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• Power Mean Inequality
  The '''Power Mean Inequality''' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] I ...ath>, the power mean with exponent <math>t</math>, where <math>t\in\mathbb{R}</math>, is defined by
  3 KB (606 words) - 23:59, 1 July 2022
• Pigeonhole Principle
  ...ts, the principle may be referred to as the '''Dirichlet box principle'''. A common phrasing of the principle uses balls and boxes and is that if <math> ...nhole principle is as follows: suppose for contradiction that there exists a way to place <math>n</math> balls into <math>k</math> boxes where <math>n>k
  11 KB (1,985 words) - 21:03, 5 August 2023
• Combination
  ...generally concerned with finding the number of combinations of size <math>r</math> from an original set of size <math>n</math> ...ons are, their various types, and how to calculate each type! It serves as a great introductory video to combinations, permutations, and counting proble
  4 KB (615 words) - 11:43, 21 May 2021
• Cauchy-Schwarz Inequality
  ...f and only if there exists a constant <math>t</math> such that <math>a_i = t b_i</math> for all <math>1 \leq i \leq n</math>, or if one list consists of ...there exists a scalar <math>t</math> such that <math>\overrightarrow{v} = t \overrightarrow{w}</math>, or if one of the vectors is zero. This formulati
  13 KB (2,048 words) - 15:28, 22 February 2024
• Ptolemy's theorem
  ...[equality condition | equality case]] of [[Ptolemy's Inequality]]. Ptolemy's theorem frequently shows up as an intermediate step in problems involving i ...ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
  7 KB (1,198 words) - 20:39, 9 March 2024
• Number theory/Advanced
  ...r theory include the [[Birch and Swinnerton-Dyer Conjecture]] and [[Fermat's Last Theorem]]. ...in some subfield (like the reals or the rationals). One also needs to add a limit point, called the point at infinity. As <math>x\to \infty</math>, the
  5 KB (849 words) - 16:14, 18 May 2021
• Divisibility rules
  ...integer]]s are [[divisibility | divisible]] by particular other [[integer]]s. All of these rules apply for [[Base number| base-10]] ''only'' -- other b https://youtu.be/6xNkyDgIhEE?t=1699
  8 KB (1,298 words) - 15:07, 23 May 2024
• Jensen's Inequality
  '''Jensen's Inequality''' is an inequality discovered by Danish mathematician Johan Jen ...[convex function]] of one real variable. Let <math>x_1,\dots,x_n\in\mathbb R</math> and let <math>a_1,\dots, a_n\ge 0</math> satisfy <math>a_1+\dots+a_n
  3 KB (623 words) - 13:10, 20 February 2024
• Combinatorial identity
  == Pascal's Identity == Pascal's Identity states that
  12 KB (1,996 words) - 12:01, 18 May 2024
• Circle
  A '''circle''' is a geometric figure commonly used in Euclidean [[geometry]]. {{asy image|<asy>unitsize(2cm);draw(unitcircle,blue);</asy>|right|A basic circle.}}
  9 KB (1,581 words) - 18:59, 9 May 2024
• Chain Rule
  ...to J</math> and <math>f:J \to \mathbb{R}</math>. Let <math>h:I \to \mathbb{R}</math> such that <math>h(x) = f(g(x)) \forall x \in I</math>. If <math>x_ ...<math>h'(x_0)</math>,<math>f'(g(x_0))</math>, and <math>g'(x_0)</math> is a matrix.)
  12 KB (2,377 words) - 11:48, 22 July 2009
• Muirhead's Inequality
  ...>p</math> [[Majorization|majorizes]] a sequence <math>q</math>, then given a set of positive reals <math>x_1,x_2,\cdots,x_n</math>: ...ath> majorizes <math>(4,2)</math> (as <math>5>4, 5+1=4+2</math>), Muirhead's inequality states that for any positive <math>x,y</math>,
  8 KB (1,346 words) - 12:53, 8 October 2023
• Fundamental Theorem of Calculus
  The '''Fundamental Theorem of Calculus''' establishes a link between the two central operations of [[calculus]]: [[derivative|diffe This section is for people who know what [[integral]]s are but don't know the Fundamental Theorem of Calculus yet, and would like to try to figu
  11 KB (2,082 words) - 15:23, 2 January 2022
• Vector
  ...y, but also most abstractly, a vector is any object which is an element of a given vector space. ...(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]].
