1,b), (a,b+1), </math> or <math>(a+1,b+1). </math>
  5 KB (897 words) - 00:21, 29 July 2022
• 2005 AIME II Problems/Problem 15
  ...h> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally [[tangent (geometry)|tangent]] to <math> w_2 </ma pair A = (-5, 12), B = (5, 12), C = (0, 0);
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2004 AIME I Problems/Problem 15
  and define a [[sequence]] as follows: <math>x_1=x</math> and <math>x_{n+1}=f(x_n)</math> ...>9</math> operations; however, not all <math>9</math> of them may be <math>A: (-1)</math> otherwise we return back to <math>x_{10} = 1</math>, contradic
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2004 AIME I Problems/Problem 14
  ...are [[positive]] [[integer]]s, and <math> c </math> is prime. Find <math> a+b+c. </math> triple Oxy = (0,0,0), A=(4*5^.5,-8,4), B=(0,-8,h), C=(Cxy.x,Cxy.y,0), D=(A.x,A.y,0), E=(B.x,B.y,0), O=(O.x,O.y,h);
  4 KB (729 words) - 01:00, 27 November 2022
• 2004 AIME I Problems/Problem 2
  ...The absolute value of the difference between the greatest element of <math>A</math> and the greatest element of <math>B</math> is <math>99</math>. Find ...e median) must be <math>2</math>. Therefore, the largest element in <math>A</math> is <math>2 + \frac{m-1}{2}</math>.
  8 KB (1,437 words) - 21:53, 19 May 2023
• 2004 AIME II Problems/Problem 7
  <math> ABCD </math> is a rectangular sheet of paper that has been folded so that corner <math> B </m pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect
  9 KB (1,501 words) - 05:34, 30 October 2023
• 2004 AIME II Problems/Problem 3
  A solid rectangular block is formed by gluing together <math> N </math> [[con ...ear that the way to minimize <math>l\cdot m\cdot n</math> is to make <math>l</math> and <math>m</math> and <math>n</math> as close together as possible,
  2 KB (377 words) - 11:53, 10 March 2014
• 1988 AIME Problems
  Let <math>\frac{m}{n}</math>, in lowest terms, be the probability that a randomly chosen positive divisor of <math>10^{99}</math> is an integer mult ...C</math>, <math>\tan \angle CAB = 22/7</math>, and the altitude from <math>A</math> divides <math>BC</math> into segments of length <math>3</math> and <
  6 KB (902 words) - 08:57, 19 June 2021
• 1992 AIME Problems
  A positive integer is called ascending if, in its decimal representation, the ...she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math>0.500</math>. During the weekend, s
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1993 AIME Problems
  ...n for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went :(a) the winner caught <math>15</math> fish;
  8 KB (1,275 words) - 06:55, 2 September 2021
• 1983 AIME Problems/Problem 11
  The solid shown has a square base of side length <math>s</math>. The upper edge is parallel to th triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);
  5 KB (865 words) - 21:11, 6 February 2023
• 1983 AIME Problems/Problem 15
  ...r arc <math>AB</math> is a rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <mat pair A=(-0.91,-0.41);
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1985 AIME Problems/Problem 4
  A small [[square (geometry) | square]] is constructed inside a square of [[area]] 1 by dividing each side of the unit square into <math>n< ...{2n^2 - 2n + 1}</math>, so the parallelogram has height <math>h = \frac{A}{l} = \frac{1}{\sqrt{2n^2 - 2n + 1}}</math>. But the height of the parallelog
  3 KB (484 words) - 21:40, 2 March 2020
• 1986 AIME Problems/Problem 14
  The shortest distances between an interior [[diagonal]] of a rectangular [[parallelepiped]], <math>P</math>, and the edges it does not m ...en <math>AC</math> and that corner, which is <math>\frac {wl}{\sqrt {w^2 + l^2}}</math>.
  2 KB (346 words) - 13:13, 22 July 2020
• 1986 AIME Problems/Problem 9
  pair C=(0,0),A=(510,0),B=IP(circle(C,450),circle(A,425)); pair Da=IP(Circle(A,289),A--B),E=IP(Circle(C,324),B--C),Ea=IP(Circle(B,270),B--C);
  11 KB (1,850 words) - 18:07, 11 October 2023
• 1988 AIME Problems/Problem 11
  ...ean [[line]] for the [[point]]s <math>w_1, w_2, \dots, w_n</math> if <math>L</math> contains points (complex numbers) <math>z_1, z_2, \dots, z_n</math> ...>, <math>w_4 = 1 + 27i</math>, and <math>w_5 = - 14 + 43i</math>, there is a unique mean line with <math>y</math>-intercept 3. Find the [[slope]] of th
  2 KB (422 words) - 00:22, 6 September 2020
• 1989 AIME Problems/Problem 15
  ...>, and line <math>XP</math> intersects line <math>YZ</math> at point <math>L</math>, then <math>\dfrac{XP}{PL} = \frac{[XPY] + [ZPX]}{[YPZ]}.</math> pair L = y/(y+z)*Y + z/(y+z)*Z;
  13 KB (2,091 words) - 00:20, 26 October 2023
• 1990 AIME Problems/Problem 12
  ...^{}_{}</math>, and <math>d^{}_{}</math> are positive integers. Find <math>a + b + c + d^{}_{}</math>. ...\cdot 12\sin 45 = 12\sqrt{2}</math> (or notice that the triangle formed is a <math>45 - 45 - 90</math> [[right triangle]]).
  6 KB (906 words) - 13:23, 5 September 2021
• 1990 AIME Problems/Problem 8
  ...n two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules: 1) The marksman first chooses a column from which a target is to be broken.
  3 KB (491 words) - 04:24, 4 November 2022
• 1991 AIME Problems/Problem 7
  ...n the Wiki front page--><onlyinclude>Find <math>A^2_{}</math>, where <math>A^{}_{}</math> is the sum of the [[absolute value]]s of all roots of the foll ...19}+\sqrt{383}}{2}\\\\x_2=\frac{\sqrt{19}-\sqrt{383}}{2}\end{array}\right\}A=|x_1|+|x_2|\Rightarrow\sqrt{383}</math>
  2 KB (285 words) - 05:15, 13 June 2022
• 1992 AIME Problems/Problem 11
  ...smallest positive integer <math>m^{}_{}</math> for which <math>R^{(m)}(l)=l^{}_{}</math>. ...> in <math>l_1</math>, and let <math>l''</math> be the reflection of <math>l'</math> in <math>l_2</math>.
  2 KB (404 words) - 19:24, 4 July 2013
• 1993 AIME Problems/Problem 12
  ...int, then <math>P_{n + 1}^{}</math> is the midpoint of <math>\overline{P_n L}</math>. Given that <math>P_7 = (14,92)\,</math>, what is <math>k + m\,</m ...right to the left, we have the following information (remembering that if a point must be <math>(0,0)</math>, we can just ignore it!):
  4 KB (611 words) - 13:59, 15 July 2023
• 1994 AIME Problems/Problem 14
  A beam of light strikes <math>\overline{BC}\,</math> at point <math>C\,</math pair B = MP("B",(0,0),NW), C = MP("C",D((1,0))), A = MP("A",expi(alpha * pi/180),N); path r = C + .4 * expi(beta * pi/180) -- C - 2*ex
  2 KB (303 words) - 00:03, 28 December 2017
• 1995 AIME Problems/Problem 12
  // projection of point A onto line BC triple projectionofpointontoline(triple A, triple B, triple C)
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1995 AIME Problems/Problem 3
  ...ing at <math>(0,0),</math> an object moves in the [[coordinate plane]] via a sequence of steps, each of length one. Each step is left, right, up, or do ...math>2!</math> ways to determine the order of the remaining two steps, for a total of <math>12</math> sequences that we have to exclude. This gives <mat
  3 KB (602 words) - 23:15, 16 June 2019
• 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 ..."\(D\)",foot(A,B,C),NE);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><a
  7 KB (1,169 words) - 15:28, 13 May 2024
• 2000 AIME I Problems/Problem 4
  The diagram shows a [[rectangle]] that has been dissected into nine non-overlapping [[square]]s ...engths from smallest to largest <math>a_1,\ldots,a_9</math>, and let <math>l,w</math> represent the dimensions of the rectangle.
  3 KB (485 words) - 00:31, 19 January 2024
• 2001 AIME I Problems/Problem 12
  A [[sphere]] is inscribed in the [[tetrahedron]] whose vertices are <math>A = (6,0,0), B = (0,4,0), C = (0,0,2),</math> and <math>D = (0,0,0).</math> triple A = (6,0,0), B = (0,4,0), C = (0,0,2), D = (0,0,0);
  6 KB (1,050 words) - 18:44, 27 September 2023
• 2001 AIME I Problems/Problem 6
  A fair die is rolled four times. The [[probability]] that each of the final t ...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
• 2003 AIME II Problems/Problem 13
  ...ne of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bu ...o return to the original vertex, the net number of clockwise steps must be a multiple of 3, i.e., <math>\#CW - \#CCW \equiv 0 \pmod{3}</math>. Since <m
  15 KB (2,406 words) - 23:56, 23 November 2023
• 2001 AIME II Problems/Problem 15
  ...th>EI = EJ = EK = 2</math>. A solid <math>S</math> is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to <ma ...ection = orthographic(camera=(1/4,2,3/4)); defaultpen(linewidth(0.7)); pen l = linewidth(0.5) + linetype("10 2");
  4 KB (518 words) - 15:01, 31 December 2021
• 2000 AIME II Problems/Problem 12
  ...ts <math>A</math>, <math>B</math> and <math>C</math> lie on the surface of a [[sphere]] with center <math>O</math> and radius <math>20</math>. It is giv ...eron's Formula]] the area of <math>\triangle ABC</math> is (alternatively, a <math>13-14-15</math> triangle may be split into <math>9-12-15</math> and <
  3 KB (532 words) - 13:14, 22 August 2020
• Integration by parts
  The purpose of integration by parts is to replace a difficult [[integral]] with an easier one. The formula is ...an "easy" (or "easier") [[derivative]] and <math>dv</math> so that it has a easy [[antiderivative]].
  1 KB (235 words) - 17:01, 11 March 2022
• Locus
  ...used most frequently in [[geometry]], to denote a set of points satisfying a certain geometric condition. ...can be defined as the locus of all points that are a certain distance from a given center.
  890 bytes (160 words) - 02:58, 22 March 2011
• Platonic solid
  A '''Platonic solid''' is a [[polyhedron]], or 3 dimensional figure, in which all [[face]]s are [[congr ...(1+5^.5)/2; currentprojection=orthographic(2,-2.2,2); pen d=linewidth(0.7),l=linetype("2 6"); triple P[];
  8 KB (1,168 words) - 22:48, 19 February 2022
• University of South Carolina High School Math Contest/1993 Exam/Problems
  If the width of a particular rectangle is doubled and the length is increased by 3, then the <cmath> \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D)
  14 KB (2,102 words) - 22:03, 26 October 2018
• Mock AIME 1 2006-2007 Problems/Problem 6
  ...integer]]s <math>a,b,c</math> where <math>\gcd(a,b,c)=1</math>, find <math>a+b+c</math>. ...or]] of the three numbers is 1, <math>a = 1, b = 5, c = 5</math> and <math>a + b + c = 011</math>.
  3 KB (460 words) - 15:52, 3 April 2012
• Mock AIME 1 2006-2007/Problems
  Let <math>\star (x)</math> be the sum of the digits of a positive integer <math>x</math>. <math>\mathcal{S}</math> is the set of pos ...\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
• 2006 Romanian NMO Problems/Grade 7/Problem 1
  ...<math>2 \cdot \frac{CN}{BC} = \frac{AM}{AB}</math>. Let <math>P</math> be a point on the line <math>AC</math>. Prove that the lines <math>MN</math> and Let <math>L</math> be a point on <math>BC</math> such that <math>N</math> is the midpoint of LC, th
  2 KB (288 words) - 21:17, 11 October 2013
• P versus NP
  .... <math>NP</math> is the class of decision problems that can be solved by a ''non-deterministic'' algorithm in polynomial time. The <math>P</math> ver 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
• 2005 AMC 10A Problems/Problem 4
  A rectangle with a [[diagonal]] of length <math>x</math> is twice as long as it is wide. What <math> \textbf{(A) } \frac{1}{4}x^2\qquad \textbf{(B) } \frac{2}{5}x^2\qquad \textbf{(C) } \f
  2 KB (265 words) - 19:07, 25 December 2022
• 2005 AMC 10A Problems/Problem 13
  <math> \textbf{(A) } 0\qquad \textbf{(B) } 7\qquad \textbf{(C) } 12\qquad \textbf{(D) } 65\qq ~JH. L
  2 KB (223 words) - 07:18, 16 July 2022
• 2005 AMC 10A Problems/Problem 25
  <math> \textbf{(A) } \frac{266}{1521}\qquad \textbf{(B) } \frac{19}{75}\qquad \textbf{(C) }\f pair A, B, C, D, E;
  6 KB (854 words) - 20:25, 24 July 2022
• Zero (constant)
  ...ashiers had to make change on a purchase of LXIV dollars with bills marked L, X, V and I when handed XC dollars.
  2 KB (371 words) - 12:55, 21 June 2023
• System of equations
  ...[[equation]]s which share the same [[variable]]s. Below is an example of a system of equations. <cmath>\left\{\begin{array}{l}a^2 + b^2 = 25\\a+b=7\end{array}\right.</cmath>
  5 KB (784 words) - 23:27, 30 July 2020
• Integral
  ...ath>c</math> is a [[real]] [[constant]]. This is because the derivative of a constant is <math>0</math>. *The integral of a function <math>f(x)</math> is written as <math>\int f(x)\,dx</math>, where
  5 KB (909 words) - 14:16, 31 May 2022
• Perpendicular bisector
  ...ch that <math>AB</math> and <math>l</math> are [[perpendicular]] and <math>l</math> passes through the [[midpoint]] of <math>AB</math>. ...hat plane. The [[set]] of lines which are perpendicular bisectors of form a plane which is the plane perpendicularly bisecting <math>AB</math>.
  2 KB (367 words) - 15:20, 1 January 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
• Roman numerals
  50 - '''L''' ...for 4, IV is used. To "subtract," a smaller-value letter is placed before a larger-value letter. This done to make numbers smaller (e.g. IX instead of
  865 bytes (140 words) - 13:58, 24 March 2019
• 2005 AMC 10A Problems/Problem 20
  <math> \textbf{(A) } \frac72\qquad \textbf{(B) } \frac{7\sqrt2}{2}\qquad \textbf{(C) } \fra ...c12</math>, so the area of <math>7</math> squares is <math>\boxed{\textbf{(A) }\frac72}</math>.
