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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 (1,315 words) - 18:18, 2 March 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 (2,149 words) - 18:44, 5 February 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);
  19 KB (3,221 words) - 01:05, 7 February 2023
• 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) \
  13 KB (1,900 words) - 22:27, 6 January 2021
• 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
  5 KB (795 words) - 16:03, 17 October 2021
• 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
• 2007 AMC 12B Problems
  Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 fee <math> \mathrm{(A) \ } 678\qquad \mathrm{(B) \ } 768\qquad \mathrm{(C) \ } 786\qquad \mathrm{
  12 KB (1,814 words) - 12:58, 19 February 2020
• Asymptote: Useful functions
  The bundled <code>markers</code> package provides a useful function for marking angles. The exact same function exists in the < ...e functions, you can replace <code>line l1, line l2</code> with <code>pair A, pair O, pair B</code>, which would mark the angle <math>\angle AOB</math>.
  4 KB (646 words) - 21:18, 26 March 2024
• Ideal
  ...two-sided ideals of rings are similar to [[normal subgroup]]s of [[group]]s. ...xa \in \alpha</math>, for all <math>x\in R</math> and <math>a\in \mathfrak{a}</math>. Symbolically, this can be written as
  8 KB (1,389 words) - 23:44, 17 February 2020
• 1951 AHSME Problems
  <math> \textbf{(A) \ } \frac {100(M - N)}{M} \qquad \textbf{(B) \ } \frac {100(M - N)}{N} \qq A rectangular field is half as wide as it is long and is completely enclosed
  23 KB (3,641 words) - 22:23, 3 November 2023
• 1998 AHSME Problems/Problem 26
  ...\circ}</math>, angles <math>B</math> and <math>D</math> are [[right angle]]s, <math>AB = 13</math>, and <math>AD = 46</math>. Then <math>AC=</math> <math>\mathrm{(A)}\ 60
  3 KB (391 words) - 14:30, 5 July 2013
• 2004 AMC 12B Problems/Problem 23
  <math>\mathrm{(A)}\ 250,\!000 Let the roots be <math>r,s,r + s</math>, and let <math>t = rs</math>. Then
  3 KB (533 words) - 14:52, 29 October 2023
• 2004 AMC 12B Problems/Problem 19
  ...d cone has horizontal bases with radii <math>18</math> and <math>2</math>. A sphere is tangent to the top, bottom, and lateral surface of the truncated <math>\mathrm{(A)}\ 6
  3 KB (520 words) - 19:12, 20 November 2023
• 2008 AMC 10A Problems
  {{AMC10 Problems|year=2008|ab=A}} ...}\ {\small\text{AM}}</math> the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
  14 KB (2,138 words) - 15:08, 18 February 2023
• 2004 AMC 12A Problems/Problem 19
  ...th>. Circles <math>B</math> and <math>C</math> are congruent. Circle <math>A</math> has radius <math>1</math> and passes through the center of <math>D</ pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0);
  5 KB (785 words) - 00:29, 31 July 2023
• 2008 AMC 12B Problems
  A basketball player made <math>5</math> baskets during a game. Each basket was worth either <math>2</math> or <math>3</math> points. <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \
  14 KB (2,199 words) - 13:43, 28 August 2020
• 2008 AMC 12B Problems/Problem 25
  ...B=11, BC=5, CD=19,</math> and <math>DA=7</math>. Bisectors of <math>\angle A</math> and <math>\angle D</math> meet at <math>P</math>, and bisectors of < <math>\textbf{(A)}\ 28\sqrt{3}\qquad \textbf{(B)}\ 30\sqrt{3}\qquad \textbf{(C)}\ 32\sqrt{3}
  12 KB (2,015 words) - 20:54, 9 October 2022
• 2008 AIME I Problems
  Of the students attending a school party, <math>60\%</math> of the students are girls, and <math>40\%</ ...l whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking, jogging, and swimming rates.
