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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) \
  982 bytes (165 words) - 00:19, 19 January 2024
• 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

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