  3 KB (414 words) - 10:25, 15 May 2023
• Euler line
  ...] <math>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain fixed orders and distan ...followed by a homothety with scale factor <math>2</math> centered at <math>A</math> brings <math>\triangle ABC \to \triangle O_AO_BO_C</math>. Let us ex
  59 KB (10,203 words) - 04:47, 30 August 2023
• 1987 IMO Problems/Problem 2
  ...umcircle]] of <math>ABC </math> again at <math>N </math>. From point <math>L </math> perpendiculars are drawn to <math>AB </math> and <math>AC </math>, ...e cyclic quadrilateral <math>ABNC</math> gives <math>\angle BNC=180-\angle A</math>. Thus, we are to prove that
  4 KB (736 words) - 15:39, 21 September 2014
• 1987 IMO Problems
  ...ath> onto itself. An element <math>i </math> in <math>S </math> is called a fixed point of the permutation <math>f </math> if <math>f(i) = i </math>.) ...umcircle]] of <math>ABC </math> again at <math>N </math>. From point <math>L </math> perpendiculars are drawn to <math>AB </math> and <math>AC </math>,
  3 KB (459 words) - 14:24, 17 September 2023
• 2006 Seniors Pancyprian/2nd grade/Problem 4
  ...adrilateral <math>ABCD</math>, that has no parallel sides, is inscribed in a circle, its sides <math>DA</math>, <math>CB</math> meet at <math>E</math> a ...ZB</math> intersect the sides of the quadrilateral at the points <math>K , L, M ,N</math> prove that
  856 bytes (129 words) - 00:10, 20 February 2020
• 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: Reference
  A '''point''' is associated with a cartesian coordinate pair in Asymptote. There are two useful functions tha ...] for the difference) in Asymptote is simply a piecewise cubic function of a parameter <math>t</math>, parameterized as <math>t</math> ranges from <math
  7 KB (1,205 words) - 21:38, 26 March 2024
• 2005 AMC 10A Problems/Problem 23
  ...ts on the circle such that <math>DC \perp AB</math> and <math>DE</math> is a second diameter. What is the ratio of the area of <math>\triangle DCE</math pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);
  6 KB (1,045 words) - 09:46, 4 April 2023
• Mock AIME 1 Pre 2005 Problems
  <math>A, B, C, D,</math> and <math>E</math> are collinear in that order such that < ...th>1 + 7 + 7^2 + \cdots + 7^{2004}</math> is divided by <math>1000</math>, a remainder of <math>N</math> is obtained. Determine the value of <math>N</ma
  6 KB (1,100 words) - 22:35, 9 January 2016
• Mock AIME 3 Pre 2005 Problems
  A function <math>f(x)</math> is defined for all real numbers <math>x</math>. ...are consonants. A string of <math>M's, O's,</math> and <math>P's</math> is a word in Zuminglish if and only if between any two <math>O's</math> there ap
  7 KB (1,135 words) - 23:53, 24 March 2019
• Mock AIME 3 Pre 2005 Problems/Problem 14
  ...h>P</math>, intersecting <math>l</math> again at <math>Q</math>. Let <math>A</math> and <math>B</math> be the intersections of <math>\omega_1</math> and <math>A_1=\sqrt{s(s-a)(s-b)(s-c)}</math>, where <math>s=\frac{a+b+c}{2}=\frac{3+6+4}{2}=\frac{13}{2}</math> we have:
  3 KB (563 words) - 02:05, 25 November 2023
• 2007 AIME I Problems/Problem 10
  In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so th ...nother row that is complementary to the first. We remove those two and use a similar argument again to show that every group of <math>6</math> rows can
  13 KB (2,328 words) - 00:12, 29 November 2023
• 2007 Cyprus MO/Lyceum/Problem 1
  <math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } -2\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ ...an simplify the equation into <math>(1)^2+1=2</math>, <math>\boxed{\mathrm{A}}</math>.
  600 bytes (94 words) - 17:04, 17 March 2013
• 2007 Cyprus MO/Lyceum/Problem 12
  ...=x+y+xy+k\ \ \ \forall x,y \in \Re</math>, where <math>k \in \Re</math> is a constant. The value of <math>f(-1)</math> is <math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \
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• 2007 Cyprus MO/Lyceum/Problem 24
  ...e a profit of 10% on its value and from the sale of the second one, he had a loss of 10% on its value. After the sale of the two televisions Costas had <math> \mathrm{(A) \ } \mathrm{profit\ }4\qquad \mathrm{(B) \ } \mathrm{neither\ profit\ nor\
  880 bytes (132 words) - 02:33, 19 January 2024
• 2007 Cyprus MO/Lyceum/Problem 21
  ...r <math>20\,\mathrm{ cm}</math> represent pulleys, that are connected with a strap. If the distances between any two pulley center points are <math>AB=3 <math> \mathrm{(A) \ } (12+20\pi)\,\mathrm{m}\qquad \mathrm{(B) \ } (12 + \pi)\,\mathrm{m}\qq
  1 KB (175 words) - 22:21, 23 July 2020
• 2007 Cyprus MO/Lyceum/Problem 22
  ...apezium with <math>\angle A= \angle D=90^\circ</math> and bases <math>AB = a</math> , <math>DC = 2a</math> . If <math>AD = 3a</math> and <math>M</math> <math> \mathrm{(A) \ } \frac{3a}{2}\qquad \mathrm{(B) \ } \frac{3a}{\sqrt{2}}\qquad \mathrm{(
  979 bytes (166 words) - 02:33, 19 January 2024
• Pascal's Theorem
  pair A=dir(20), B=dir(40), D(L(D(IP(D(L(A,B,0.5,7)),D(L(E,D,0.5,1.5)))),
  4 KB (712 words) - 21:57, 25 April 2020
• Circumradius
  ...very triangle is [[cyclic]], every triangle has a circumscribed circle, or a [[circumcircle]]. ==Formula for a Triangle==
  4 KB (729 words) - 16:52, 19 February 2024
• Surface area
  ...the sum of the areas of its six [[square]] [[face]]s; the surface area of a [[tetrahedron]] is the sum of the area of its four [[triangle | triangular] ...ctangular prism, the surface area is <math>2\cdot (lw+hw+lh)</math>, where l,w, and h are the length, width and height, respectively.
  1 KB (209 words) - 01:34, 25 January 2016
• LaTeX:Layout
  ...his allows you to use the same code in many source files by just including a single line in each source file. ...font size, which is 10pt by default but can be increased to 11pt or 12pt. A reference on other options for this command can be found [http://www.nada.k
  30 KB (5,171 words) - 10:16, 4 April 2021
• Dodecagon
  A '''dodecagon''' is a 12-sided [[polygon]]. The sum of its internal [[angle]]s is <math>1800^{\ci A regular dodecagon can be seen below:
  1 KB (219 words) - 13:08, 15 June 2018
• 2005 AMC 12A Problems/Problem 22
  A rectangular box <math> P </math> is inscribed in a sphere of radius <math>r</math>. The surface area of <math>P</math> is 384, <math>\mathrm{(A)}\ 8\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 12\qquad \mathrm{(D)}\ 14\q
  2 KB (334 words) - 10:20, 16 September 2022
• Green's Theorem
  '''Green's Theorem''' is a result in [[real analysis]]. It is a special case of [[Stokes' Theorem]].
  2 KB (381 words) - 12:12, 30 May 2019
• 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
• 2000 PMWC Problems
  <math>y</math> is a number that has <math>8</math> different factors (including the number <mat 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
• 1997 PMWC Problems/Problem I6
  ...>6</math> small ones and had <math>\textdollar 5</math> left. How much was a small exercise book? <math>l=10</math>
  660 bytes (105 words) - 14:23, 20 April 2014
• 1997 PMWC Problems/Problem I11
  A rectangle <math>ABCD</math> is made up of five small congruent rectangles a pair A=(0,0),B=(0,-2.5),C=(3,-2.5),D=(3,0);
  944 bytes (154 words) - 12:44, 13 August 2014
• 1961 IMO Problems
  \quad x + y + z \!\!\! &= a \; \, \\ ...</math> and <math>b </math> are constants. Give the conditions that <math>a </math> and <math>b </math> must satisfy so that <math>x, y, z </math> (the
  3 KB (425 words) - 21:18, 20 August 2020
• 1961 IMO Problems/Problem 6
  ...o not form a triangle.) What is the locus of point <math>G</math> as <math>A', B', C'</math> range independently over the plane <math>\epsilon</math>? ...\prime B^\prime C^\prime</math>. It is obvious that the centroid of <math>A^\prime B^\prime C^\prime</math> can be any point on plane <math>\epsilon</m
  2 KB (301 words) - 23:29, 18 July 2016
• 1999 USAMO Problems/Problem 5
  ...ut producing SOS then the game is a draw. Prove that the second player has a winning strategy. ...n S in <math>l + 3</math> and if <math>l < m</math> he fills an S in <math>l-3</math>.
  2 KB (433 words) - 13:35, 4 July 2013
• 2006 Cyprus MO/Lyceum/Problem 12
  <math>\mathrm{(A)}\ 8\qquad\mathrm{(B)}\ 0\qquad\mathrm{(C)}\ 11\qquad\mathrm{(D)}\ 5\qquad\ {{CYMO box|year=2006|l=Lyceum|num-b=11|num-a=13}}
  701 bytes (92 words) - 16:51, 23 November 2016
• 2006 Cyprus MO/Lyceum/Problem 17
  ...Gamma</math> is equilateral triangle of side <math>\alpha</math> and <math>A\Delta=BE=\frac{\alpha}{3}</math>. The measure of the angle <math>\angle\Gam <math>\mathrm{(A)}\ 60^\circ\qquad\mathrm{(B)}\ 50^\circ\qquad\mathrm{(C)}\ 40^\circ\qquad\m
  789 bytes (123 words) - 22:00, 30 November 2015
• 2006 Cyprus MO/Lyceum/Problem 19
  ...[isosceles triangle]] with<math> AB=A\Gamma=\sqrt2</math> and <math>\angle A=45^\circ</math>. If <math>B\Delta</math> is an [[altitude]] of the [[triang <math>\mathrm{(A)}\ \frac{4\sqrt3-\pi}{6}\qquad\mathrm{(B)}\ 4\left(\sqrt2-\frac{\pi}{3}\rig
  1 KB (214 words) - 23:44, 22 December 2016
• 2006 Cyprus MO/Lyceum/Problem 29
  {{CYMO box|year=2006|l=Lyceum|num-b=28|num-a=30}}
  99 bytes (12 words) - 23:19, 19 December 2019
• 1960 IMO Problems/Problem 6
  ...ne of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies i a) Prove that <math>V_1 \neq V_2</math>;
  7 KB (1,214 words) - 18:49, 29 January 2018
• Asymptote: Useful functions
  The bundled <code>markers</code> package provides a useful function for marking angles. The exact same function exists in the < Label ''L''="", int ''n''=1, real ''radius''=0, real ''space''=0,<br/>
  4 KB (646 words) - 21:18, 26 March 2024
• Asymptote: CSE5
  ...of two paths without common points, which may be useful if you try to draw a picture just from its verbal description and do not want to make any calcul CSE5 contains doubles of each function; a function with a full name and a function with an abbreviated name.
  4 KB (607 words) - 15:09, 30 June 2020
• 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>. It is well known that the excircle opposite <math>A</math> is tangent to <math>\overline{BC}</math> at the point <math>D_2</mat
  11 KB (2,091 words) - 08:35, 16 November 2017
• L'Hôpital's Rule
  '''L'Hopital's Rule''' is a theorem dealing with [[limit]]s that is very important to [[calculus]]. The definition of a derivative is <math>f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}</ma
  2 KB (475 words) - 15:04, 24 March 2022
• Squeeze Theorem
  ...so called the '''Sandwich Theorem''' or the '''Squeeze Play Theorem''') is a relatively simple [[theorem]] that deals with [[calculus]], specifically [[ ...as <math>x</math> approaches <math>S</math>, then <math>\lim_{x\to S}f(x)=L</math>.
  2 KB (439 words) - 15:36, 1 December 2015
• De Longchamps point
  label("L",(34,50),(1,1)); </asy>|right|The de Longchamps <br />point (<math>L</math>) is the the <br />[[orthocenter]] (<math>H</math>) reflected <br />
  10 KB (1,780 words) - 09:23, 17 November 2022
• Graph (graph theory)
  ...[[empty set | nonempty]] [[set]] of [[vertex|vertices]] that are joined by a number (possibly zero) of [[edge]]s. Graphs are frequently represented gra ...' graphs: there is at most one edge joining two vertices, no edge may join a vertex to itself, and the edges are not directed. For graphs with multiple
  8 KB (1,428 words) - 10:26, 27 August 2020
• 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
• 2007 AMC 10A Problems/Problem 21
  ...n a [[cube]] that has a [[surface area]] of <math>24</math> square meters. A second cube is then inscribed within the sphere. What is the surface area i <math>\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 9 \qquad
  4 KB (609 words) - 14:41, 3 December 2023
• 1951 AHSME Problems/Problem 2
  A rectangular field is half as wide as it is long and is completely enclosed <math>(\mathrm{A})\ \frac{x^2}2 \qquad (\mathrm{B})\ 2x^2 \qquad (\mathrm{C})\ \frac{2x^2}9
  743 bytes (121 words) - 12:19, 5 July 2013
• 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
• Construction
  ...to mark off segments, draw circles and arcs, and draw straight lines) are a branch of [[geometry]] that rely on the use of basic geometrical [[axiom]]s A '''compass''' is a tool that can draw circles and arcs of circles.
  3 KB (443 words) - 20:52, 28 August 2014
• 2007 AMC 10A Problems/Problem 18
  ...f its sides has length <math>4</math>, and each two consecutive sides form a right angle. Suppose that <math>\overline{AG}</math> and <math>\overline{CH ..., D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1,1), J=(0,1), K=(0,2), L=(1,2);
  6 KB (867 words) - 00:17, 20 May 2023
• 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
• 1976 IMO Problems/Problem 2
  ...math>P_{k+1}(x)</math>. Let <math>P_1(r)=s</math>, where <math>s</math> is a real root of <math>P_k(x)</math>. We have that <math>-2<s<2</math>, so <mat ...th>. However, <math>|s|<2</math> from our inductive hypothesis, so this is a contradiction. Therefore <math>P_{k+1}(x)</math> has no double roots. This
  3 KB (596 words) - 16:19, 28 July 2015
• Shoelace Theorem
  The '''Shoelace Theorem''' is a nifty formula for finding the [[area]] of a simple [[polygon]] given the [[Cartesian coordinate system|coordinates]] of ...<math>(x_n, y_n)</math>, listed in clockwise order. Then the area (<math>A</math>) of <math>P</math> is
  8 KB (1,358 words) - 15:32, 22 February 2024
• 0.999...
  ...on>0</math>, there exists an integer <math>N>0</math> such that <math>|a_n-L|<\epsilon</math> for all integers greater than <math>N</math>. So in fact, ...r. But how do we know that there are no 'gaps' so it could possibly not be a real number. But lets prove it.
  3 KB (577 words) - 20:04, 4 February 2023
• 2008 USAMO Problems/Problem 3
  (''Gabriel Carroll'') Let <math>n</math> be a positive integer. Denote by <math>S_n</math> the set of points <math>(x, y) .../math> paths (a partition of <math>S_n</math> into <math>m</math> paths is a set <math>\mathcal{P}</math> of <math>m</math> nonempty paths such that eac
  9 KB (1,585 words) - 01:00, 14 August 2014
• Normal subgroup
  ...al subgroup''' <math>{\rm H}</math> of a [[group]] <math>{\rm G}</math> is a [[subgroup]] of <math>{\rm G}</math> for which the relation "<math>xy^{-1} <math>N</math> is said to be a normal subgroup of a group <math>G</math> if <math>aNa^{-1}=N</math>.Note that this means <math>
  15 KB (2,840 words) - 12:22, 9 April 2019
• Free magma
  ...'free magma''' is [[magma]] structure that is as general as possible&mdash;a magma generated from an initial set with no constraints or relations. The free magma generated from a [[set]] <math>X</math> is constructed as follows.
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• How should I prepare?
  ...' are an excellent resource to help prepare for math contests. They cover a broad range of topics, from algebra to geometry to number theory to combina ...tests to see if the books are on the right level for you. '''Alcumus''' is a good resource even if you do not own any of the AoPS books.
  13 KB (1,926 words) - 11:22, 30 November 2023
• 2003 USAMO Problems/Problem 6
  ...numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices. ...ath>b+d+f</math> must be odd. WLOG let <math>a+c+e</math> be odd and <math>a\ge c\ge e \ge 0</math>.