  9 KB (1,536 words) - 00:46, 26 August 2023
• 2008 AIME II Problems
  ...te which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin ...by <math>14</math> cm. Ten slices are cut from the cheese. Each slice has a width of <math>1</math> cm and is cut parallel to one face of the cheese. T
  7 KB (1,167 words) - 21:33, 12 August 2020
• 2008 AIME I Problems/Problem 11
  ...math>B</math>, and <math>AABAA</math>, while <math>BBAB</math> is not such a sequence. How many such sequences have length 14? ...math>n-2</math> ending in a <math>B</math>. Thus, we have the [[recursion]]s
  7 KB (1,173 words) - 22:39, 28 November 2023
• 2008 AIME I Problems/Problem 14
  ...a</math>. Point <math>P</math> is the foot of the perpendicular from <math>A</math> to line <math>CT</math>. Suppose <math>\overline{AB} = 18</math>, an ...- (9,0), A= O + (9,0), C=A+(18,0), T = 9 * expi(-1.2309594), P = foot(A,C,T);
  8 KB (1,333 words) - 00:18, 1 February 2024
• Mock AIME 5 Pre 2005 Problems
  &\text {ii. all ten digits, a through j are all distinct.}\\ &\text {iii.} a + b + c + d + e + f + g + h + i + j = 45\end{align*}</cmath>
  6 KB (909 words) - 07:27, 12 October 2022
• Mock AIME 1 2007-2008 Problems
  ...to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent? ...1+i)^{17} - (1-i)^{17}</math>, where <math>i=\sqrt{-1}</math>. Find <math>|S|</math>.
  6 KB (992 words) - 14:15, 13 February 2018
• 2008 AIME II Problems/Problem 2
  ...mark at exactly the same time. How many minutes has it taken them?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> ...</math> be the time Rudolph takes disregarding breaks and <math>\frac{4}{3}r</math> be the time Jennifer takes disregarding breaks. We have the equation
  3 KB (432 words) - 11:53, 19 June 2019
• 2008 AIME II Problems/Problem 7
  Let <math>r</math>, <math>s</math>, and <math>t</math> be the three roots of the equation Find <math>(r + s)^3 + (s + t)^3 + (t + r)^3</math>.
  7 KB (1,251 words) - 19:18, 2 January 2024
• 2008 AIME II Problems/Problem 9
  ...h> units in the positive <math>x</math>-direction. Given that the particle's position after <math>150</math> moves is <math>(p,q)</math>, find the great ...the x-axis. If <math>(x', y')</math> is the position of the particle after a move from <math>P</math>, then we have two equations for <math>x'</math> an
  5 KB (725 words) - 22:37, 28 January 2024
• 2008 AIME II Problems/Problem 11
  real r=44-6*35^.5; pair A=(0,96),B=(-28,0),C=(28,0),X=C-(64/3,0),Y=B+(4*r/3,0),P=X+(0,16),Q=Y+(0,r),M=foot(Q,X,P);
  6 KB (1,065 words) - 20:12, 9 August 2022
• 1974 USAMO Problems/Problem 5
  pair C=(0,1), A=(1,6), B=(4,1), D=(1.5,2); draw(D--A--B--C--D--B);
  7 KB (1,221 words) - 18:57, 3 July 2013
• 2008 AMC 10A Problems/Problem 15
  <math>\mathrm{(A)}\ 120\qquad\mathrm{(B)}\ 130\qquad\mathrm{(C)}\ 140\qquad\mathrm{(D)}\ 150 ...d as <math>I</math>, and set Han's speed as <math>H</math>. Therefore, Jan's speed is <math>H+5.</math>
  6 KB (1,033 words) - 15:19, 1 July 2021
• 2008 USAMO Problems/Problem 1
  <cmath> k_0 k_1 \dotsc k_n = a^2+a+1, </cmath> ...on <math>n</math>. For our base case, <math>n=0</math>, we may let <math>a</math> be positive integer.
  11 KB (1,964 words) - 03:38, 17 August 2019
• 2008 USAMO Problems/Problem 5
  ...<math>y</math> and write <math>y - x</math> in its place. Prove that after a finite number of such operations, we can end up with at least one <math>0</ ...gers <math>A = \left < a_1, a_2, a_3 \right ></math> so that <math>R \cdot A = 0</math> continues to hold. (For example, if we replace <math>r_1</math>
  2 KB (407 words) - 23:37, 21 July 2020
• 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
• Sylow Theorems
  ...eory of [[finite]] [[group]]s. They give a partial converse to [[Lagrange's Theorem]], and are one of the most important results in the field. They ar :'''Theorem.''' Every finite group contains a [[Sylow p-subgroup |Sylow <math>p</math>-subgroup]].