  7 KB (1,265 words) - 13:22, 14 July 2021
• Logarithm
  ...would be a pain to have to calculate any time you wanted to use it (say in a comparison of large numbers). Its natural logarithm though (partly due to ...ly 7 digits before the decimal point. Comparing the logs of the numbers to a given precision can allow easier comparison than computing and comparing th
  4 KB (680 words) - 12:54, 16 October 2023
• Roots of unity
  ...ts of unity come up when we examine the [[complex number|complex]] [[root]]s of the [[polynomial]] <math> x^n=1 </math>. ...making <math>r^n=1\Rightarrow r=1</math> (magnitude is always expressed as a positive real number). This leaves us with <math>e^{ni\theta} = e^{2\pi ik
  3 KB (558 words) - 21:36, 11 December 2011
• Newton's Sums
  ...ficient way of finding the sums of [[root]]s of a [[polynomial]] raised to a power. They can also be used to derive several [[factoring]] [[identity|id Consider a polynomial <math>P(x)</math> of degree <math>n</math>,
  4 KB (690 words) - 13:11, 20 February 2024
• Rational approximation
  ...n [[real number]] <math>x</math> can be approximated by [[rational number]]s. Of course, since the rationals are dense on the real line, we, surely, can ...th> can be approximated by a rational number <math>\frac{p}{q}</math> with a given denominator <math>q\ge 1</math> with an error not exceeding <math>\fr
  7 KB (1,290 words) - 12:18, 30 May 2019
• Trigonometry
  ...s and angles of triangles through the '''trigonometric functions'''. It is a fundamental branch of mathematics, and its discovery paved the way towards ...[[Law of Sines]] and the [[Law of Cosines]]; many more, such as [[Stewart's Theorem]], are most easily proven using trigonometry. In algebra, expressio
  8 KB (1,217 words) - 20:15, 7 September 2023
• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  ...ric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> that says: ..._1}+\cdots+\frac{1}{x_n}} \ge \sqrt[n_4]{\frac{x_1^{n_4}+\cdots+x_a^{n_4}}{a}}</cmath> where <math>n_1>1,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math
  5 KB (912 words) - 20:06, 14 March 2023
• Goldbach Conjecture
  ...every [[even integer]] greater than two is the sum of two [[prime number]]s. The conjecture has been tested up to 400,000,000,000,000. Goldbach's conjecture is one of the oldest unsolved problems in [[number theory]] and
  7 KB (1,201 words) - 16:59, 19 February 2024
• Power of a Point Theorem
  ...e lengths of the [[line segment]]s formed when two [[line]]s [[intersect]] a [[circle]] and each other. ...of the lines is [[tangent line|tangent]] to the circle while the other is a [[secant line|secant]] (middle figure). In this case, we have <math> AB^2 =
  5 KB (827 words) - 17:30, 21 February 2024
• Majorization
  We say a [[nonincreasing]] [[sequence]] of [[real number]]s <math> a_1, \ldots ,a_n</math> '''majorizes''' another nonincreasing sequen ...also say that this is the case if and only if for all <math> t \in \mathbb{R} </math>,
  2 KB (288 words) - 22:48, 5 July 2023
• 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
• 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 https://youtu.be/7an5wU9Q5hk?t=777
  15 KB (2,396 words) - 20:24, 21 February 2024
• Modular arithmetic/Intermediate
  ...or <math>a \equiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. ...Z}_n</math> for short). This structure gives us a useful tool for solving a wide range of number-theoretic problems, including finding solutions to [[D
  14 KB (2,317 words) - 19:01, 29 October 2021
• 2006 AIME I Problems
  In quadrilateral <math> ABCD , \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <mat ...f <math> \mathcal{A}. </math> Find the number of possible values of <math> S. </math>
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 13
  ...st integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect square]]. ...ldots,</math> and <math>2^{n-1}-2^{n-2} = 2^{n-2}</math> elements of <math>S</math> that are divisible by <math>2^1</math> but not by <math>2^2</math>.
  10 KB (1,702 words) - 00:45, 16 November 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 8
  ...area of rhombus <math> \mathcal{T}</math>. Given that <math> K </math> is a [[positive integer]], find the number of possible values for <math> K</math pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6),
  5 KB (730 words) - 15:05, 15 January 2024
• 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
• Set
  ...ms, see [[Zermelo-Fraenkel Axioms]]. In this article we shall present just a brief discussion of the most common properties of sets and operations relat ...g: <math>\{1,4,5,3,24,4,4,5,6,2\}</math> Such an entity is actually called a multiset.