  5 KB (739 words) - 13:39, 4 July 2013
• 2001 IMO Shortlist Problems/N1
  Suppose that there is such a number <math>n</math>. Let <math>a</math> be the number of digits of <math>(n + 8)!</math>, and let <math>b</m <math>8 \cdot 10^{a-1} \leqslant (n + 8)! < 9 \cdot 10^{a-1}</math>
  3 KB (440 words) - 08:14, 1 April 2022
• 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 ...24, 2000 at the [[Collège de France]]. [[Timothy Gowers]] first presented a lecture titled ''The Importance of Mathematics'' as an introduction. After
  13 KB (1,969 words) - 17:57, 22 February 2024
• 1990 AHSME Problems/Problem 26
  Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. T <math>\textbf{(A) } 1 \qquad
  3 KB (402 words) - 23:17, 23 September 2023
• 1995 IMO Problems/Problem 1
  ...h>XY</math> meets <math>BC</math> at <math>Z</math>. Let <math>P</math> be a point on the line <math>XY</math> other than <math>Z</math>. The line <math Think about Radical Axis, Power of a Point and Radical Center.
  5 KB (847 words) - 19:03, 12 October 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
• 2001 USAMO Problems/Problem 6
  Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its in '''Lemma''': If <math>ABCD</math> is an isosceles trapezoid, then <math>a + c = b + d</math>.
  2 KB (302 words) - 18:11, 22 February 2016
• 1992 USAMO Problems/Problem 4
  ...<math>C</math>, and <math>P</math> is tangent to the sphere through <math>A'</math>, <math>B'</math>, <math>C'</math>, and <math>P</math>. Prove that ...ause the base angles are equal. Thus, <math>AP=BP</math>. Similarly, <math>A'P=B'P</math>. Thus, <math>AA'=BB'</math>. By symmetry, <math>BB'=CC'</math>
  5 KB (807 words) - 18:37, 25 June 2021
• 1972 USAMO Problems/Problem 2
  A given tetrahedron <math>ABCD</math> is isosceles, that is, <math>AB=CD, AC= In this case, the tetrahedron is not actually a tetrahedron, so this maximum isn't actually attainable.
  8 KB (1,458 words) - 23:42, 27 February 2022
• 1974 USAMO Problems/Problem 4
  A father, mother and son hold a family tournament, playing a two person board game with no ties. The tournament rules are: ...ath>L</math> denote a game the father loses, and let <math>x</math> denote a game the father does not participate in. Then, with the knowledge that the
  6 KB (1,037 words) - 23:42, 11 May 2018
• 1962 IMO Problems/Problem 6
  ...h>I</math>. Extend <math>AI</math> to meet the circumcircle again at <math>L</math>. Then extend <math>LO</math> so it meets the circumcircle again at < ...1}{2}\angle ABC + \angle CBL = \frac {1}{2}\angle ABC + \frac {1}{2}\angle A</math>. Therefore, <math>BIL</math> is isosceles, and <math>IL = BL</math>.
  2 KB (308 words) - 06:29, 16 December 2023
• 1985 AJHSME Problems/Problem 19
  If the [[length]] and width of a [[rectangle]] are each increased by <math>10\% </math>, then the [[perimete <math>\text{(A)}\ 1\% \qquad \text{(B)}\ 10\% \qquad \text{(C)}\ 20\% \qquad \text{(D)}\ 2
  1 KB (219 words) - 09:04, 22 January 2023
• 2004 AMC 10B Problems/Problem 21
  <math> \mathrm{(A) \ } 3722 \qquad \mathrm{(B) \ } 3732 \qquad \mathrm{(C) \ } 3914 \qquad \m ...are <math>A=\{ 3k+1 : 0\leq k < 2004 \}</math> and <math>B=\{ 7l+9 : 0\leq l<2004\}</math>.
  2 KB (357 words) - 16:20, 5 May 2024
• 2004 AMC 10B Problems/Problem 4
  A standard six-sided die is rolled, and <math>P</math> is the product of the <math> \mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 12 \qquad \mathrm{(C) \ } 24 \qquad \mathrm{(
  2 KB (348 words) - 23:42, 16 January 2024
• 2004 AMC 12B Problems/Problem 14
  pair C=(0,0), A=(0,5), B=(12,0), M=(0,4), N=(4,0); pair J=intersectionpoint(A--B, M--(M+rotate(90)*(B-A)) );
  7 KB (1,083 words) - 22:41, 23 November 2020
• 2009 AMC 10A Problems/Problem 16
  ...and <math>|c-d|=4</math>. What is the sum of all possible values of <math>|a-d|</math>? \mathrm{(A)}\ 9
  3 KB (524 words) - 16:26, 23 June 2023
• 2009 AMC 10A Problems/Problem 21
  ...portions containing a ring of congruent circles that are circumscribed by a larger circle, In the figure shown, the number of smaller circles is four. \mathrm{(A)}\ 3-2\sqrt2
  3 KB (474 words) - 12:50, 29 September 2023
• 2009 AMC 10B Problems/Problem 25
  ...om and independently for each face. What is the probability that there is a continuous stripe encircling the cube? <math>\mathrm{(A)}\frac 18\qquad
  4 KB (637 words) - 04:52, 2 July 2022
• 2009 AMC 10B Problems/Problem 22
  ...pieces as shown in this top view, where <math>M</math> is the midpoint of a top edge. The piece whose top is triangle <math>B</math> contains <math>c</ label("$A$",(-0.4,0.7));
  12 KB (1,868 words) - 03:36, 30 September 2023
• Mock AIME 1 Pre 2005 Problems/Problem 10
  ...h>ABCDEFG</math> at <math>A</math>, and <math>P</math> is a point on <math>l</math> such that triangle <math>AOP</math> is isosceles. Let <math>p</math> {{Mock AIME box|year=Pre 2005|n=1|num-b=9|num-a=11|source=14769}}
  540 bytes (96 words) - 00:28, 23 December 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 5
  ...AC = 450</math> and <math>BC = 300</math>. Points <math>K</math> and <math>L</math> are located on <math>\overline{AC}</math> and <math>\overline{AB}</m pair A=(0,0), B=(30*sqrt(331),0), C, K, L, M, P;
  8 KB (1,224 words) - 19:52, 7 March 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
• 2000 AMC 8 Problems
  <math>\mathrm{(A)}\ 15\qquad\mathrm{(B)}\ 16\qquad\mathrm{(C)}\ 17\qquad\mathrm{(D)}\ 21\qqu <math>\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qqua
  15 KB (2,165 words) - 03:32, 13 April 2024
• 2008 IMO Problems/Problem 6
  ...s a circle <math>\omega</math> tangent to ray <math>BA</math> beyond <math>A</math> and to the ray <math>BC</math> beyond <math>C</math>, which is also L be the foot of the projection of K to DC.
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• Radical axis
  The theory of radical axis is a priceless geometric tool that can solve formidable geometric problems fairl ...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
• 1968 AHSME Problems/Problem 35
  MP("a",(5,0),S); MP("L",(-8,2),S);
  3 KB (597 words) - 01:52, 16 August 2023
• 2009 USAMO Problems/Problem 5
  pair A = (-.6, .8), B = (.6, .8), C = (.9, -sqrt(.19)), D = (-.9, -sqrt(.19)), G = draw(A--B--C--D--cycle); draw(B--D);
  6 KB (973 words) - 19:24, 18 October 2018
• 2009 IMO Problems/Problem 2
  ...ctively, and let <math>\Gamma</math> be the circle passing through <math>K,L</math> and <math>M</math>. Suppose that the line <math>PQ</math> is tangent dot("A", (40, 100), 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 ...ctively, and let <math>\Gamma</math> be the circle passing through <math>K,L</math> and <math>M</math>. Suppose that the line <math>PQ</math> is tangent
  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
• Tower law
  ...>[L:K]</math> and <math>[K:F]</math> are both [[finite]] then so is <math>[L:F]</math> and <cmath>[L:F] = [L:K][K:F].</cmath>
  3 KB (567 words) - 08:42, 21 August 2009
• Substitution
  '''Substitution''' is when one replaces all instances of a variable (or expression) with another equivalent variable (or expression). ...s. It is generally introduced in a first year high school algebra class. A solution generally exists when the number of equations is exactly equal to
  3 KB (428 words) - 16:53, 10 April 2020
• 2010 AMC 12A Problems/Problem 7
  ...town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower ho <math>\textbf{(A)}\ 0.04 \qquad \textbf{(B)}\ \frac{0.4}{\pi} \qquad \textbf{(C)}\ 0.4 \qqua
  1 KB (198 words) - 18:08, 28 June 2021
• 2010 AMC 12A Problems/Problem 21
  <math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \ [Quick note: Since we don't know <math>a</math>, <math>b</math>, and <math>c</math>, we really don't even need the l
  5 KB (969 words) - 21:33, 22 June 2022
• 2010 AMC 12A Problems/Problem 12
  In a magical swamp there are two species of talking amphibians: toads, whose sta Chris: "LeRoy is a frog."
  2 KB (413 words) - 20:08, 31 January 2024
• 2010 AMC 12B Problems
  <math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ A big <math>L</math> is formed as shown. What is its area?
  12 KB (1,845 words) - 13:00, 19 February 2020
• Characteristic polynomial
  The '''characteristic polynomial''' of a linear [[operator]] refers to the [[polynomial]] whose roots are the [[eige ...<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
• Eigenvalue
  ...or space]]; in particular, it is a subspace of the domain of the map <math>L</math>.
  821 bytes (138 words) - 19:32, 2 March 2010
• Proofs without words
  rt(a^2 - ab + b^2) rt(b^2 - bc + c^2) < rt(a^2 + ac + c^2) ...ck(pair A, pair B, pair ticklength = (0.15,0)){ draw(A--B ^^ A-ticklength--A+ticklength ^^ B-ticklength--B+ticklength); }
  55 KB (7,986 words) - 17:04, 20 December 2018
• Linearly independent
  ...ath>v_1, v_2, \ldots, v_n</math> in a [[vector space]] <math>V</math> over a field <math>K</math> are '''linearly independent''' if there do not exist s ...rly independent vectors, that is, if <math>\{v_1, \ldots, v_n\}</math> are a basis, then <math>\{v_1, \ldots, v_n, w\}</math> for any vector <math>w \in
  2 KB (300 words) - 23:35, 16 March 2010
• 2010 AIME I Problems/Problem 11
  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)); draw(Arc((8,7.67),A,G,CW),dark,EndArrow(8)); draw(B--C--G--cycle,linetype("4 4"));
  4 KB (636 words) - 16:46, 25 November 2023
• 2010 AIME I Problems/Problem 13
  [[Rectangle]] <math>ABCD</math> and a [[semicircle]] with diameter <math>AB</math> are coplanar and have nonoverl label("$A$",(-16.43287,-9.3374),NE/2);
  10 KB (1,418 words) - 23:05, 20 October 2021
• 2007 IMO Problems
  (a) Prove that, for any real numbers <math>x_1\le x_2\le \cdots\le x_n</math>, ...h> such that <math>ABCD</math> is a parallelogram and <math>BCED</math> is a cyclic quadrilateral.
  3 KB (505 words) - 09:24, 10 September 2020
• 2010 AIME II Problems
  A point <math>P</math> is chosen at random in the interior of a unit square <math>S</math>. Let <math>d(P)</math> denote the distance from ...where <math>a</math> and <math>b</math> are integers satisfying <math>1\le a < b \le 20</math>. Find the greatest positive integer <math>n</math> such t
  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
• 2010 USAMO Problems/Problem 4
  Let <math>ABC</math> be a triangle with <math>\angle A = 90^{\circ}</math>. Points <math>D</math> void ldot(pair p, string l, pair dir=p) { dot(p); label(l, p, unit(dir)); }
  7 KB (1,230 words) - 19:47, 31 January 2024
• 2010 USAJMO Problems/Problem 1
  ...Find with proof the smallest <math>n</math> such that <math>P(n)</math> is a multiple of <math>2010</math>. ...> is closed under multiplication and a non-square times a square is always a non-square.
  12 KB (2,338 words) - 20:30, 13 February 2024
• 2010 USAMO Problems/Problem 1
  Let <math>AXYZB</math> be a convex pentagon inscribed in a semicircle of diameter Since <math>XY</math> is a chord of the circle with diameter <math>AB</math>,
  13 KB (2,178 words) - 14:14, 11 September 2021
• 1995 AJHSME Problems
  <math>\text{(A)}\ 4\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 4 <math>\text{(A)}\ 8 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 14 \qquad \text{(D)}\ 16 \qqu
  14 KB (2,096 words) - 18:29, 2 January 2023
• Scholarships by State
  ...ships.asp Junior League of Rockford Outstanding Volunteer Scholarship] for a high school senior in Winnebago County. * [[The James L. and Nellie M. Westlake Scholarship Program]]
  9 KB (1,107 words) - 15:32, 16 July 2010
• American Society of Civil Engineers Maine Section
  ...he award is based on academic performance, extracurricular activities, and a statement of interest in civil engineering. Applications are available from Leslie L. Corrow, P.E.
  540 bytes (75 words) - 12:29, 20 June 2010
• 2010 AMC 12B Problems/Problem 2
  A big <math>L</math> is formed as shown. What is its area? <math>\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\
  802 bytes (110 words) - 10:58, 4 July 2013
• 2010 IMO Problems/Problem 4
  ...> meet again its circumcircle <math>\Gamma</math> at <math>K</math>, <math>L</math>, respectively <math>M</math>. The tangent line at <math>C</math> to ...thout loss of generality]], suppose that <math>AS > BS</math>. By Power of a Point, <math>SP^2 = SC^2 = SB \cdot SA</math>, so <math>\overline{SP}</math
  6 KB (895 words) - 18:32, 2 April 2021
• 2010 AMC 12B Problems/Problem 21
  Let <math>a > 0</math>, and let <math>P(x)</math> be a polynomial with integer coefficients such that <math>P(1) = P(3) = P(5) = P(7) = a</math>,
  7 KB (1,166 words) - 00:04, 26 February 2024
• 2010 IMO Problems/Problem 2
  ...Let <math>E</math> be a point on arc <math>BDC</math>, and <math>F</math> a point on the segment <math>BC</math>, such that <math>\angle BAF=\angle CAE Observation 3. As a result, we have <math>\angle{KID} = \angle{IJD} = \angle{DAE} = \angle{FAD}
  3 KB (525 words) - 14:52, 16 July 2023
• 2010 IMO Problems/Problem 6
  ...ldots</math> be a sequence of positive real numbers, and <math>s</math> be a positive integer, such that So for solving This Problem, we need to take a assumption that,
  4 KB (786 words) - 08:46, 12 March 2024
• 2010 AMC 10A Problems/Problem 7
  ...ile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last por \mathrm{(A)}\ 1
  3 KB (370 words) - 00:16, 29 June 2022
• 2010 AMC 10B Problems/Problem 7
  ...le has side lengths <math>10</math>, <math>10</math>, and <math>12</math>. A rectangle has width <math>4</math> and area equal to the <math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\
  2 KB (318 words) - 09:00, 1 September 2022
• 2010 AMC 10A Problems/Problem 21
  ...three positive integer roots. What is the smallest possible value of <math>a</math>? <math>\textbf{(A)}\ 78 \qquad \textbf{(B)}\ 88 \qquad \textbf{(C)}\ 98 \qquad \textbf{(D)}\
  4 KB (664 words) - 10:44, 28 March 2024
• 2010 IMO Problems
  ...ft\lfloor a\right\rfloor</math> is greatest integer not greater than <math>a.</math> ...Let <math>E</math> be a point on arc <math>BDC</math>, and <math>F</math> a point on the segment <math>BC</math>, such that <math>\angle BAF=\angle CAE
  4 KB (603 words) - 09:22, 10 September 2020
• 2004 AMC 8 Problems
  ...ter length represents <math>72</math> kilometers. How many kilometers does a <math>17</math>-centimeter length represent? <math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 102\qquad\textbf{(C)}\ 204\qquad\textbf{(D)}\ 864\q
  13 KB (1,860 words) - 19:58, 8 May 2023
• 2006 AMC 8 Problems
  <math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qqu <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 19\qquad
  16 KB (2,215 words) - 19:18, 10 April 2024
• '''Hölder's Inequality,''' a generalization of the '''Cauchy-Schwarz inequality''', states that, ...can also be generalized further to <math> n</math> sets of variables with a similar form.