  11 KB (2,071 words) - 12:25, 9 April 2019
• Routh's Theorem
  ...be the intersection of <math>CF</math> and <math>AD</math>. Then, '''Routh's Theorem''' states that <cmath>[GHI]=\dfrac{(rst-1)^2}{(rs+r+1)(st+s+1)(tr+t+1)}[ABC]</cmath>
  2 KB (267 words) - 00:02, 24 March 2021
• Category (category theory)
  A category, <math>\mathcal{C}</math>, is a mathematical object consisting of: * A [[class]], <math>\text{Ob}(\mathcal{C})</math> of objects.
  5 KB (792 words) - 19:01, 7 April 2012
• 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
• 2002 USAMO Problems
  ...^{2002} </math>. Prove that it is possible to color every subset of <math>S </math> either blue or red so that the following conditions hold: (a) the union of any two red subsets is red;
  3 KB (486 words) - 09:21, 14 May 2021
• 2002 AMC 12A Problems/Problem 7
  ...</math> arc of circle B. What is the ratio of circle A's area and circle B's area? <math>\textbf{(A)}\ 4/9 \qquad \textbf{(B)}\ 2/3 \qquad \textbf{(C)}\ 5/6 \qquad \textbf{(D)
  3 KB (492 words) - 14:46, 31 January 2024
• 2002 AMC 12A Problems/Problem 11
  <math>\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ ...in, we find <math>d=12</math>. From <math>d=rt</math>, we find that <math>r</math>, our answer, is <math>\boxed{\textbf{(B) }48 }</math>.
  2 KB (396 words) - 00:19, 1 February 2024
• 2021 GMC 10B
  <math>\textbf{(A)} ~99 \qquad\textbf{(B)} ~100 \qquad\textbf{(C)} ~101 \qquad\textbf{(D)} ~1 ...an area of <math>\frac{\pi}{\sqrt{2}}</math> is <math>r</math>. Find <math>r^{2}</math>
  11 KB (1,695 words) - 14:33, 7 March 2022
• 1971 IMO Problems/Problem 6
  Let <math>A = (a_{ij})(i, j = 1, 2, \cdots, n)</math> be a square matrix whose elements are non-negative integers. Suppose that whenev ...ecome smaller. So we have<cmath> T \ge (n - S)^2 + S^2 \ge 2(\frac{n - S + S}{2})^2 = \frac{n^2}{2}</cmath>by AM - QM inequality.
  6 KB (1,192 words) - 14:14, 29 January 2021
• 2009 AMC 12A Problems/Problem 19
  ...ns were <math>A</math> and <math>B</math>, respectively. Each polygon had a side length of <math>2</math>. Which of the following is true? ...B\qquad \textbf{(D)}\ A </math> <math>= \frac {7}{5}B\qquad \textbf{(E)}\ A = \frac {49}{25}B</math>
  4 KB (630 words) - 21:27, 30 December 2023
• 2009 AMC 12A Problems/Problem 16
  A circle with center <math>C</math> is tangent to the positive <math>x</math> <math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \
  2 KB (314 words) - 21:07, 16 January 2020
• 2001 AMC 12 Problems/Problem 11
  ...tly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white \textbf{(A) }\frac {3}{10}
  2 KB (423 words) - 12:21, 11 June 2022
• 2002 AMC 12A Problems/Problem 24
  ...d pairs of real numbers <math>(a,b)</math> such that <math>(a+bi)^{2002} = a-bi</math>. \text{(A) }1001
  4 KB (743 words) - 19:54, 14 March 2024
• 2009 AMC 10A Problems/Problem 6
  ...micircle but outside the circle is shaded. What fraction of the semicircle's area is shaded? \mathrm{(A)}\ \frac{1}{2}
  2 KB (380 words) - 09:21, 8 June 2021
• 2009 AMC 12B Problems/Problem 20
  ...h> pyramids and a new polyhedron <math>R</math>. How many edges does <math>R</math> have? <math>\mathrm{(A)}\ 200\qquad
  9 KB (1,567 words) - 13:43, 19 August 2023
• 2009 AMC 10B Problems
  ...nt muffin or a <math>75</math>-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? <math>\text{(A) } 1\qquad\text{(B) } 2\qquad\text{(C) } 3\qquad\text{(D) } 4\qquad\text{(E
  15 KB (2,262 words) - 00:53, 18 June 2021
• 2009 AMC 10B Problems/Problem 24
  real r=5, R=6; path t=r*dir(0)--r*dir(20)--R*dir(20)--R*dir(0);
  3 KB (546 words) - 04:26, 16 January 2023
• 2009 AIME I Problems/Problem 2
  ...s a complex number <math>z</math> with imaginary part <math>164</math> and a positive integer <math>n</math> such that Let <math>z = a + 164i</math>.