  11 KB (2,021 words) - 00:00, 17 July 2011
• Newman's Tauberian Theorem
  '''Newman's Tauberian Theorem''' is a [[tauberian theorem]] Let <math>f:(0,+\infty)\to\mathbb C</math> be a bounded function. Assume that
  6 KB (1,034 words) - 07:55, 12 August 2019
• 2005 AIME I Problems/Problem 4
  ...members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Fi ...the number of students is <math>n(n + 7)</math> which must be 5 more than a perfect square, so <math>n \leq 14</math>. In fact, when <math>n = 14</mat
  8 KB (1,248 words) - 11:43, 16 August 2022
• 2005 AIME I Problems/Problem 6
  ...ath>1 + ir</math> and <math>1 - ir</math>. Their product is <math>P = 1 + r^2 = 1 + \sqrt{2006}</math>. <math>44^2 = 1936 < 2006 < 2025 = 45^2</math> If you think of each part of the product as a quadratic, then <math>((x-1)^2+\sqrt{2006})</math> is bound to hold the two
  4 KB (686 words) - 01:55, 5 December 2022
• 2005 AIME II Problems/Problem 14
  pair A = origin; pair C = rotate(15,A)*(A+dir(-50));
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2004 AIME II Problems/Problem 15
  ...eated <math>8</math> more times. After the last fold, the strip has become a stack of <math>1024</math> unit squares. How many of these squares lie belo ...number of squares below the <math>n</math> square after the final fold in a strip of length <math>2^{k}</math>.
  6 KB (899 words) - 20:58, 12 May 2022
• 2004 AIME II Problems/Problem 14
  ...th> 7+77+777+7+7=875 </math> could be obtained from eight <math> 7 </math>'s in this way. For how many values of <math> n </math> is it possible to inse ...</math>. Then the question is asking for the number of values of <math>n = a + 2b + 3c</math>.
  11 KB (1,857 words) - 21:55, 19 June 2023
• 1990 AIME Problems
  ...neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. ...9}{58}</math> as large as each interior angle of <math>P_2^{}</math>. What's the largest possible value of <math>s_{}^{}</math>?
  6 KB (870 words) - 10:14, 19 June 2021
• 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 The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught <math
  8 KB (1,275 words) - 06:55, 2 September 2021
• 1997 AIME Problems
  ...math> rectangles, of which <math>s</math> are squares. The number <math>s/r</math> can be written in the form <math>m/n,</math> where <math>m</math> an ...he two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah sho
  7 KB (1,098 words) - 17:08, 25 June 2020
• 2001 AIME I Problems
  ...than the mean of <math>\mathcal{S}</math>. Find the mean of <math>\mathcal{S}</math>. ...and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>.
  7 KB (1,212 words) - 22:16, 17 December 2023
• 2002 AIME I Problems
  ...plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right- ...gram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the r
  8 KB (1,374 words) - 21:09, 27 July 2023
• 2001 AIME II Problems
  ...rom left to right, each pair of consecutive digits of <math>N</math> forms a perfect square. What are the leftmost three digits of <math>N</math>? Each of the 2001 students at a high school studies either Spanish or French, and some study both. The numb
  8 KB (1,282 words) - 21:12, 19 February 2019
• 1983 AIME Problems/Problem 14
  ...<math>P</math>, one of the points of intersection, a line is drawn in such a way that the chords <math>QP</math> and <math>PR</math> have equal length. ...l("$P$",P,1.5*dir(80)); label("$R$",R,NE); label("12",waypoint(O1--O2,0.4),S);</asy>
  13 KB (2,151 words) - 17:48, 27 May 2024
• 1983 AIME Problems/Problem 15
  ...r arc <math>AB</math> is a rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <mat pair A=(-0.91,-0.41);
  20 KB (3,497 words) - 15:37, 27 May 2024
• 1984 AIME Problems/Problem 8
  ...math> is a root of <math>z^6+z^3+1</math>, then <math>0=(r^3-1)(r^6+r^3+1)=r^9-1</math>. The polynomial <math>x^9-1</math> has all of its roots with [[a ...\circ </math>, then <math>r^6+r^3+1=3</math>. (When we multiplied by <math>r^3 - 1</math> at the beginning, we introduced some extraneous solutions, and
  3 KB (430 words) - 19:05, 7 February 2023
• 1984 AIME Problems/Problem 9
  ...math> is <math>12 \mbox { cm}^2</math>. These two faces meet each other at a <math>30^\circ</math> angle. Find the [[volume]] of the tetrahedron in <mat path3 rightanglemark(triple A, triple B, triple C, real s=8)
  6 KB (947 words) - 20:44, 26 November 2021
• 1986 AIME Problems/Problem 13
  ...tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by <tt>TH</tt>, <tt>HH</tt>, and etc. For ex ...tches to <tt>H</tt> four times; hence it follows that our string will have a structure of <tt>THTHTHTH</tt>.