  1 KB (197 words) - 12:12, 29 October 2016
• 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
• 1998 AJHSME Problems
  <math>\text{(A)}\ \dfrac{6}{x} \qquad \text{(B)}\ \dfrac{6}{x+1} \qquad \text{(C)}\ \dfrac ...text{b}\cdot \text{c}</math>, what is the value of <math>\begin{tabular}{r|l}3&4 \\ \hline 1&2\end{tabular}</math>?
  14 KB (1,920 words) - 19:31, 31 January 2024
• Hölder's Inequality
  Note that with two sequences <math>\mathbf{a}</math> and <math>\mathbf{b}</math>, and <math>\lambda_a = \lambda_b = 1/2< ...nces of nonnegative reals, and let <math>\{ \lambda_i \}_{i=1}^n</math> be a sequence of nonnegative reals such that <math>\sum \lambda = 1</math>. The
  4 KB (762 words) - 00:11, 18 June 2023
• 1999 AMC 8 Problems/Problem 15
  ...ee letters. The first is chosen from the set {C,H,L,P,R}, the second from {A,I,O}, and the third from {D,M,N,T}. <math>\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 40 \qq
  3 KB (516 words) - 14:50, 21 December 2022
• 2003 AMC 12A Problems/Problem 14
  Points <math>K, L, M,</math> and <math>N</math> lie in the plane of the square <math>ABCD</ma pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5);
  3 KB (483 words) - 17:26, 22 June 2023
• 2011 AMC 10A Problems/Problem 4
  <math> \textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\q From here it is obvious that <math>Y - X = 102 - 10 = \boxed{92 \ \mathbf{(A)}}</math>.
  2 KB (323 words) - 12:07, 19 October 2023
• 2011 AMC 10A Problems/Problem 16
  <math>\text{(A)}\,3\sqrt2 \qquad\text{(B)}\,2\sqrt6 \qquad\text{(C)}\,\frac{7\sqrt2}{2} \q We can change the insides of the square root into a perfect square and then simplify.
  4 KB (558 words) - 20:43, 21 August 2023
• 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 10A Problems/Problem 22
  Each vertex of convex pentagon <math>ABCDE</math> is to be assigned a color. There are <math>6</math> colors to choose from, and the ends of each <math> \textbf{(A)}\ 2520\qquad\textbf{(B)}\ 2880\qquad\textbf{(C)}\ 3120\qquad\textbf{(D)}\
  14 KB (2,425 words) - 09:13, 5 November 2023
• Named after Guillaume de l'Hopital; however, it is believed that the rule was actually discovered by J ...> case, the limit <cmath> \lim_{x \to a} \cfrac {f(x)}{g(x)} = \lim_{x \to a} \cfrac {f'(x)}{g'(x)} </cmath> where <math> f'(x) </math> and <math> g'(x)
  4 KB (764 words) - 23:10, 2 January 2012
• 2012 IMO Problems/Problem 1
  ...the lines <math>AB</math> and <math>AC</math> at <math>K</math> and <math>L</math>, respectively. The lines <math>LM</math> and <math>BJ</math> meet at .../math>. Since <math>AK</math> and <math>AL</math> are tangents from <math>A</math> to the circle <math>J</math>, <math>AK = AL</math>. Then, we have <
  7 KB (1,189 words) - 01:22, 19 November 2023
• 2011 AIME I Problems/Problem 4
  ...ngle <math>A</math> intersects <math> \overline{BC} </math> at point <math>L</math>, and the angle bisector of angle <math>B</math> intersects <math> \o pair A,B,C,D,K,L,M,N,P,Q;
  6 KB (1,068 words) - 18:52, 2 August 2023
• 2011 AIME I Problems
  ...olution in jar <math>C</math> is added to jar B. At the end both jar <math>A</math> and jar <math>B</math> contain solutions that are <math>50\%</math> .../math> the <math>y</math>-axis. In the new coordinate system, point <math>A</math> is on the positive <math>x</math>-axis, and point <math>B</math> is
  10 KB (1,634 words) - 22:21, 28 December 2023
• 2011 AIME I Problems/Problem 1
  ...remainder of the solution in jar C is added to jar B. At the end both jar A and jar B contain solutions that are 50% acid. Given that <math>m</math> a Jar A contains <math>\frac{11}{5}</math> liters of water, and <math>\frac{9}{5}</
  4 KB (786 words) - 16:43, 5 February 2022
• 2011 AIME I Problems/Problem 3
  .../math> the <math>y</math>-axis. In the new coordinate system, point <math>A</math> is on the positive <math>x</math>-axis, and point <math>B</math> is ...nt <math>A=(24,-1)</math>, we may write the point-slope equation for <math>L</math> as <math>y+1=\frac{5}{12}(x-24)</math>.
  4 KB (699 words) - 16:01, 13 April 2024
• Files (Python)
  ...d whether we want to read or write to the [[open function]], which returns a file object. For more details see the open function article. Any file we read from must already exist, but if a file we try to write to doesn't exist Python will create it for us. Note a
  3 KB (483 words) - 00:24, 18 April 2011
• 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 Note that the lines <math>l(P_1), l(P_2), l(P_3)</math> are <cmath>y=2a_1x-a_1^2, y=2a_2x-a_2^2, y=2a_3x-a_3^2,</cmath>
  15 KB (2,593 words) - 13:37, 29 January 2021
• KGS math club/solution 11 6
  Here is a set of relative speeds of the horses, to solve the problem: ...t.head).iterate(>>= (\l->[a:l|a<-[1..headl-1],all(\b->a*(a+1)`mod`(b-a)==0)l])).map(:[])$[1..]
  749 bytes (70 words) - 10:55, 11 May 2011
• 2000 AMC 8 Problems/Problem 6
  ...s are drawn with the side lengths as labeled. The area of the shaded <math>L</math>-shaped region is pair A,B,C,D;
  2 KB (365 words) - 21:02, 28 July 2023
• 2011 AMC 10B Problems/Problem 6
  <math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 39 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 57 x &= \boxed{\textbf{(A)} 30}
  1 KB (220 words) - 05:34, 25 June 2022
• 2011 AMC 10B Problems/Problem 9
  pair A=(0,0), B=(5,0), C=(1.8,2.4), D=(5-4sqrt(3)/3,0), E=(5-4sqrt(3)/3,sqrt(3)); pair[] ps={A,B,C,D,E};
  4 KB (587 words) - 22:08, 31 August 2023
• 2011 AMC 10B Problems/Problem 16
  A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. W pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)),
  4 KB (606 words) - 06:14, 25 June 2022
• 2011 AMC 10B Problems/Problem 10
  <math> \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 11 \q ...rac{10^{10}}{10^9 + 10^8 + \ldots + 10 + 1}.</cmath> Using the formula for a geometric series, we have <cmath>10^9 + 10^8 + \ldots + 10 + 1 = \dfrac{10^
  2 KB (230 words) - 22:43, 31 August 2023
• Olympiad books
  Here is a list of '''Olympiad Books''' that have Olympiad-level problems used to trai *''Inequalities: A Mathematical Olympiad Approach'' - '''Radmila Bulajich Manfrino, Jose Anton
  17 KB (2,261 words) - 00:30, 22 April 2024
• 2007 AMC 10B Problems/Problem 21
  <math>\textbf{(A) } \frac{3}{2} \qquad\textbf{(B) } \frac{60}{37} \qquad\textbf{(C) } \frac{ Find the length of the altitude of <math>\triangle ABC.</math> Since it is a right triangle, the area of <math>\triangle ABC</math> is <math>\frac{1}{2}
  2 KB (335 words) - 16:17, 18 February 2024
• 2011 AMC 12B Problems/Problem 14
  ...to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math> <math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \t
  4 KB (619 words) - 19:15, 11 May 2021
• Block walking
  ...r right or down. We start off with some easier means of this problem from A to B by labeling closer distances. Let's label these points with letters. label("L", (3,3), NE);
  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
• 1998 AJHSME Problems/Problem 2
  ...text{b}\cdot \text{c}</math>, what is the value of <math>\begin{tabular}{r|l}3&4 \\ \hline 1&2\end{tabular}</math>? <math>\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qqua
  797 bytes (126 words) - 15:40, 20 October 2016
• 2003 AMC 10B Problems/Problem 20
  pair A=(0,0), B=(5,0), C=(5,3), D=(0,3); draw(A--B--C--D--cycle);
  5 KB (898 words) - 17:26, 3 July 2023
• 1999 AMC 8 Problems/Problem 4
  for (int a = 0; a < 6; ++a) dot((4*a,3*b));
  1 KB (196 words) - 21:02, 14 April 2017
• 2015 IMO Problems/Problem 4
  ...le of triangle <math>BDF</math> and the segment <math>AB</math>. Let <math>L</math> be the second point of intersection of the circumcircle of triangle ...etric wrt <math>AO</math>, a diameter of <math>\Omega</math> through <math>A</math>. So are <math>F</math> and <math>G</math>, as <math>AF=AG</math>. Th
  3 KB (502 words) - 23:58, 5 October 2015
• 2000 AMC 8 Problems/Problem 16
  In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 t <math>\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500
  948 bytes (143 words) - 20:05, 15 April 2023
• Simson line
  In [[geometry]], given a [[triangle]] ABC and a point P on its [[circumcircle]], the three closest points to P on lines AB, 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
• Autograph
  Autograph is a piece of code written in [[Asymptote]] that graphs functions. The user must This code is currently developed by PythonNut, a AoPS user. The changes are kept on his blog.
  17 KB (2,910 words) - 15:01, 2 September 2011
• 2011 IMO Problems
  ...inct positive integers which achieve the largest possible value of <math>n(A)</math>. ...d the line now rotates clockwise about <math>Q</math>, until it next meets a point of <math>S</math>. This process continues indefinitely.
  4 KB (682 words) - 00:12, 18 February 2021
• 1966 IMO Problems/Problem 6
  ...<math>BC, CA, AB</math> of triangle <math>ABC</math>, any points <math>K, L,M</math>, respectively, are selected. Prove that the area of at least one o ...hs of sides <math>BC</math>, <math>CA</math>, and <math>AB</math> be <math>a</math>, <math>b</math>, and <math>c</math>, respectively. Let <math>BK=d</m
  2 KB (397 words) - 00:37, 17 May 2015
• 1999 AMC 8 Problems/Problem 22
  In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is <math>\text{(A)}\ \frac{3}{8} \qquad \text{(B)}\ \frac{1}{2} \qquad \text{(C)}\ \frac{3}{4
  963 bytes (160 words) - 22:28, 24 January 2024
• 2002 AMC 8 Problems/Problem 10
  ...nd by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and for ( int y = 0; y &l;= 5; ++y )
  2 KB (253 words) - 22:32, 17 February 2024
• 2000 AMC 8 Problems/Problem 25
  ...\overline{BC} </math> and <math> \overline{CD} </math> are joined to form a triangle, the area of that triangle is pair A,B,C,D;
  3 KB (484 words) - 13:59, 22 October 2023
• 1999 AMC 8 Problems/Problem 25
  label("$A$",(0,0),SW); label("$L$",(5.25,4.5),S);
  4 KB (655 words) - 05:21, 12 March 2024
• 1997 AJHSME Problems/Problem 12
  pair H,I,J,K,L; H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0);
  1 KB (188 words) - 12:37, 2 May 2021
• 1992 AJHSME Problems/Problem 11
  The bar graph shows the results of a survey on color preferences. What percent preferred blue? for (int a = 1; a <= 6; ++a)
  2 KB (226 words) - 00:09, 5 July 2013
• 1997 AHSME Problems/Problem 5
  A rectangle with perimeter <math>176</math> is divided into five congruent re <math> \mathrm{(A)\ } 35.2 \qquad \mathrm{(B) \ }76 \qquad \mathrm{(C) \ } 80 \qquad \mathrm
  1 KB (231 words) - 14:12, 5 July 2013
• 2008 AMC 8 Problems/Problem 2
  <math>\textbf{(A)}\ 8671 \qquad ...ath>, and <math>E=1</math>. Therefore, the answer is <math>\boxed{\textbf{(A)}\ 8671}</math>
  596 bytes (87 words) - 16:23, 2 January 2024
• 1992 AJHSME Problems/Problem 21
  Northside's Drum and Bugle Corps raised money for a trip. The drummers and bugle players kept separate sales records. According for (int a = 1; a <= 6; ++a)
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• 1993 AJHSME Problems/Problem 13
  ...block letters is painted in black with strokes <math>1</math> unit wide on a <math>5</math> by <math>15</math> rectangular white sign with dimensions as <math>\text{(A)}\ 30 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 34 \qquad \text{(D)}\ 36 \qq
  1 KB (177 words) - 04:55, 25 November 2019
• 1995 AJHSME Problems/Problem 18
  The area of each of the four congruent L-shaped regions of this 100-inch by 100-inch square is 3/16 of the total are <math>\text{(A)}\ 25 \qquad \text{(B)}\ 44 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 62 \qq
  895 bytes (142 words) - 12:53, 12 November 2017
• 2003 AMC 8 Problems/Problem 18
  ...one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following 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''' Let <math>x_1,x_2,\dots,x_n</math> be a sequence of integers, such that <math>-1\leq x_i\leq 2</math>, for <math>i=
  25 KB (4,154 words) - 16:27, 2 September 2011
• 2006 AMC 8 Problems/Problem 10
  ...which <math> w </math> is the width and <math> l </math> is the length of a rectangle with area 12. What should his graph look like? <math> \textbf{(A)} </math> <asy>
  2 KB (227 words) - 20:15, 19 March 2019
• 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
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• 1985 AHSME Problems/Problem 17
  ...parallel lines <math>L</math> and <math>L'</math> that pass through <math>A</math> and <math>C</math> and are perpendicular to <math>DB</math>. The are real x=sqrt(6), y=sqrt(3), a=0.4;
  2 KB (340 words) - 22:39, 19 March 2024
• 2010 AMC 8 Problems
  <math> \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qq If <math>a @ b = \frac{a\times b}{a+b}</math> for <math>a,b</math> positive integers, then what is <math>5 @ 10</math>?