  5 KB (850 words) - 12:53, 23 December 2023
• 2009 AIME I Problems/Problem 12
  ...cle having <math>\overline{CD}</math> as a diameter. Let <math>I</math> be a point outside <math>\triangle ABC</math> such that <math>\overline{AI}</mat ...h>. Hence <math>AQ = 144, BP = 1225, AB = 1369</math> and the radius <math>r = OD = 210</math>.
  12 KB (1,970 words) - 22:53, 22 January 2024
• Cauchy's Integral Formula
  '''Cauchy's Integral Formula''' is a fundamental result in [[complex analysis]]. It states that if <math>U</math> is a subset of
  4 KB (689 words) - 17:19, 18 January 2024
• 2009 AIME II Problems
  ...ng a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finis ...ath>b</math>, and <math>c</math> are positive real numbers such that <math>a^{\log_3 7} = 27</math>, <math>b^{\log_7 11} = 49</math>, and <math>c^{\log_
  8 KB (1,366 words) - 21:33, 3 January 2021
• 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
• Radical axis
  ...n should peruse this material carefully, as it may contain the keys to one's future success. ...rtains to that situation. I hope after you read this text, you will become a better math student, armed with another tool to solve difficult problems. B
  10 KB (1,797 words) - 02:05, 24 October 2023
• 2009 AIME II Problems/Problem 15
  ...<math>t</math> is not divisible by the square of any prime. Find <math>r+s+t</math>. ...c{3}{5}</math>, point <math>B</math> lies between <math>M</math> and <math>A</math> on the semicircular arc. We will first compute the length of <math>\
  11 KB (1,849 words) - 19:43, 2 January 2023
• 1990 AJHSME Problems
  <math>\text{(A)}\ 947 \qquad \text{(B)}\ 1037 \qquad \text{(C)}\ 1047 \qquad \text{(D)}\ 1 <math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad
  15 KB (2,059 words) - 15:03, 6 October 2021
• 1990 AJHSME Problems/Problem 10
  ...ext{C}</math>. If this sum equals the sum of the dates behind <math>\text{A}</math> and <math>\text{B}</math>, then the letter is for(int a=0; a<6; ++a)
  1 KB (231 words) - 10:51, 8 September 2019
• Base introduction
  Bases. These confuse a lot of us. Hopefully this will make it better. ...as a subscript. Every number in base 10 (our normal numbers) does not have a subscript.
  3 KB (552 words) - 20:57, 9 July 2009
• 2009 USAMO Problems/Problem 5
  ...}</math> and <math>\overline{BC}</math> at points <math>R</math> and <math>S</math>, respectively. Prove that quadrilateral <math>PQRS</math> is cyclic We will use directed angles in this solution. Extend <math>QR</math> to <math>T</math> as follows:
  6 KB (973 words) - 19:24, 18 October 2018
• 2009 USAMO Problems/Problem 6
  ...r <math>r</math> such that <math>(s_i - s_j)r</math> and <math>(t_i - t_j)/r</math> are integers for all <math>i</math> and <math>j</math>. ...ac{c_2}{d_2} - \frac{c_1}{d_1})</math> is an integer. Now we can set <math>r = \frac{b_1 b_2}{d_1 d_2}</math>, because the least common denominator of <
  2 KB (463 words) - 12:19, 21 August 2020
• Semisimple module
  A '''semisimple module''' is, informally, a [[module]] that is not far removed from [[simple module]]s.