  4 KB (772 words) - 21:09, 7 May 2024
• 1987 AIME Problems/Problem 6
  ...XY = WZ</math>, <math>PQ = PQ</math> and the [[area]]s of the [[trapezoid]]s <math>PQZW</math> and <math>PQYX</math> are the same, then the heights of t ...is equal to <math>[PXYQ]=\frac{19}{2}* \frac{XY+87}{2}=\frac{19}{2}* \frac{a+b+106}{2}</math>. Setting these equal to each other and solving gives <math
  3 KB (530 words) - 07:46, 1 June 2018
• 1988 AIME Problems/Problem 13
  ...ath> and <math>b</math> are integers such that <math>x^2 - x - 1</math> is a factor of <math>ax^{17} + bx^{16} + 1</math>. 0 = ax^{17} + bx^{16} + 1 = a(F_{17}\cdot x + F_{16}) + b(F_{16}\cdot x + F_{15}) + 1 &\Longrightarrow (a
  10 KB (1,585 words) - 03:58, 1 May 2023
• 1989 AIME Problems/Problem 14
  ...lex number]] of the form <math>r+si</math>, where <math>r</math> and <math>s</math> are integers, can be uniquely expressed in the base <math>-n+i</math <center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center>
  2 KB (408 words) - 17:28, 16 September 2023
• 1989 AIME Problems/Problem 11
  ...ach between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique [[mode]] (most frequent value). Let <math>D</math> be the difference ...be <math>M</math>, and the sum of the numbers <math>\neq x</math> be <math>S</math>. Then
  5 KB (851 words) - 18:01, 28 December 2022
• 1990 AIME Problems/Problem 7
  .../math> can be written in the form <math>ax+2y+c=0_{}^{}</math>. Find <math>a+c_{}^{}</math>. pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17);
  8 KB (1,319 words) - 11:34, 22 November 2023
• 1992 AIME Problems/Problem 5
  ...^{}_{}</math> are not necessarily distinct. To write the elements of <math>S^{}_{}</math> as fractions in lowest terms, how many different numerators ar ...ere aren't any numbers which are multiples of <math>37^2</math>, so we can't get numerators which are multiples of <math>37</math>. Therefore <math>648
  2 KB (277 words) - 20:45, 4 March 2024
• 1993 AIME Problems/Problem 13
  ...enny and Kenny can see each other again. If <math>t\,</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? ...100).</math> Then at time <math>t</math>, they end up at points <math>(-50+t,100)</math> and <math>(-50+3t,-100).</math>
  8 KB (1,231 words) - 20:06, 26 November 2023
• 1995 AIME Problems/Problem 5
  For certain real values of <math>a, b, c,</math> and <math>d_{},</math> the equation <math>x^4+ax^3+bx^2+cx+d Since the [[coefficient]]s of the [[polynomial]] are real, it follows that the non-real roots must com
  3 KB (451 words) - 15:02, 6 September 2021
• 1997 AIME Problems/Problem 12
  ...math>f</math> defined by <math>f(x)= \frac{ax+b}{cx+d}</math>, where <math>a</math>,<math>b</math>,<math>c</math> and <math>d</math> are nonzero real nu ...in the domain. Substituting the function definition, we have <math>\frac {a\frac {ax + b}{cx + d} + b}{c\frac {ax + b}{cx + d} + d} = x</math>, which r
  11 KB (2,063 words) - 22:59, 21 October 2023
• 2000 AIME I Problems/Problem 15
  A stack of <math>2000</math> cards is labelled with the integers from <math>1 ...r logic as Solution 1, we find that 1999 has position <math>1024</math> in a <math>2048</math> card stack, where the fake cards towards the front.