  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
• 2011 AMC 8 Problems/Problem 19
  pair A,B,C,D,E,F,G,H,I,J,K,L; A=(0,0);
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• 1970 Canadian MO Problems/Problem 6
  ...ruct a circle with centre <math>C</math> such that the tangents from <math>A</math> and <math>B</math> are parallel. ...ct lines <math>l</math> parallel to the line <math>MC</math> through <math>A</math>, and line <math>k</math> parallel to <math>MC</math> through <math>B
  2 KB (293 words) - 21:24, 21 December 2011
• 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
• 2011 AMC 8 Problems/Problem 7
  ..., C=(0,4), D=(4,4), F=(2,0), G=(3,0), H=(1,4), I=(2,4), J=(3,4), K=(0,-2), L=(4,-2), M=(0,-6), O=(0,-4), P=(4,-4), Q=(2,-2), R=(2,-6), T=(6,4), U=(10,0) draw(C--H--(1,0)--A--cycle,linewidth(1.6)); draw(M--O--Q--R--cycle,linewidth(1.6)); draw(A_1--V
  4 KB (557 words) - 00:30, 4 January 2023
• 2011 AMC 8 Problems/Problem 9
  Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of he ...f); label("M",(-1.39,5.32),SE*lsf); label("I",(-1.34,4.93),SE*lsf); label("L",(-1.36,4.51),SE*lsf); label("E",(-1.37,4.11),SE*lsf); label("S",(-1.39,3.7
  2 KB (279 words) - 20:27, 17 April 2021
• 1950 AHSME Problems/Problem 40
  ...> \frac {x^2-1}{x-1}</math> as <math>x</math> approaches <math>1</math> as a limit is: <math>\textbf{(A)}\ 0 \qquad
  1 KB (205 words) - 13:43, 4 April 2024
• 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 19
  ...me amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two he <math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qqu
  3 KB (422 words) - 17:11, 21 August 2021
• 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 12A Problems/Problem 22
  Distinct planes <math>p_1,p_2,....,p_k</math> intersect the interior of a cube <math>Q</math>. Let <math>S</math> be the union of the faces of <math> <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
• 1989 AHSME Problems/Problem 13
  pair a = (0,0),b= (6,0),c=(0,1),d=(6,1); pair e = t*a,f=t*b,g=t*c,h=t*d;
  2 KB (242 words) - 07:51, 22 October 2014
• 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
• 2012 AMC 12B Problems/Problem 21
  <math> \textbf{(A)}\ 29\sqrt{3} \qquad\textbf{(B)}\ \frac{21}{2}\sqrt{2}+\frac{41}{2}\sqrt{3} pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*di
  9 KB (1,451 words) - 10:01, 27 December 2022
• 2012 AMC 10B Problems/Problem 15
  In a round-robin tournament with 6 teams, each team plays one game against each <math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\t
  2 KB (307 words) - 20:04, 13 July 2021
• 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 ...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
• 2007 iTest Problems/Problem 15
  Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and ...one of its sides coincides with a side of the square. Then "circumscribe" a circle around the
  3 KB (539 words) - 21:01, 29 July 2018
• 2012 AIME II Problems/Problem 4
  ...tially heading south, and Cao bikes in a straight line across the field to a point <math>D</math> on the south edge of the field. Cao arrives at point < label("$a$", (0.6,0),dir(-90));
  5 KB (796 words) - 00:17, 20 January 2024
• 2012 AIME II Problems/Problem 10
  ...integers <math>n</math> less than <math>1000</math> for which there exists a positive real number <math>x</math> such that <math>n=x\lfloor x \rfloor</m We know that <math>x</math> cannot be irrational because the product of a rational number and an irrational number is irrational (but <math>n</math>
  3 KB (541 words) - 13:52, 7 May 2023
• 2008 iTest Problems/Problem 6
  ...ght triangle with legs 5 and 12. Find the least integer greater than <math>L</math>. ...by <math>2</math> and dividing both sides by <math>13</math>, we get <math>L = \frac{60}{13}</math>. The least integer greater than <math>\frac{60}{13}<
  915 bytes (135 words) - 00:52, 22 June 2018
• 1988 USAMO Problems/Problem 4
  ...ta IAB</math>, <math>\Delta IBC</math>, and <math>\Delta ICA</math> lie on a circle whose center is the circumcenter of <math>\Delta ABC</math>. ...>, and <math>O_b</math>, respectively. It then suffices to show that <math>A</math>, <math>B</math>, <math>C</math>, <math>O_a</math>, <math>O_b</math>,
  3 KB (504 words) - 19:25, 14 October 2021
• 1974 AHSME Problems/Problem 30
  A line segment is divided so that the lesser part is to the greater part as t <math> \mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }2R \qquad \mathrm{(C) \ } R^{-1} \qquad \mathr
  1 KB (229 words) - 19:22, 19 March 2014
• 2006 SMT/Geometry Problems
  Given a cube, determine the ratio of the volume of the octahedron formed by connect ...ath> and <math> X </math> is on the same side of line <math> AB </math> as a given point <math> C </math>.
  5 KB (837 words) - 12:22, 27 May 2012
• 2010 AMC 8 Problems/Problem 3
  ....8), E=(1,-5.6), F=(1,-5.4), G=(1,-5.2), H=(1,-5), J=(1,-4.8), K=(1,-4.6), L=(1,-4.4), M=(1,-4.2), N=(1,-4), P=(1,-3.8), Q=(1,-3.6), R=(1,-3.4), S=(1,-3 ...D(B--(8,-2),linewidth(0.4)); D(B--S_1); D(T--R_1); D(N--Q_1); D(H--P_1); D(A--O_1); D(C_1--E_1); D(E_1--F_1); D(F_1--D_1); D(D_1--C_1); D(G_1--L_2); D(L
  5 KB (847 words) - 19:59, 5 February 2023
• 2006 SMT/Geometry Problems/Problem 4
  ...ath> and <math> X </math> is on the same side of line <math> AB </math> as a given point <math> C </math>. label("$A$",(-1,0),SW);
  4 KB (622 words) - 15:07, 28 May 2012
• 1952 AHSME Problems
  If the radius of a circle is a rational number, its area is given by a number which is: ...ational} \qquad \textbf{(C)\ } \text{integral} \qquad \textbf{(D)\ } \text{a perfect square }\qquad \textbf{(E)\ } \text{none of these} </math>
  23 KB (3,556 words) - 15:35, 30 December 2023
• 1980 AHSME Problems
  <math>\text{(A)} \ 12 \qquad \text{(B)} \ 13 \qquad \text{(C)} \ 14 \qquad \text{(D)} \ 15 The degree of <math>(x^2+1)^4 (x^3+1)^3</math> as a polynomial in <math>x</math> is
  15 KB (2,302 words) - 10:47, 30 April 2021
• 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
• 1989 AHSME Problems/Problem 5
  Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpi for(int a:x){
  2 KB (295 words) - 07:43, 22 October 2014
• 2012 IMO Problems
  ...the lines <math>AB</math> and <math>AC</math> at <math>K</math> and <math>L</math>, respectively. The lines <math>LM</math> and <math>BJ</math> meet at (The excircle of <math>ABC</math> opposite the vertex <math>A</math> is the circle that is tangent to the line segment <math>BC</math>,
  5 KB (834 words) - 11:00, 24 February 2021
• Bolzano-Weierstrass Theorem
  ...hat every bounded [[sequence]] of real numbers <math>(a_n)</math> contains a convergent subsequence. ...<math>N</math> such that for each <math>k\ge N</math> that <math>\mathcal{L}(I_k)<\epsilon</math>. Since <math>a_{n_k}</math> and <math>x</math> are c
  1 KB (270 words) - 17:18, 12 June 2022
• 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
• 2012 AMC 8 Problems/Problem 5
  pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; A=(4,0);
  2 KB (208 words) - 20:16, 3 January 2024
• 2004 AMC 8 Problems/Problem 23
  pair J=(-3,2); pair K=(-3,-2); pair L=(3,-2); pair M=(3,2); draw(J--K--L--M--cycle);
  2 KB (308 words) - 17:26, 25 July 2021
• 2005 AMC 8 Problems/Problem 22
  ...detergent in three different sized boxes: small (S), medium (M) and large (L). The medium size costs 50% more than the small size and contains 20% less <math>\text{(A)}\, SML \qquad \text{(B)}\, LMS \qquad \text{(C)}\, MSL \qquad \text{(D)}\,
  1 KB (185 words) - 20:36, 15 January 2024
• 2013 AMC 12A Problems
  {{AMC12 Problems|year=2013|ab=A}} <math>\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \
  14 KB (2,206 words) - 19:31, 15 May 2024
• 2013 AMC 12A Problems/Problem 11
  real s=1/2,m=5/6,l=1; pair A=origin,B=(l,0),C=rotate(60)*l,D=(s,0),E=rotate(60)*s,F=m,G=rotate(60)*m;
  2 KB (322 words) - 20:12, 24 November 2020
• 2013 AMC 10A Problems/Problem 4
  A softball team played ten games, scoring <math>1</math>, <math>2</math>, <ma <math> \textbf{(A)}\ 35 \qquad\textbf{(B)}\ 40 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 50
  2 KB (303 words) - 21:40, 4 July 2023
• 2013 AMC 12B Problems/Problem 1
  On a particular January day, the high temperature in Lincoln, Nebraska, was <mat <math>\textbf{(A)}\ -13 \qquad \textbf{(B)}\ -8 \qquad \textbf{(C)}\ -5 \qquad \textbf{(D)}\
  934 bytes (141 words) - 23:30, 15 April 2022
• 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
  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</ <cmath>b^2+a^2+2\cdot a\cdot b\cdot \cos(\angle ADC)=9.\qquad (6)</cmath>
  13 KB (2,116 words) - 23:24, 21 March 2024
• 2003 AMC 12B Problems/Problem 10
  pair A = (5,0), B = A+10*dir(72), C = (0,r+h), E = (-5,0), D = E+10*dir(108); draw(A--B--C--D--E--cycle);
  1 KB (237 words) - 23:06, 3 February 2020
• 2012 IMO Problems/Problem 5
  ...segment <math>AX</math> such that <math>BK=BC</math>. Similarly, let <math>L</math> be the point on the segment <math>BX</math> such that <math>AL=AC</m ...umcircle of triangle <math>ABC</math>, the circle with its center as <math>A</math> and radius as <math>AC</math>, and the circle with its center as <ma
  2 KB (437 words) - 08:30, 20 November 2023
• 2014 AIME II Problems/Problem 11
  ...me positive integers, and <math>b</math> is a positive integer. Find <math>a+b+c</math>. ...ath>\overline{PM}\parallel\overline{CD}</math>. Thus, <math>APME</math> is a parallelogram and <math>AE = PM = \frac{CD}{2}</math>. We can then use coor
  6 KB (1,059 words) - 18:24, 20 January 2024
• 2014 AIME II Problems/Problem 14
  In <math>\triangle{ABC}, AB=10, \angle{A}=30^\circ</math> , and <math>\angle{C=45^\circ}</math>. Let <math>H, D,</ma ...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
• 1996 USAMO Problems/Problem 3
  ...ABC</math> and the interior of its reflection <math>A'B'C'</math> in <math>l</math> has area more than <math>\frac{2}{3}</math> the area of triangle <ma Let the triangle be <math>ABC</math>. Assume <math>A</math> is the largest angle. Let <math>AD</math> be the altitude. Assume <m
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• Symmetry
  A proof utilizes '''symmetry''' if the steps to prove one thing is identical Denote <math>A' = \omega \cap BC, C' = \omega \cap AB.</math>
  19 KB (3,292 words) - 13:04, 13 May 2024
• 2013 IMO Problems/Problem 4
  ...M</math>, and let <math>Y</math> be the point such that <math>WY</math> is a diameter of <math>\omega_2</math>. Prove that <math>X, Y</math> and <math>H Draw a good diagram, or use the one below. What do you notice? (In particular, wha
  11 KB (1,991 words) - 01:31, 19 November 2023
• 1967 AHSME Problems/Problem 11
  <math>\textbf{(A)}\ 0\qquad <cmath> AC = \sqrt{l^2 + w^2} </cmath>
  849 bytes (126 words) - 20:00, 28 September 2023
• 2016 AMC 10A Problems
  {{AMC10 Problems|year=2016|ab=A}} <math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\
  14 KB (2,104 words) - 22:26, 16 September 2022
• 1995 USAMO Problems/Problem 4
  (a) <math>m - n </math> divides <math>q_m - q_n</math> for <math>m>n \geq 0</m (b) There is a polynomial <math>P</math> such that <math>|q_n|<P(n)</math> for all <math>n
  3 KB (502 words) - 15:27, 11 May 2018
• 1969 IMO Problems/Problem 5
  ...ld be collinear. Also no points lie inside the area of ABC because if such a point D did exist, then the angle DAC would be less than BAC - contradictio ...s, you can create a convex quadrilateral containing two of them and two of A,B,C.
  2 KB (394 words) - 13:44, 29 January 2021
• 1971 IMO Problems/Problem 5
  ...th>m</math> points in <math>S</math> which are at unit distance from <math>A</math>. ...ean distance between the points is <math>1</math>) and this will follow as a consequence.
  4 KB (749 words) - 14:09, 29 January 2021
• 1973 IMO Problems/Problem 4
  A soldier needs to check on the presence of mines in a region having the ...this problem must go from <math>A</math> to a point on <math>c_2</math> to a point on <math>c_1</math>. Let us first find the shortest possible path. We
  3 KB (502 words) - 23:48, 12 December 2022
• 1974 IMO Problems/Problem 2
  ...tric mean of <math>AD</math> and <math>DB</math> if and only if <math>\sin{A}\sin{B} \leq \sin^2 (\frac{C}{2})</math>. Let a point <math>D</math> on the side <math>AB</math>.
  3 KB (553 words) - 15:56, 29 January 2021
• 1974 IMO Problems/Problem 5
  ...a+b+d}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{a+c+d}</cmath> where <math>a, b, c, d,</math> are arbitrary positive numbers. ...{c+d} > S > \frac{a}{a+b+c+d}+\frac{b}{a+b+c+d}+\frac{c}{a+b+c+d}+\frac{d}{a+b+c+d} = 1.</cmath> We will now prove that <math>S</math> can reach any ran
  2 KB (400 words) - 16:00, 29 January 2021
• 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
• 2013 AMC 8 Problems/Problem 18
  Isabella uses one-foot cubical blocks to build a rectangular fort that is <math>12</math> feet long, <math>10</math> feet wi triple A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P;
  3 KB (412 words) - 22:30, 18 December 2023
• 1994 AHSME Problems
  <math> \textbf{(A)}\ 13^{13} \qquad\textbf{(B)}\ 13^{36} \qquad\textbf{(C)}\ 36^{13} \qquad\t A large rectangle is partitioned into four rectangles by two segments paralle
  14 KB (2,124 words) - 13:39, 19 February 2020
• 1952 AHSME Problems/Problem 17
  ...give a discount of <math> 20\% </math> of the marked price and still make a profit of <math> 20\% </math> of the selling price. The per cent of the lis <math> \textbf{(A) \ }20 \qquad \textbf{(B) \ }100 \qquad \textbf{(C) \ }125 \qquad \textbf{
  1 KB (186 words) - 01:44, 3 January 2014
• 1981 AHSME Problems
  <math> \textbf{(A)}\ \sqrt{2}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\ <math> \textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ \sqrt{5}\qquad\textbf{(C)}\ 3\qquad\textbf{(
  17 KB (2,633 words) - 15:44, 16 September 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 7
  ...ath>, <math>y</math>, <math>a</math>, and <math>b</math> satisfy <math>x < a</math> and <math>y < b</math>. How many of the following inequalities must <math>\textbf{(I)}\ x+y < a+b\qquad</math>
  2 KB (263 words) - 16:35, 8 September 2021
• 2014 AMC 10A Problems/Problem 10
  Five positive consecutive integers starting with <math>a</math> have average <math>b</math>. What is the average of <math>5</math> c ...(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}\ a+6\qquad\textbf{(E)}\ a+7</math>
  2 KB (320 words) - 00:07, 27 June 2023
• 2014 AMC 10A Problems/Problem 13
  pair A = rotate(60,B)*C; pair E = rotate(270,A)*B;
  7 KB (1,040 words) - 02:07, 8 October 2022
• 2014 AMC 10A Problems/Problem 14
  ...two perpendicular lines intersecting at the point <math>A(6,8)</math> have a sum of zero. What is the area of <math>\triangle APQ</math>? <math> \textbf{(A)}\ 45\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qqu
  6 KB (1,001 words) - 13:07, 25 July 2022
• 2014 AMC 10A Problems/Problem 15
  David drives from his home to the airport to catch a flight. He drives <math>35</math> miles in the first hour, but realizes th <math>\textbf{(A) }140\qquad
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• 2014 AMC 10A Problems/Problem 16
  pair A = (0,2); draw(A--B--C--D--cycle);
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• 2014 AMC 10A Problems/Problem 17
  <math> \textbf{(A)}\ \dfrac16\qquad\textbf{(B)}\ \dfrac{13}{72}\qquad\textbf{(C)}\ \dfrac7{36 ...(2, 2, 4); (3, 3, 6)</math> have <math>3</math> ways to rearrange them for a total of <math>9</math> ways. Adding them up, we get <math>45</math> ways.