  6 KB (1,183 words) - 15:02, 18 August 2009
• 1994 IMO Problems/Problem 3
  ...2k\}</math> whose base 2 representation has precisely three <math>1</math>s. * (a) Prove that, for each positive integer <math>m</math>, there exists at leas
  7 KB (1,298 words) - 19:59, 9 February 2024
• Asymptote: Olympiad Package Part 1
  ...er and over and over again. Thus, the olympiad package is a lifesaver. Let's look at this simple bit of code: (Note, the import olympiad command is not real r=2.5, a=3.5;
  1 KB (209 words) - 01:48, 7 January 2010
• 1993 AJHSME Problems
  Which pair of numbers does NOT have a product equal to <math>36</math>? <math>\text{(A)}\ \{ -4,-9\} \qquad \text{(B)}\ \{ -3,-12\} \qquad \text{(C)}\ \left\{ \df
  14 KB (1,797 words) - 11:13, 28 December 2022
• 2010 AMC 12A Problems/Problem 7
  ...and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan m <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 \ ...gative, all <math>3</math> zeros must be double roots because the function's degree is <math>6</math>.
  5 KB (969 words) - 21:33, 22 June 2022
• 2010 AMC 10A Problems
  {{AMC10 Problems|year=2010|ab=A}} Mary’s top book shelf holds five books with the following widths, in centimeters:
  13 KB (1,902 words) - 11:20, 5 March 2023
• Characteristic polynomial
  ...[operator]] refers to the [[polynomial]] whose roots are the [[eigenvalue]]s of the operator. It carries much information about the operator. ...det(tI - A)</math>, which is a <math>n</math>th degree polynomial in <math>t</math>. Here, <math>I</math> refers to the <math>n\times n</math> [[identit
  19 KB (3,412 words) - 14:57, 21 September 2022
• 2010 AIME I Problems
  ...h>p</math> be the probability that exactly one of the selected divisors is a perfect square. The probability <math>p</math> can be expressed in the form ...mes 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}</math> is divided by <math>1000</math>.
  8 KB (1,243 words) - 21:58, 10 August 2020
• 2010 AIME I Problems/Problem 13
  .../math>, respectively. Line <math>\ell</math> divides region <math>\mathcal{R}</math> into two regions with areas in the ratio <math>1: 2</math>. Suppose label("$A$",(-16.43287,-9.3374),NE/2);
  10 KB (1,418 words) - 23:05, 20 October 2021
• Mock AIME 2 2010 Answer Key
  ...an be easily proved through strong induction. Starting from 2010, which is a multiple of 15, we must first purge 1 lemming. We can then purge 4 lemmings ...atisfies <math>b_1+b_2+\ldots+b_{10}\equiv 0 \pmod{3}</math>, there exists a corresponding <math>(a_1, a_2, \ldots, a_{10})</math> such that <math>a_1+a
  36 KB (6,214 words) - 20:22, 13 July 2023
• 2010 AIME II Problems
  ...math> denote the distance from <math>P</math> to the closest side of <math>S</math>. The probability that <math>\frac{1}{5}\le d(P)\le\frac{1}{3}</math> ...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 14
  ...<math>r</math> is not divisible by the square of any prime. Find <math>p+q+r</math>. ...o use Power of a Point on <math>BA</math> and <math>CD</math>. By Power of a Point Theorem, <math>CP\cdot PD=1\cdot 2=BP\cdot PA</math>. Since <math>BP+
  10 KB (1,507 words) - 00:31, 19 November 2023
• 2010 AIME II Problems/Problem 7
  ...2w-4</math>, where <math>i^2=-1</math>. Find <math>|a+b+c|</math>.<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> == Solution (vieta's) ==
  4 KB (735 words) - 02:41, 27 June 2022
• 2010 AIME II Problems/Problem 10
  ...econd-degree [[polynomial]]s <math>f(x)</math> with integer [[coefficient]]s and integer zeros for which <math>f(0)=2010</math>. ...guishable. In one case, <math>(a,r,s) = (2010,1,1)</math>, we have <math>r=s</math>. The other <math>80</math> cases are double counting, so there are <
  6 KB (972 words) - 16:01, 16 September 2023
• 1993 USAMO Problems/Problem 2
  Let <math>ABCD</math> be a convex quadrilateral such that diagonals <math>AC</math> and <math>BD</math pair A,B,C,D;
  2 KB (446 words) - 08:09, 10 April 2023
• 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>. ...s. We claim that the relation <math>R = \{(j, k)\in [n]\times[n]\mid jk\in S\}</math> is an equivalence relation on <math>[n]</math>.