  15 KB (2,673 words) - 19:16, 6 January 2024
• 2000 AIME I Problems/Problem 14
  ...r</math> times as large as angle <math>APQ,</math> where <math>r</math> is a positive real number. Find <math>\lfloor 1000r \rfloor</math>. pair A,B,C,P,Q;
  8 KB (1,275 words) - 03:04, 27 February 2022
• 2001 AIME I Problems/Problem 7
  ...), I=incenter(A,B,C), D=IP((0,I.y)--(20,I.y),A--B), E=IP((0,I.y)--(20,I.y),A--C); D(MP("A",A,N)--MP("B",B)--MP("C",C)--cycle); D(MP("I",I,NE)); D(MP("E",E,NE)--MP("D",D
  9 KB (1,540 words) - 08:31, 1 December 2022
• 2003 AIME I Problems/Problem 13
  ...math>2</math> representation has more <math>1</math>'s than <math>0</math>'s. Find the [[remainder]] when <math> N </math> is divided by <math>1000</mat ...</math> notation, with <math>k+1</math> of the digits being <math>1</math>'s.
  4 KB (651 words) - 19:42, 7 October 2023
• 2002 AIME II Problems/Problem 12
  ...<math>c</math> are positive integers. Find <math>\left(p+q+r+s\right)\left(a+b+c\right)</math>. ...y=\frac{2x}{3}</math>. We can find the number of such paths using a Pascal's Triangle type method below, computing the number of paths to each point tha
  7 KB (1,127 words) - 13:34, 19 June 2022
• 2001 AIME II Problems/Problem 7
  ...ine{ST}</math> with <math>S</math> on <math>\overline{PR}</math> and <math>T</math> on <math>\overline{QR}</math>, such that <math>\overline{ST}</math> pair P = (0,0), Q = (90, 0), R = (0, 120), S=(0, 60), T=(45, 60), U = (60,0), V=(60, 40), O1 = (30,30), O2 = (15, 75), O3 = (70, 10
  7 KB (1,112 words) - 02:15, 26 December 2022
• 2000 AIME II Problems/Problem 13
  ...<math>r</math> are relatively prime, and <math>r>0</math>. Find <math>m+n+r</math>. Thus <math>r=\frac{-1+\sqrt{161}}{40}</math>, and so the final answer is <math>-1+161+40
  6 KB (1,060 words) - 17:36, 26 April 2024
• 2006 USAMO Problems/Problem 1
  ...ath> be a prime number and let <math>s</math> be an integer with <math>0 < s < p </math>. Prove that there exist integers <math>m</math> and <math>n</ma <math> \left\{ \frac{sm}{p} \right\} < \left\{ \frac{sn}{p} \right\} < \frac{s}{p} </math>
  3 KB (506 words) - 17:54, 22 June 2023
• 2006 USAMO Problems/Problem 3
  Let <math>f(x)</math> be a non-constant polynomial in <math>x</math> of degree <math>d</math> with Let <math>g(n) = f(n^2)</math>, then <math>g(n)</math> is a polynomial of degree <math>2</math> or
  9 KB (1,699 words) - 13:48, 11 April 2020
• 2006 USAMO Problems/Problem 6
  ...th>, <math>SBF</math>, <math>TCF</math>, and <math>TDE</math> pass through a common point. ...imilarity, and <math>X</math> is the center of spiral similarity for <math>A,E,B,</math> and <math>F</math>.
  5 KB (986 words) - 22:46, 18 May 2015
• Dedekind domain
  A '''Dedekind domain''' is a [[integral domain]] <math>R</math> satisfying the following properties: * <math>R</math> is a [[noetherian]] [[ring]].
  9 KB (1,648 words) - 16:36, 14 October 2017
• Mock AIME 1 2006-2007 Problems/Problem 9
  ...h>m</math> and <math>n</math> are [[relatively prime]] [[positive integer]]s, compute the sum of the positive [[prime number | prime]] [[divisor | facto ..._{10}+a_{11}+a_{12}+...</math>. If the sum of all distinct values of <math>S</math> is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math>
  5 KB (744 words) - 19:46, 20 October 2020
• Mock AIME 1 2006-2007 Problems/Problem 13
  ...se variables - but there is no obvious way of computing them. We will show a different solution. ...\alpha\beta\gamma=r</math>. Then we can write <math>P(x)=x^3 - px^2 - qx - r</math>.