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• 2014 AMC 10A Problems/Problem 18
  A square in the coordinate plane has vertices whose <math>y</math>-coordinate <math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qqu
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• 2014 AMC 10A Problems/Problem 22
  ...ine{AB}=20</math> and <math>\overline{BC}=10</math>. Let <math>E</math> be a point on <math>\overline{CD}</math> such that <math>\angle CBE=15^\circ</ma <math> \textbf{(A)}\ \dfrac{20\sqrt3}3\qquad\textbf{(B)}\ 10\sqrt3\qquad\textbf{(C)}\ 18\qqua
  12 KB (1,821 words) - 18:16, 29 October 2023
• 2014 AMC 10A Problems/Problem 23
  ...ate a new shape with area <math>B</math>. What is the ratio <math>\frac{B}{A}</math>? real L = 0.05;
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• 1994 AHSME Problems/Problem 2
  A large rectangle is partitioned into four rectangles by two segments paralle <math> \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 21
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• 1994 AHSME Problems/Problem 23
  In the <math>xy</math>-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at Label l;
  2 KB (393 words) - 04:22, 28 May 2021
• 2000 PMWC Problems/Problem I14
  A copy machine has a following enlargement/reduction buttons: <cmath>\begin{tabular}{|l|l|l|l|l|l|l|}
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• 2014 USAJMO Problems/Problem 2
  ...riangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60^{\circ}</math>, and let <math>O</math> and <math>H</math> denote the cir (a) Prove that line <math>OH</math> intersects both segments <math>AB</math> a
  7 KB (1,197 words) - 19:40, 28 February 2023
• 2018 AIME I Problems/Problem 14
  Let <math>SP_1P_2P_3EP_4P_5</math> be a heptagon. A frog starts jumping at vertex <math>S</math>. From any vertex of the heptag ...arrow P_3 \leftrightarrow E</math>. We can count the number of left/right (L/R) paths of length <math>\le 11</math> that start at <math>S</math> and end
  4 KB (649 words) - 17:29, 22 December 2023
• 2007 iTest Problems/Problem 12
  <math>\text{(A) }\frac{2}{3}\qquad \text{(L) }2007\qquad</math>
  3 KB (500 words) - 04:44, 10 June 2018
• 2014 AMC 10B Problems/Problem 20
  <math> \textbf {(A) } 8 \qquad \textbf {(B) } 10 \qquad \textbf {(C) } 12 \qquad \textbf {(D) ...th> is negative. Then we have <math>x^2<51</math>. We know that if we find a positive number that works, it's parallel negative will work. Therefore, we
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• 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> pair A=(1,0), B=(0,0), C=(0,1), D=(1,1), E=(2-sqrt(3),0), F=(2-sqrt(3),1), G=(1,sq
  8 KB (1,307 words) - 02:07, 1 January 2023
• Mock AIME 3 2006-2007 Problems/Problem 11
  <math>2r\cos(a)(r\cos(a)+r\sin(a))=6+r^2</math> <math>2\cos(a)(\cos(a)+\sin(a))=\frac{6}{r^2}+1</math>
  4 KB (699 words) - 01:53, 30 April 2022
• 2014 AIME I Problems/Problem 4
  Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west dir ...arly, the second train has to cover a distance equal to its own length, at a rate of <math>r_2 + \dfrac{1}{3}</math>. Since the times are equal and <mat
  5 KB (784 words) - 13:59, 30 November 2021
• 2014 AIME I Problems/Problem 11
  ...ns and independently of the other moves. The probability the token ends at a point on the graph of <math>|y|=|x|</math> is <math>\frac{m}{n}</math>, whe ...sformation <math>(x, y)\rightarrow (x+y, x-y)</math>. Then we can see that a movement up, right, left, or down in the old coordinates adds the vectors <
  11 KB (1,677 words) - 23:54, 4 February 2022
• 2014 AIME I Problems/Problem 13
  ...nts <math>\overline{EG}</math> and <math>\overline{FH}</math> intersect at a point <math>P</math>, and the areas of the quadrilaterals <math>AEPH, BFPE, pair A = (0,sqrt(850));
  9 KB (1,404 words) - 21:07, 13 October 2023
• 1991 AHSME Problems/Problem 22
  MP("A",(-4/3,8*sqrt(2)/3),W);MP("A'",(4/3,8*sqrt(2)/3),E); ...>\overline{PA'B'}</math> are common tangents with <math>A</math> and <math>A'</math> on the smaller circle <math>B</math> and <math>B'</math> on the lar
  2 KB (295 words) - 19:09, 11 October 2016
• 1969 AHSME Problems/Problem 35
  .../math> and <math>y=m</math>, where <math>-6<m<6</math>. Let <math>r=[L(-m)-L(m)]/m</math>. Then, as <math>m</math> is made arbitrarily close to zero, th <math>\text{(A) arbitrarily close to } 0\quad\\
  2 KB (399 words) - 13:19, 27 July 2018
• 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
• 1980 AHSME Problems/Problem 29
  <cmath>\begin{array}{l} x^2-3xy+2y^2-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array} < <math>\text{(A)} \ 0 \qquad
  975 bytes (122 words) - 10:14, 15 June 2021
• 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 ...of the line <math>y = \frac{2}{3} x + 4</math>, then an equation for <math>L</math> is:
  17 KB (2,512 words) - 18:30, 12 October 2023
• 2007 iTest Problems/Problem 13
  <math>\text{(A) } 3 \quad \text{(L) } 14 \quad
  1 KB (212 words) - 14:35, 4 August 2018
• 2007 iTest Problems/Problem 14
  <math>\text{(A) } 0 \quad \text{(L) } 11 \quad
  2 KB (265 words) - 21:23, 4 February 2023
• 2001 IMO Problems
  ...at <math>\angle C \geq \angle B+30^{\circ}</math>. Prove that <math>\angle A+\angle COP < 90^{\circ}</math>. ...ath>a,b,c</math> be positive real numbers. Prove that <math>\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1</math>
  2 KB (343 words) - 21:52, 15 April 2021
• 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
• 1972 AHSME Problems
  <math>\text{(A) I and II}\qquad If a dealer could get his goods for <math>8</math>% less while keeping his selli
  17 KB (2,732 words) - 13:54, 20 February 2020
• 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. <math>\text{(A) }0\qquad
  2 KB (393 words) - 17:01, 10 June 2018
• 2007 iTest Problems/Problem 17
  <math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad \text{(L) }\frac{4}{7}\qquad
  2 KB (266 words) - 21:30, 4 February 2023
• 2007 iTest Problems/Problem 16
  <math>\text{(A) }1\qquad \text{(L) }311\qquad
  3 KB (420 words) - 20:28, 17 June 2018
• 2007 iTest Problems/Problem 21
  James writes down fifteen 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{(L) } 11 \quad
  929 bytes (137 words) - 22:05, 10 January 2019
• 2007 iTest Problems/Problem 25
  <math>\text{(A) } 0\qquad \text{(L) } 11\qquad
  2 KB (270 words) - 14:35, 29 July 2018
• 2007 iTest Problems/Problem 23
  <math>\text{(A) } 0\quad \text{(L) } 6\quad
  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
• Mock AIME 5 2005-2006 Problems/Problem 2
  ...he line tangent to <math>\omega_2</math> at <math>A</math> intersect <math>l</math> at <math>C</math>. Find the area of <math>\triangle ABC</math>. {{Mock AIME box|year=2005-2006|n=5|source=76847|num-b=1|num-a=3}}
  752 bytes (117 words) - 21:16, 8 October 2014
• 2007 IMO Problems/Problem 4
  ...h>BC</math> is <math>K</math> and the midpoint of <math>AC</math> is <math>L</math>. Prove that the triangles <math>RPK</math> and <math>RQL</math> have 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
  ...}</math>. Point <math>H</math> is the foot of the perpendicular from <math>A</math> to <math>BD</math>. Points <math>S</math> and <math>T</math> lie on pair A = (0,1), C=(0,-1);
  6 KB (1,071 words) - 03:58, 8 September 2018
• 2014 IMO Problems/Problem 4
  label("$A$", (1.880000000000002,3.760000000000004), NE * labelscalefactor); Let <math>L</math> be the midpoint of <math>BC</math>. Easy angle chasing gives <math>\
  13 KB (1,803 words) - 12:00, 19 May 2024
• 1965 AHSME Problems
  <math>\textbf{(A)}\ 0 \qquad A regular hexagon is inscribed in a circle. The ratio of the length of a side of the hexagon to the length of the
  19 KB (2,873 words) - 18:57, 16 August 2023
• 1963 AHSME Problems
  <math>\textbf{(A)}\ (0,0)\qquad <math>\textbf{(A)}\ -14 \qquad
  20 KB (3,122 words) - 14:17, 20 February 2020
• 2008 UNCO Math Contest II Problems
  Let <math>S = {a,b,c,d}</math> be a set of four positive integers. If pairs of distinct elements of <math>S</ma ...btained: <math>5, 10, 11, 13, 14, 19.</math> Determine the values of <math>a,
  6 KB (871 words) - 22:13, 7 November 2014
• 2008 UNCO Math Contest II Problems/Problem 6
  Points <math>A</math> and <math>B</math> are on the same side of line <math>L</math> in the plane. <math>A</math> is <math>5</math> units away
  2 KB (252 words) - 22:28, 18 March 2018
• 2007 UNCO Math Contest II Problems/Problem 9
  A circle is inscribed in an equilateral triangle whose side your answer as an exact multiple of <math>\pi</math> (and not as a
  3 KB (412 words) - 18:49, 29 January 2018
• 1988 AHSME Problems/Problem 9
  ...), E=(32,0), F=(48,0), G=(48,16), H=(32,16), I=(0,8), J=(10,8), K=(10,16), L=(32,6), M=(40,6), N=(40,16); draw(A--B--C--D--A^^E--F--G--H--E^^I--J--K^^L--M--N);
  2 KB (367 words) - 08:24, 12 July 2022
• 1987 AHSME Problems/Problem 13
  ...nto a roll for cash registers by wrapping it <math>600</math> times around a cardboard tube of diameter <math>2</math> cm, forming a roll <math>10</math> cm in diameter. Approximate the length of the paper in
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• 1986 AHSME Problems/Problem 2
  ...of the line <math>y = \frac{2}{3} x + 4</math>, then an equation for <math>L</math> is: <math>\textbf{(A)}\ y = \frac{1}{3} x + 8 \qquad
  884 bytes (137 words) - 18:01, 1 April 2018
• 1986 AHSME Problems/Problem 6
  Using a table of a certain height, two identical blocks of wood are placed as shown in Figure <math>\textbf{(A) }28\text{ inches}\qquad\textbf{(B) }29\text{ inches}\qquad\textbf{(C) }30\
  2 KB (221 words) - 18:11, 1 April 2018
• 2016 AMC 12A Problems
  {{AMC12 Problems|year=2016|ab=A}} <math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\
  15 KB (2,348 words) - 17:20, 19 January 2024
• 2014 AMC 8 Problems
  <math>\textbf{(A) }-10\qquad\textbf{(B) }-6\qquad\textbf{(C) }0\qquad\textbf{(D) }6\qquad \t Paul owes Paula 35 cents and has a pocket full of 5-cent coins, 10-cent coins, and 25-cent coins that he can u
  13 KB (1,957 words) - 12:08, 13 January 2024
• 2014 AMC 8 Problems/Problem 15
  ...math>12</math> equal arcs, marked the letters <math>A</math> through <math>L</math> as seen below. What is the number of degrees in the sum of the angle ...let = {"$A$","$B$","$C$","$D$","$E$","$F$","$G$","$H$","$I$","$J$","$K$","$L$"};
  2 KB (280 words) - 11:10, 2 July 2023
• 2015 AMC 10A Problems/Problem 10
  <math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\t ...of it. For example, if we start with an <math>a</math>, we can only place a <math>c</math> or <math>d</math> next to it. Unfortunately, after that ste
  3 KB (525 words) - 19:17, 10 August 2023
• 2015 AMC 10A Problems/Problem 19
  <math> \textbf{(A) }\dfrac{5\sqrt{2}}{3}\qquad\textbf{(B) }\dfrac{50\sqrt{3}-75}{4}\qquad\tex ...ath> right triangle and a <math>30-60-90</math> right triangle by dropping a perpendicular from <math>D</math> to side <math>AC</math>. Let <math>F</mat
  9 KB (1,513 words) - 23:44, 13 May 2024
• 2015 AMC 12B Problems
  <math>\textbf{(A)}\; -2 \qquad\textbf{(B)}\; \dfrac{1}{16} \qquad\textbf{(C)}\; \dfrac{7}{4} Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishe
  13 KB (2,064 words) - 13:39, 1 October 2022
• 2015 AMC 12B Problems/Problem 7
  ...which it has rotational symmetry is <math>R</math> degrees. What is <math>L+R</math> ? <math>\textbf{(A)}\; 24 \qquad\textbf{(B)}\; 27 \qquad\textbf{(C)}\; 32 \qquad\textbf{(D)}\;
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• 2015 AMC 12B Problems/Problem 9
  ...erson to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is <math>\tfrac{1}{2}</math>, indepe <math>\textbf{(A)}\; \dfrac{1}{2} \qquad\textbf{(B)}\; \dfrac{3}{5} \qquad\textbf{(C)}\; \df
  2 KB (302 words) - 16:19, 7 September 2017
• 2015 AMC 10B Problems/Problem 11
  ...the positive integers less than <math>100</math>, each of whose digits is a prime number, one is selected at random. What is the probability that the s <math>\textbf{(A)} \dfrac{8}{99}\qquad \textbf{(B)} \dfrac{2}{5}\qquad \textbf{(C)} \dfrac{9
  2 KB (270 words) - 17:13, 2 August 2022
• 2015 AIME I Problems/Problem 7
  ...and <math>\overline{AB}</math>, respectively, so that <math>KLMN</math> is a square. The area of <math>KLMN</math> is 99. Find the area of <math>FGHJ</m pair A,B,C,D,E,F,G,H,J,K,L,M,N;
  6 KB (1,105 words) - 21:02, 9 November 2023
• 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 ...ers <math>a</math> and <math>c</math>. Can you tell me the values of <math>a</math> and <math>c</math>?"
  8 KB (1,326 words) - 19:15, 13 January 2024
• 1998 IMO Problems/Problem 3
  First we must <math>d</math>etermine gener<math>a</math>l values for <math>d(n)</math>: ...a2 * .. * pc ^ ac</math>, if <math>d</math> is an ar<math>b</math>itr<math>a</math>ry divisor of <math>n</math> then <math>d</math> must have the same p
  3 KB (566 words) - 23:48, 18 November 2023
• 1998 IMO Problems/Problem 5
  BC, CA, and AB at K, L, and M , respectively. The line through B parallel {{IMO box|year=1998|num-b=4|num-a=6}}
  335 bytes (61 words) - 23:50, 18 November 2023
• 2015 AIME II Problems/Problem 7
  pair A,B,C,E,F,P,Q,R,S; A=(48/5,36/5);
  7 KB (1,180 words) - 14:08, 14 February 2023
• 2015 AIME II Problems/Problem 15
  ...xternal tangent of the two circles. A line <math>\ell</math> through <math>A</math> intersects <math>\mathcal{P}</math> again at <math>D</math> and inte pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689);
  31 KB (5,086 words) - 19:15, 20 December 2023
• 2016 AIME I Problems
  ...and <math>1</math> satisfy <math>S(a)S(-a)=2016</math>. Find <math>S(a)+S(-a)</math>. ...th> is directly proportional to <math>k</math>. The probability of rolling a <math>7</math> with this pair of dice is <math>\frac{m}{n}</math>, where <m
  8 KB (1,360 words) - 12:19, 29 January 2022
• 2016 AMC 10A Problems/Problem 24
  A quadrilateral is inscribed in a circle of radius <math>200\sqrt{2}</math>. Three of the sides of this quadr <math>\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\te
  24 KB (3,861 words) - 14:17, 26 April 2024
• 2015 USAMO Problems/Problem 4
  ...d <math>(j, k)</math> and moving them to <math>(i, k)</math> and <math>(j, l)</math> respectively. ...ling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.
  4 KB (770 words) - 20:09, 10 March 2016
• 2015 USAMO Problems
  ...n segment <math>\overline{PQ}</math>, show that <math>M</math> moves along a circle. ...math>S</math> is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set <math>T \subseteq S</ma
  4 KB (629 words) - 13:49, 22 November 2023
• 2015 USAJMO Problems
  ...way to continue with a finite sequence of moves so as to obtain in the end a constant sequence. ...n segment <math>\overline{PQ}</math>, show that <math>M</math> moves along a circle.
  3 KB (565 words) - 16:42, 5 August 2023
• 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
• 1997 JBMO Problems
  ...y 9 points inside a square of side length 1 we can always find 3 that form a triangle with area less than <math>\frac{1}{8}</math>. ...math> and <math>CI</math> meet <math>MN</math> at <math>K</math> and <math>L</math> respectively. Prove that <math>AI+BI+CI>BC+KL</math>.
  2 KB (245 words) - 22:44, 14 September 2023
• 2016 AMC 12A Problems/Problem 15
  ...vely, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math <math>\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\text
  8 KB (1,255 words) - 09:05, 5 September 2022
• 2016 AMC 12A Problems/Problem 19
  ...> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HT <math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\
  5 KB (881 words) - 13:34, 7 October 2023
• 2016 AMC 10B Problems/Problem 12
  <math>\textbf{(A)}\ 0.2\qquad\textbf{(B)}\ 0.4\qquad\textbf{(C)}\ 0.5\qquad\textbf{(D)}\ 0.7 ~Alternate solve by JH. L
  2 KB (308 words) - 04:35, 4 November 2022
• 2016 AMC 10B Problems/Problem 19
  label("$A$",(0,4), NW); <math>\textbf{(A)}~\frac{\sqrt{13}}{16} \qquad
  11 KB (1,608 words) - 03:18, 23 January 2023
• 2016 AMC 10B Problems/Problem 8
  <math>\textbf{(A)}\ 0 \qquad So the answer is <math>\textbf{(A)}\ \boxed{0}</math>.
  2 KB (366 words) - 12:46, 2 July 2023
• 2016 AMC 10B Problems/Problem 22
  ...</math>, <math>B</math> beat <math>C</math>, and <math>C</math> beat <math>A?</math> <math>\textbf{(A)}\ 385 \qquad
  6 KB (1,034 words) - 00:30, 31 December 2022
• 2016 AMC 10B Problems/Problem 15
  ...<math>3\times3</math> array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share <math>\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\t
  2 KB (378 words) - 21:13, 18 June 2022
• 2016 AMC 12B Problems/Problem 11
  <math>\textbf{(A)}\ 30 \qquad Label l;
  3 KB (441 words) - 19:04, 25 July 2023
• 2016 AMC 12B Problems/Problem 20
  ...</math>, <math>B</math> beat <math>C</math>, and <math>C</math> beat <math>A?</math> <math>\textbf{(A)}\ 385 \qquad
  3 KB (571 words) - 16:29, 15 July 2023
• AP Calculus
  ...lus AB is supposed to be roughly equal to the first semester and a half of a typical year-long introductory, single-variable college calculus course, wh **L'Hôpital's rule (will be added to AB in 2017)
  5 KB (665 words) - 15:27, 3 March 2016
• 2016 AIME I Problems/Problem 6
  ...ath>AB</math> at <math>L</math>. The line through <math>C</math> and <math>L</math> intersects the circumscribed circle of <math>\triangle ABC</math> at real c=8.1,a=5*(c+sqrt(c^2-64))/6,b=5*(c-sqrt(c^2-64))/6;
  14 KB (2,397 words) - 20:04, 27 August 2023
• 2016 AIME I Problems/Problem 14
  ...in the coordinate plane are a circle radius <math>\frac{1}{10}</math> and a square with sides of length <math>\frac{1}{5}</math> whose sides are parall ...<math>l</math> with an integral <math>x</math>-coordinate are, since <math>l</math> has the equation <math>y = \frac{3x}{7}</math>:
  6 KB (967 words) - 21:08, 22 November 2022
• 2016 AIME I Problems/Problem 3
  ...r of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no ...5,-0.85),G=B-(0,1.1),H=F-(0,0.6),I=E-(0,0.6),J=D-(0,1.1),K=C-(0,1.4),L=C+K-A;
  4 KB (636 words) - 13:00, 22 December 2020
• 2016 AIME I Problems/Problem 9
  Triangle <math>ABC</math> has <math>AB=40,AC=31,</math> and <math>\sin{A}=\frac{1}{5}</math>. This triangle is inscribed in rectangle <math>AQRS</ma ...by drawing triangle ABC, where <math>A</math> is obtuse. Therefore, angle A is acute. Let angle <math>CAS=n</math> and angle <math>BAQ=m</math>. Then,
  9 KB (1,526 words) - 02:31, 29 December 2021
• 2016 AIME II Problems/Problem 7
  ...n a side of <math>ABCD</math> and each vertex of <math>EFGH</math> lies on a side of <math>IJKL</math>. Find the difference between the largest and smal ...math>12^{2} \cdot 14, 11 \cdot 12 \cdot 14, 11^{2} \cdot 14</math> forming a geometric progression).
  2 KB (316 words) - 00:10, 2 September 2021
• 2016 AIME II Problems/Problem 14
  ...planes). 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
• 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
• 2015 USAJMO Problems/Problem 6
  ...d <math>(j, k)</math> and moving them to <math>(i, k)</math> and <math>(j, l)</math> respectively. ...ling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.
  4 KB (769 words) - 20:24, 16 March 2019
• 2016 USAMO Problems/Problem 5
  ...BC</math>. Let <math>R</math> be the point such that <math>NPQR</math> is a rhombus. It follows that <math>OD\perp QR</math>. ...n the two triangles in 90 degrees, <math>OI\perp MR</math>. However, <math>l</math> is parallel to the bisector of <math>MNR</math>, which is perpendicu
  8 KB (1,449 words) - 00:09, 12 October 2023
• 1986 IMO Problems
  ...h> in the set <math>\{2,5,13,d\}</math> such that <math>ab-1</math> is not a perfect square. ...ath> such that <math>P_{k+1}</math> is the image of <math>P_k</math> under a rotation center <math>A_{k+1}</math> through an angle <math>120^o</math> cl
  3 KB (482 words) - 00:04, 30 January 2021
• CSS and HTML fundamentals
  ...ages, headings, bold text, lists, images, and basically all the content of a page with HTML. ...layout, size, and much more! In order to use CSS, however, you must know a little bit about HTML.
  7 KB (1,174 words) - 21:20, 14 November 2016
• 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 10A Problems/Problem 3
  <math>\textbf{(A)}\ 72\qquad\textbf{(B)}\ 78\qquad\textbf{(C)}\ 90\qquad\textbf{(D)}\ 120\qq ~JH. L
  3 KB (384 words) - 23:41, 13 June 2022
• 2017 AMC 10A Problems/Problem 19
  ...o sit next to Eric. How many ways are there for the five of them to sit in a row of <math>5</math> chairs under these conditions? <math> \textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qqu
  7 KB (1,196 words) - 01:31, 31 January 2023
• 2017 AMC 12A Problems/Problem 8
  <math> \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 20 \ Let the length <math>AB</math> be <math>L</math>. Then, we see that the region is just the union of the cylinder with
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• 2017 AMC 10A Problems/Problem 22
  ...verline{AC}</math> of equilateral triangle <math>ABC</math> are tangent to a circle at points <math>B</math> and <math>C</math> respectively. What fract <math>\textbf{(A) } \frac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \textbf{(B) } \frac{\sqrt{3}}{
  6 KB (993 words) - 20:28, 23 February 2024
• 2017 AMC 12A Problems/Problem 21
  A set <math>S</math> is constructed as follows. To begin, <math>S = \{0,10\}< <math> \textbf{(A)}\ 4
  2 KB (372 words) - 10:48, 27 March 2023
• 2017 AMC 10A Problems/Problem 16
  ...get their names from how many minutes it takes them to run one lap around a circular race track: Horse <math>k</math> runs one lap in exactly <math>k</ <math>\textbf{(A) }2 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }5 \qqua
  4 KB (588 words) - 09:54, 13 July 2023
• 2017 AMC 10B Problems/Problem 14
  <math>\textbf{(A)}\ \frac{1}{5}\qquad\textbf{(B)}\ \frac{2}{5}\qquad\textbf{(C)}\ \frac{3}{5 Note that the patterns for the units digits repeat, so in a sense we only need to find the patterns for the digits <math>0-9</math> .
  4 KB (665 words) - 01:17, 4 December 2023
• 2017 AMC 10B Problems/Problem 21
  <math>\textbf{(A)}\ \sqrt{5}\qquad\textbf{(B)}\ \frac{11}{4}\qquad\textbf{(C)}\ 2\sqrt{2}\qq ...>AD = BD = CD = 5</math>, and <math>[ADB] = [ADC] = 12</math>. Since <math>A = rs,</math> we have <math>r = \frac As</math>, so the inradius of <math>\t
  4 KB (601 words) - 00:34, 8 August 2023
• 2017 AIME I Problems/Problem 11
  ...rangements of the <math>9</math> numbers <math>1, 2, 3, \dots, 9</math> in a <math>3 \times 3</math> array. For each such arrangement, let <math>a_1</ma ...ut there are only 4 numbers less than 5 in <math>1, 2, 3, \dots, 9</math>, a Contradiction. Thus, if <math>5 \in \{a_1, a_2, a_3\}</math>, then <math>m
  6 KB (1,019 words) - 12:40, 24 January 2024
• 2017 USAMO Problems/Problem 5
  ...integers. Find all real numbers <math>c > 0</math> such that there exists a labeling of the lattice points <math>( x, y ) \in \mathbf{Z}^2</math> with ...ly verify that these labelings satisfy the required conditions. Therefore, a labeling as desired exists for all <math>0 < c\le \sqrt[4]{2}.</math>
  8 KB (1,495 words) - 12:19, 17 July 2023
• 2017 USAMO Problems/Problem 3
  ...<math>\triangle MAN</math> intersect at points <math>L_1</math> and <math>L.</math> Prove that <math>\Omega</math> passes through the midpoint of eithe ...the orthocenter of <math>\triangle DMX \implies</math> the points <math>X, A,</math> and <math>S</math> are collinear.
  4 KB (711 words) - 18:24, 8 May 2023
• 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
• 2017 UNCO Math Contest II Problems/Problem 2
  pair A=dir(60),B=dir(120),C=dir(180),D=dir(240),E=dir(300),F=dir(360),O=(0,0); ...sqrt(3))*A,H=(2/sqrt(3))*B,I=(2/sqrt(3))*C,J=(2/sqrt(3))*D,K=(2/sqrt(3))*E,L=(2/sqrt(3))*F;
  1 KB (195 words) - 18:22, 16 January 2023
• 1972 AHSME Problems/Problem 30
  label("L",(3.75,h/2),W); ...t one corner touches the opposite side. The length in inches of the crease L in terms of angle <math>\theta</math> is
  3 KB (431 words) - 19:52, 23 June 2021
• 1997 JBMO Problems/Problem 3
  ...math> and <math>CI</math> meet <math>MN</math> at <math>K</math> and <math>L</math> respectively. Prove that <math>AI+BI+CI>BC+KL</math>. ...=(0,0), A=(50,120), C=(140,0), N=(25,60), M=(95,60), I=(60,40), K=(90,60), L=(20,60);
  2 KB (305 words) - 19:14, 7 August 2018
• 2020 AMC 12B Problems
  <math>\textbf{(A) }5 \qquad \textbf{(B) }4 + \sqrt{7} + \sqrt{10} \qquad \textbf{(C) } 10 \q ...^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)}</cmath><math>\textbf{(A) } 1 \qquad \textbf{(B) } \frac{9951}{9950} \qquad \textbf{(C) } \frac{4780
  16 KB (2,417 words) - 01:03, 28 April 2022
• MIE 2016/Day 1/Problem 2
  (a) <math>0\leq k\leq 2</math> <math>\left.\begin{array}{l}x > 0\\x > 7\\x < -2\end{array}\right\}x\in(-2,0)\cup(7,+\infty)</math>
  778 bytes (140 words) - 19:29, 8 January 2018
• 2018 AMC 10A Problems/Problem 23
  ...h> units. In the corner where those sides meet at a right angle, he leaves a small unplanted square <math>S</math> so that from the air it looks like th pair A, B, C, D, F;
  16 KB (2,454 words) - 13:30, 23 September 2023
• 2018 AMC 12A Problems/Problem 23
  In <math>\triangle PAT,</math> <math>\angle P=36^{\circ},</math> <math>\angle A=56^{\circ},</math> and <math>PA=10.</math> Points <math>U</math> and <math> <math>\textbf{(A) } 76 \qquad
  12 KB (1,878 words) - 22:11, 23 October 2021
• 2018 AMC 12B Problems/Problem 15
  <math>\textbf{(A) } 96 \qquad \textbf{(B) } 97 \qquad \textbf{(C) } 98 \qquad \textbf{(D) } ...h>C.</math> Since <math>\underline{ABC}\equiv0\pmod3,</math> we need <math>A+B+C\equiv0\pmod3</math> by the divisibility rule for <math>3.</math>
  8 KB (1,244 words) - 08:19, 30 May 2023
• 1963 AHSME Problems/Problem 33
  ...given line and <math>4</math> units from it. A possible equation for <math>L</math> is: <math>\textbf{(A)}\ y =\frac{3}{4}x+1\qquad
  2 KB (381 words) - 13:52, 21 June 2018
• 1977 AHSME Problems/Problem 19
  <math>\textbf{(A) }PQRS\text{ is a parallelogram}\\ \textbf{(B) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a rhombus}\\
  2 KB (305 words) - 08:38, 28 June 2018
• 2019 IMO Problems
  ''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>'' ...at <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly b
  4 KB (713 words) - 08:59, 10 September 2020
• 1964 AHSME Problems/Problem 13
  A circle is inscribed in a triangle with side lengths <math>8, 13</math>, and <math>17</math>. Let the made by a point of tangency, be <math>r</math> and <math>s</math>, with <math>r<s</ma
  2 KB (306 words) - 18:26, 23 July 2019
• 1973 AHSME Problems/Problem 12
  The average (arithmetic mean) age of a group consisting of doctors and lawyers in 40. If the doctors average 35 an <math> \textbf{(A)}\ 3: 2\qquad\textbf{(B)}\ 3: 1\qquad\textbf{(C)}\ 2: 3\qquad\textbf{(D)}\
  1 KB (174 words) - 13:58, 20 February 2020
• 2018 IMO Problems/Problem 3
  ...h> where w.l.o.g. <math>a>b</math>. Then <math>a-b=M_n</math>. Since <math>a\leq M_{n+1}</math> and <math>b\geq m_{n+1}</math>, we obtain that <math>M_{ ...+2+3+\cdots+2018</math> and <math>\{m_1,m_2,m_3,\dots,m_{2018}\}</math> is a permutation of <math>\{1,2,3,\dots,2018\}</math>. Also, this implies that t
  4 KB (626 words) - 01:45, 19 November 2023
• 2018 IMO Problems/Problem 6
  A convex quadrilateral <math>ABCD</math> satisfies <math>AB\cdot CD=BC \cdot The poinx <math>X</math> is inside <math>ABCD,</math> so points <math>E,A,X,C</math> follow in this order.