  3 KB (568 words) - 15:50, 3 April 2012
• Mock AIME 1 2006-2007 Problems/Problem 14
  ...</math> with a variable radius that is tangent to <math>AB</math> at <math>T</math>. Let <math>P</math> be the point such that circle <math>O</math> is ...ee below an attempted solution to understand why this problem doesn't have a solution:
  3 KB (541 words) - 17:32, 22 November 2023
• Mock AIME 1 2006-2007/Problems
  ...^{7}</math>. If <math>m</math> is the number of elements in <math>\mathcal{S}</math>, compute <math>\star(m)</math>. ...\sqrt{7}</math>, <math>CA=1</math>, and <math>AB=3</math>. If <math>\angle A=\frac{\pi}{n}</math> where <math>n</math> is an integer, find the remainder
  8 KB (1,355 words) - 14:54, 21 August 2020
• 1971 Canadian MO Problems/Problem 4
  .../math> such that the two polynomials <math>x^2+ax+1</math> and <math>x^2+x+a</math> have at least one root in common. Let this root be <math>r</math>. Then we have
  2 KB (346 words) - 23:20, 18 January 2023
• 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
• 1987 IMO Problems/Problem 6
  ..., then we can conclude that <math>m</math> is a prime. Since there must be a factor of <math>m</math> less than <math>\sqrt{m}</math>. ...t let <math>r=\lfloor\sqrt{n/3}\rfloor</math>, then we can write <math>n=3(r+\epsilon)^2(0\leq\epsilon< 1)</math>, so <math>h=6r\epsilon+3\epsilon^2\leq
  2 KB (430 words) - 13:03, 24 February 2024
• Euler line
  ...fact of Euclidean [[geometry]]. Certain fixed orders and distance [[ratio]]s hold among these points. In particular, <math>\overline{OGNH}</math> and < ...s examine what else this transformation, which we denote as <math>\mathcal{S}</math>, will do.
  59 KB (10,203 words) - 04:47, 30 August 2023
• 2003 AMC 10A Problems
  {{AMC10 Problems|year=2003|ab=A}} <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \
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• 2003 AMC 10A Problems/Problem 17
  ...rcumscribed circle. What is the radius, in inches, of the circle? <!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> <math> \mathrm{(A) \ } \frac{3\sqrt{2}}{\pi}\qquad \mathrm{(B) \ } \frac{3\sqrt{3}}{\pi}\qqu
  2 KB (414 words) - 13:48, 4 April 2024
• 1985 IMO Problems/Problem 1
  A [[circle]] has center on the side <math>AB</math> of the [[cyclic quadrilat ...>O</math> be the center of the circle mentioned in the problem. Let <math>T</math> be the second intersection of the circumcircle of <math>CDO </math>
  4 KB (684 words) - 07:28, 3 October 2021
• 2001 AMC 8 Problems
  ...class is making a golf trophy. He has to paint <math>300</math> dimples on a golf ball. If it takes him <math>2</math> seconds to paint one dimple, how <math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D)
  13 KB (1,994 words) - 13:04, 18 February 2024
• 2002 IMO Shortlist Problems/A3
  ...x + d </math>, where <math> a, b, c, d </math> are [[integer]]s and <math> a \neq 0 </math>. Suppose that <math> xP(x) = yP(y) </math> for infinitely m a(x^4 - y^4) + b(x^3 - y^3) + c(x^2 - y^2) + d(x - y) = 0
  4 KB (662 words) - 19:51, 8 September 2009
• Asymptote: Reference
  A '''point''' is associated with a cartesian coordinate pair in Asymptote. There are two useful functions tha ...ewise cubic function of a parameter <math>t</math>, parameterized as <math>t</math> ranges from <math>0</math> to the number of nodes (say <math>n</math
  7 KB (1,205 words) - 21:38, 26 March 2024
• 2007 AIME II Problems/Problem 10
  .../math> is the set of all elements of <math>S</math> which are not in <math>A.</math>) **<math>A</math> must have either 0 or 6 elements, probability: <math>\frac{2}{2^6} =
  8 KB (1,367 words) - 11:48, 23 October 2022
• 2002 USAMO Problems/Problem 2
  Let <math>ABC </math> be a triangle such that \left( \cot \frac{A}{2} \right)^2 + \left( 2 \cot \frac{B}{2} \right)^2 + \left( 3 \cot \frac{C
  2 KB (298 words) - 22:32, 6 April 2016
• 2002 USAMO Problems/Problem 4
  ...ber]]s. Determine all functions <math> f : \mathbb{R} \rightarrow \mathbb{R} </math> such that ...math>, then we obtain <math>f(x^2) = xf(x) </math>. Therefore the problem's condition becomes
  3 KB (490 words) - 22:09, 18 September 2018
• Mock AIME 1 Pre 2005 Problems
  ...tive integers with three distinct digits. Compute the remainder when <math>S</math> is divided by <math>1000</math>. <math>A, B, C, D,</math> and <math>E</math> are collinear in that order such that <
  6 KB (1,100 words) - 22:35, 9 January 2016
• Mock AIME 3 Pre 2005 Problems/Problem 5
  ...>P's</math> is a word in Zuminglish if and only if between any two <math>O's</math> there appear at least two consonants. Let <math>N</math> denote the ...te the number of <math>n</math>-letter words ending in a vowel followed by a constant (<tt>VC</tt> - the only other combination, two vowels, is impossib
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• 2007 AIME I Problems/Problem 13
  ...oints of <math>AE</math>, <math>BC</math>, and <math>CD</math>. The plane's intersection with the pyramid has an area that can be expressed as <math>\s Note first that the intersection is a [[pentagon]].