  8 KB (1,407 words) - 01:47, 19 November 2023
• 2002 Indonesia MO Problems
  <math>\left\{\begin{array}{l}x+y+z = 6\\x^2 + y^2 + z^2 = 12\\x^3 + y^3 + z^3 = 24\end{array}\right.</ma ...f arc <math>AB</math> that contains <math>C</math>. Let <math>E</math> be a point on <math>AC</math> such that <math>DE</math> is perpendicular to <mat
  2 KB (386 words) - 00:08, 4 August 2018
• 2002 Indonesia MO Problems/Problem 3
  <math>\left\{\begin{array}{l}x+y+z = 6\\x^2 + y^2 + z^2 = 12\\x^3 + y^3 + z^3 = 24\end{array}\right.</ma <cmath>a^3 - 6a^2 + 12a - 8 = 0</cmath>
  2 KB (353 words) - 23:38, 13 July 2022
• Mock Geometry AIME 2011 Problems/Problem 13
  ...qrt{b},</math> where <math>a,b</math> are positive integers. What is <math>a+b</math>? ...s case of law of sines. First part of solution is angle chasing. Let <math>L</math> be intersection of angle bisector <math>\ell</math> with <math>BC</m
  3 KB (463 words) - 15:49, 12 July 2023
• 2004 Indonesia MO Problems
  A trough, if filled with cold water tap, will be full in 14 minutes. To empty ...th> and <math> c</math>. Show that all these tangent points are located on a circle.
  3 KB (478 words) - 13:45, 28 July 2018
• 2004 Indonesia MO Problems/Problem 5
  Given a system of equations: <math>\left\{\begin{array}{l}x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 = 1\\4x_1 + 9x_2 + 16x_3
  1 KB (171 words) - 18:14, 28 July 2018
• 2013 Indonesia MO Problems
  In a <math>4 \times 6</math> grid, all edges and diagonals are drawn (see attach ...ines <math>ME, MF</math> intersects <math>\omega</math> at points <math>K, L</math> respectively. Prove that <math>AM, BL, CK</math> are concurrent.
  4 KB (648 words) - 00:29, 13 August 2018
• 2001 JBMO Problems/Problem 2
  Let <math>ABC</math> be a triangle with <math>\angle C = 90^\circ</math> and <math>CA \ne CB</math>. label("A",(0,40),NW);
  3 KB (394 words) - 00:27, 14 August 2018
• 2009 Indonesia MO Problems
  .../math> be the largest integer that is not more than <math> x</math>. Given a sequence of positive integers <math> a_1,a_2,a_3,\ldots</math> such that <m For every triangle <math> ABC</math>, let <math> D,E,F</math> be a point located on segment <math> BC,CA,AB</math>, respectively. Let <math> P
  3 KB (565 words) - 23:33, 26 August 2018
• 2014 Indonesia MO Problems
  ...1,2,\ldots,9</math> once each such that the sum of any two numbers sharing a side is prime? Let <math>ABCD</math> be a trapezoid (quadrilateral with one pair of parallel sides) such that <math>A
  3 KB (540 words) - 23:40, 26 August 2018
• 2006 Indonesia MO Problems
  ...ath> be positive integers. If <math> 30|a+b+c</math>, prove that <math> 30|a^5+b^5+c^5</math>. ...he set of all triangles <math> ABC</math> which have property: <math> \tan A,\tan B,\tan C</math> are positive integers. Prove that all triangles in <ma
  3 KB (439 words) - 12:39, 4 September 2018
• 2005 Indonesia MO Problems
  Let <math> n</math> be a positive integer. Determine the number of triangles (non congruent) with in (a) Prove that <math> \sqrt{k}</math> is rational.
  3 KB (485 words) - 00:31, 5 September 2018
• 1991 IMO Problems/Problem 6
  ...1, x_2,\ldots</math> of real numbers is said to be ''bounded'' if there is a constant <math>C</math> such that <math>|x_i| \leq C</math> for every <mat ...uence <math>x_0,x_1,x_2,\ldots</math> such that <math>|x_i-x_j|\cdot |i-j|^a\geq 1</math> for every pair of distinct nonnegative integers <math>i,j</mat
  6 KB (1,068 words) - 03:20, 24 January 2024
• 2006 iTest Problems/Problem 18
  <math>\text{(A) Ptolemy; Archimedes}\qquad <math>\text{(G) L'Hopital; Goldbach}\qquad
  2 KB (252 words) - 12:04, 27 November 2018
• 2006 iTest Problems/Problem 15
  <math>\text{(A) }37\qquad \text{(L) }48\qquad
  1 KB (197 words) - 23:31, 27 November 2018
• 2006 iTest Problems/Problem 16
  ...wins four games) will require the full seven games to determine a winner? <math>\text{(A) }\frac{1}{16}\qquad
  2 KB (277 words) - 13:03, 29 November 2018
• 2006 iTest Problems/Problem 13
  <math>\text{(A) } 615 \quad \text{(L) } 1715 \quad
  2 KB (216 words) - 23:35, 3 November 2023
• 2006 iTest Problems/Problem 17
  <math>\text{(A) }\frac{2305}{2401}\qquad \text{(L) }\frac{48}{49}\qquad</math>
  2 KB (290 words) - 18:27, 30 November 2018
• 2006 iTest Problems/Problem 14
  ...</math> is the smallest positive integer such that <math>2^x</math> leaves a remainder of <math>1</math> when divided by <math>5</math>, <math>7</math>, <math>\text{(A) } 15 \quad
  2 KB (238 words) - 20:27, 30 November 2018
• 2006 iTest Problems/Problem 26
  ...> and perimeter <math>P</math>. The largest possible value of <math>\tfrac A{P^2}</math> can be expressed as <math>\tfrac mn</math>, where <math>m</math Let <math>l</math> and <math>w</math> be the length and width of the rectangle, respect
  2 KB (303 words) - 19:40, 1 December 2018
• 2019 AIME II Problems/Problem 7
  pair A, B, C, D, E, F, G, H, I, J, K, L; A = intersectionpoints(Circle(B,120),Circle(C,180))[0];
  7 KB (1,053 words) - 14:58, 14 January 2024
• 2019 AIME II Problems/Problem 15
  pair A,B,C,P,Q,X,Y,O; A = dir(180+theta);
  7 KB (1,115 words) - 03:11, 7 January 2024
• 1988 IMO Problems
  ...ne assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly <math> \frac {n}{2}</math> zeros? A function <math> f</math> defined on the positive integers (and taking posit
  3 KB (499 words) - 13:29, 2 August 2021
• 2020 AMC 10B Problems
  <math>\textbf{(A)}\ -20 \qquad\textbf{(B)}\ -3 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 5 \ <math>\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 28 \qquad\textbf{(D)}\ 40
  16 KB (2,496 words) - 20:09, 19 May 2024
• 2019 USAJMO Problems
  ...with an asterisk <math>(*)</math>, the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement ...owls contains an apple, and each of the last <math>b</math> bowls contains a pear.
  4 KB (652 words) - 14:21, 10 March 2024
• 2019 USAMO Problems
  ...with an asterisk <math>(*)</math>, the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement Let <math>\mathbb{N}</math> be the set of positive integers. A function <math>f:\mathbb{N}\to\mathbb{N}</math> satisfies the equation <cma
  3 KB (495 words) - 13:47, 22 November 2023
• 2019 USAMO Problems/Problem 4
  Let <math>n</math> be a nonnegative integer. Determine the number of ways that one can choose <math ...,l}</math> whenever <math>0\leq i\leq k\leq n</math> and <math>0\leq j\leq l\leq n</math>.
  7 KB (1,288 words) - 19:17, 26 April 2023
• 2018 UNM-PNM Statewide High School Mathematics Contest II Problems
  UNM - PNM STATEWIDE MATHEMATICS CONTEST L. February 3, 2018. Second Round. Three Hours ...positive integers <math>a</math> such that <math>a < 100</math> and <math>a^3 + 23</math> is divisible by <math>24</math>.
  4 KB (600 words) - 02:25, 20 January 2019
• 2003 Pan African MO Problems/Problem 4
  Let <math>\mathbb{N}_0=\{0,1,2 \cdots \}</math>. Does there exist a function <math>f: \mathbb{N}_0 \to \mathbb{N}_0</math> such that: Let <math>n = 5^a \cdot b</math>, where <math>a,b</math> are integers and <math>b</math> is relatively prime to 5. Notice
  1 KB (213 words) - 13:59, 27 January 2020
• 2019 AMC 10A Problems/Problem 7
  <math>\textbf{(A) } 4 \qquad\textbf{(B) } 4\sqrt{2} \qquad\textbf{(C) } 6 \qquad\textbf{(D) ...ll be <math>s = \sqrt{2} + \sqrt{20}</math>, so the area reduces nicely to a difference of squares, making it <math>\boxed{\textbf{(C) }6}</math>.
  7 KB (1,079 words) - 22:24, 10 November 2023
• 2019 AMC 10A Problems/Problem 14
  For a set of four distinct lines in a plane, there are exactly <math>N</math> distinct points that lie on two or <math>\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) }
  4 KB (718 words) - 18:32, 30 October 2022
• 2019 AIME I Problems
  ...>J</math>. The value of <math>B - J</math> is at least <math>2</math> with a probability that can be expressed in the form <math>\tfrac{m}{n}</math>, wh ...ath>PR=15</math>, <math>QR=20</math>, and <math>PQ=25</math>. Points <math>A</math> and <math>B</math> lie on <math>\overline{PQ}</math>, points <math>C
  8 KB (1,331 words) - 06:57, 4 January 2021
• 2019 AMC 10B Problems/Problem 4
  ...> such that <math>a,b,c</math> form an arithmetic progression pass through a common point. What are the coordinates of that point? <math>\textbf{(A) } (-1,2)
  4 KB (597 words) - 10:24, 24 June 2023
• 2019 AMC 12B Problems
  <math>\textbf{(A) } \frac{5}{8} \qquad \textbf{(B) } \frac{4}{5} \qquad \textbf{(C) } \frac{ ...h>n-2</math> is prime." Which of the following values of <math>n</math> is a counterexample to this statement?
  16 KB (2,477 words) - 15:41, 9 September 2023
• 2019 AMC 12B Problems/Problem 10
  ...ing along any portion of a road more than once. (Paula is allowed to visit a city more than once.) pair A = (j,i);
  8 KB (1,306 words) - 22:18, 1 February 2020
• 2019 AMC 12B Problems/Problem 13
  ...1,2,3....</math> What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?<br> <math>\textbf{(A) } \frac{1}{4} \qquad\textbf{(B) } \frac{2}{7} \qquad\textbf{(C) } \frac{1}
  11 KB (1,795 words) - 00:24, 26 March 2023
• 2019 AIME I Problems/Problem 6
  ...h>, <math>MN = 65</math>, and <math>KL = 28</math>. The line through <math>L</math> perpendicular to side <math>\overline{KN}</math> intersects diagonal ...<math>\angle LNK=\beta</math>. Let <math>P</math> be the project of <math>L</math> onto line <math>NK</math>. Note <math>\angle KLP=\beta</math>.
  11 KB (1,717 words) - 20:11, 19 January 2024
• 2019 AIME II Problems/Problem 5
  ...ch ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are <math>N</math> ways for ...however, each ambassador/assistant team picks a gap between the seats (A...L) and the ambassador sits in the even seat while the assistant sits in the o
  8 KB (1,329 words) - 17:00, 27 January 2024
• 1988 IMO Problems/Problem 5
  ...ACD</math> intersect the sides <math> AB, AC</math> at the points <math> K,L</math> respectively. If <math> E</math> and <math> E_1</math> dnote the are ...a: Through the incenter <math>I</math> of <math>\triangle{ABC}</math> draw a line that meets the sides <math>AB</math> and <math>AC</math> at <math>P</m
  3 KB (578 words) - 11:38, 30 January 2021
• 2017 IMO Problems/Problem 4
  ...>JST</math> intersects <math>\ell</math> at two distinct points. Let <math>A</math> be the common point of <math>\Gamma</math> and <math>\ell</math> tha <math>K'T</math> is the image of <math>\Omega</math> at this inversion, <math>l = AR</math> is tangent line to <math>\Omega</math> at <math>R,</math> so <m
  4 KB (681 words) - 01:41, 19 November 2023
• 2006 iTest Problems/Problem 20
  <math>\textbf{(A) }2\qquad \textbf{(L) }18\qquad</math>
  4 KB (334 words) - 12:49, 15 January 2020
• 2006 iTest Problems/Problem 19
  <math>\textbf{(A) }7\qquad \textbf{(L) }18\qquad</math>
  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 15
  The following is a power of a point solution to this menace of a problem: ...r*dir(270+phi-57), B=r*dir(theta-57), H=extension(A,foot(A,B,C),B,foot(B,C,A));
  16 KB (2,678 words) - 22:45, 27 November 2023
• 2019 USAJMO Problems/Problem 5
  Let <math>n</math> be a nonnegative integer. Determine the number of ways that one can choose <math ...,l}</math> whenever <math>0\leq i\leq k\leq n</math> and <math>0\leq j\leq l\leq n</math>.
  4 KB (747 words) - 02:29, 14 May 2020
• 2005 Indonesia MO Problems/Problem 7
  ...> be the intersection of <math> AA_2</math> and <math> BB_1</math>, <math> L</math> be the intersection of <math> BB_2</math> and <math> CC_1</math>, <m ...be <math>(2d_1, -2d_2)</math>, where all variables are rational and <math>a, b_2, c, d_2 \ge 0</math>.
  5 KB (959 words) - 20:57, 12 October 2019
• Math M Addicts
  ...gularly every Saturday (non-holidays). Math-M-Addicts is also constructing a school program, called Group Z, to build mathematical foundations in school ...l, there are five tiers, increasing in prestige and level: Y, S, L, I, and A. Students are encouraged to go in that order, but students are permitted to
  825 bytes (133 words) - 10:56, 29 November 2019
• 2006 Indonesia MO Problems/Problem 5
  ...ath> G</math> is the centroid of triangle <math> ABC</math>. A line <math> l</math> passes through <math> G</math>, intersecting line <math> AB</math> a pair B=(0,0),A=(20,90),C=(100,0),M=(50,0),G=(40,30);
  3 KB (489 words) - 18:48, 14 December 2019
• 2019 IMO Problems/Problem 5
  ...T</math> on the other. Harry has <math>n</math> of these coins arranged in a line from left to right. He repeatedly performs the following operation: (a) Show that, for each initial configuration, Harry stops after a finite number of operations.
  10 KB (1,760 words) - 01:51, 19 November 2023
• 2020 AMC 10A Problems/Problem 10
  ...6</math>, and <math>343</math> cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except <math> \textbf{(A)}\ 644\qquad\textbf{(B)}\ 658\qquad\textbf{(C)}\ 664\qquad\textbf{(D)}\ 720
  5 KB (846 words) - 21:39, 10 November 2023
• 2020 AMC 10A Problems/Problem 19
  .... How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and mo pair A = (0.082, 0.378);
  8 KB (1,269 words) - 05:36, 4 November 2022
• 2020 AMC 12A Problems/Problem 12
  Line <math>l</math> in the coordinate plane has equation <math>3x-5y+40=0</math>. This l <math>\textbf{(A) } 10 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 20 \qquad \textbf{(D) }
  7 KB (1,145 words) - 20:27, 5 November 2023
• 2020 AMC 12A Problems/Problem 25
  The number <math>a=\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively <cmath> \lfloor x \rfloor \cdot \{x\} = a \cdot x^2</cmath>
  9 KB (1,541 words) - 17:13, 21 July 2022
• 2020 AMC 12B Problems/Problem 25
  ...<math>[0, a]</math> and <math>[0, 1]</math>, respectively, and let <math>P(a)</math> be the probability that What is the maximum value of <math>P(a)?</math>
  8 KB (1,412 words) - 06:17, 30 December 2023
• Nichomauss' Theorem
  ==A Visual Proof== ...process to <math>n</math> unit cubes and <math>n</math> cuboids with <math>l=w=n</math>, <math>h=1</math>, which gets us to <math>(1+2+...+n)^2=1^3+2^3+
  1 KB (207 words) - 20:21, 17 November 2020

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