  7 KB (1,034 words) - 21:56, 22 September 2022
• 2007 AIME II Problems/Problem 15
  .../math>, and <math>\omega</math> is [[externally tangent]] to <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math>. If the sides pair A,B,C,X,Y,Z,P,Q,R;
  11 KB (2,099 words) - 17:51, 4 January 2024
• 1961 IMO Problems/Problem 2
  ...>b</math>, and <math>c</math> be the lengths of a triangle whose area is ''S''. Prove that <math>a^2 + b^2 + c^2 \ge 4S\sqrt{3}</math>
  5 KB (860 words) - 13:12, 13 February 2024
• 2007 USAMO Problems/Problem 1
  ...m Vandervelde'') Let <math>n</math> be a [[positive]] [[integer]]. Define a [[sequence]] by setting <math>a_1 = n</math> and, for each <math>k>1</math> ...integers, so <math>b_{k+1} \le b_k</math>. As the <math>b_k</math>'s form a [[non-increasing]] sequence of positive integers, they must eventually beco
  6 KB (1,204 words) - 20:06, 23 August 2023
• 2007 USAMO Problems/Problem 4
  (''Reid Barton'') An ''animal'' with <math>n</math> ''cells'' is a connected figure consisting of <math>n</math> equal-sized [[Square (geometr ...two or more dinosaurs. Find with proof the [[maximum]] number of cells in a primitive dinosaur.
  10 KB (1,878 words) - 14:56, 30 June 2021
• 2007 USAMO Problems/Problem 6
  ...ga</math>. Let <math>P_A</math> and <math>Q_A</math> denote the [[center]]s of <math>\omega_A</math> and <math>\Omega_A</math>, respectively. Define p <cmath>8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,</cmath>
  7 KB (1,274 words) - 15:11, 31 August 2017
• 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
• 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
• LaTeX:Commands
  | <math>a^{i+1}_3</math>||a^{i+1}_3||<math>x^{3^2}</math>||x^{3^2} Notice that we can apply both a subscript and a superscript at the same time. For subscripts or superscripts with more than
  12 KB (1,898 words) - 15:31, 22 February 2024
• 2007 AMC 12A Problems/Problem 9
  ...both choices require the same amount of time. What is the [[ratio]] of Yan's distance from his home to his distance from the stadium? <math>\mathrm{(A)}\ \frac 23\qquad \mathrm{(B)}\ \frac 34\qquad \mathrm{(C)}\ \frac 45\qquad
  5 KB (804 words) - 14:55, 21 August 2022
• 2005 AMC 12A Problems/Problem 6
  ...had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met? (\mathrm {A}) \ 4 \qquad (\mathrm {B}) \ 5 \qquad (\mathrm {C})\ 6 \qquad (\mathrm {D})
  1,006 bytes (166 words) - 21:18, 3 July 2013
• 1988 IMO Problems/Problem 6
  ...>b</math> be positive integers such that <math>ab + 1</math> divides <math>a^{2} + b^{2}</math>. Show that \frac {a^{2} + b^{2}}{ab + 1}
  4 KB (720 words) - 12:26, 7 April 2024

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