  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 <math>AB</math>. Denote by <math>P, Q, R, S</math> the feet of the perpendiculars from <math>Y</math> onto
  13 KB (2,178 words) - 14:14, 11 September 2021
• 1984 USAMO Problems/Problem 2
  ...(\text{i})\quad</math> For which positive integers <math>n</math> is there a finite set <math>S_n</math> of <math>n</math> distinct positive integers su ...sitive integers such that the geometric mean of any finite subset of <math>S</math> is an integer?
  4 KB (855 words) - 03:12, 23 October 2022
• 2010 AMC 12B Problems/Problem 19
  A high school basketball game between the Raiders and Wildcats was tied at th <math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\
  6 KB (990 words) - 19:14, 25 March 2024
• 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
• 2019 IMO Problems/Problem 6
  ...perpendicular to <math>EF</math> meets <math>\omega</math> again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumci ...t lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.
  5 KB (792 words) - 01:52, 19 November 2023
• 1996 AJHSME Problems
  <math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad <math>\text{(A)}\ \text{Jose} \qquad \text{(B)}\ \text{Thuy} \qquad \text{(C)}\ \text{Kare
  13 KB (1,880 words) - 13:35, 19 February 2020
• Python reference
  This page is intended to serve as a brief, beginners reference to the [[Python]] programming language. In Python, there is no need to specify a type when storing information in a variable. It is basic and straightforward to store information.
  28 KB (4,762 words) - 21:20, 12 June 2023
• 2006 AMC 8 Problems
  <math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qqu ...answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score?
  16 KB (2,215 words) - 19:18, 10 April 2024
• 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
• 2010 AMC 10B Problems/Problem 22
  <math>\textbf{(A)}\ 1930 \qquad \textbf{(B)}\ 1931 \qquad \textbf{(C)}\ 1932 \qquad \textbf{ ...ions such that the red bag is empty is equal to <math>2^7</math>, since it's equivalent to distributing the <math>7</math> candies into <math>2</math> b
  4 KB (750 words) - 22:59, 18 January 2024
• 2011 AMC 12A Problems/Problem 12
  ...<math>A.</math> How many hours did it take the power boat to go from <math>A</math> to <math>B</math>? \textbf{(A)}\ 3 \qquad
  5 KB (928 words) - 23:01, 14 April 2022
• 1958 AHSME Problems
  <math> \textbf{(A)}\ 5\qquad <math> \textbf{(A)}\ y - x\qquad \textbf{(B)}\ x - y\qquad \textbf{(C)}\ \frac {y - x}{xy}\qq
  25 KB (3,872 words) - 14:21, 20 February 2020
• 2011 AMC 10A Problems
  {{AMC10 Problems|year=2011|ab=A}} A cell phone plan costs <math>\textdollar 20</math> each month, plus <math>5<
  13 KB (1,903 words) - 18:09, 19 April 2021
• 1993 AHSME Problems
  ...<math>c</math>, define <math>\boxed{a,b,c}</math> to mean <math>a^b-b^c+c^a</math>. Then <math>\boxed{1,-1,2}</math> equals <math>\text{(A)} \ -4 \qquad \text{(B)} \ -2 \qquad \text{(C)} \ 0 \qquad \text{(D)} \ 2 \
  20 KB (2,814 words) - 08:15, 27 June 2021
• 2012 IMO Problems/Problem 1
  ...tersection of the lines <math>AF</math> and <math>BC</math>, and let <math>T</math> be the point of intersection of the lines <math>AG</math> and <math> .../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
  ...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 11
  ...e the sum of the elements in <math>R</math>. Find the remainder when <math>S</math> is divided by 1000. ...<math>2^i \equiv 2^j \pmod{1000}</math>. All that is left is to find <math>S</math> in mod <math>1000</math>. After some computation:
  11 KB (1,668 words) - 22:10, 24 February 2023
• 2011 AIME I Problems/Problem 13
  ..., <math>s</math>, and <math>t</math> are positive integers. Find <math>r+s+t</math>. ...h>a^2+b^2+c^2=1</math>. The distance from a point <math>(X,Y,Z)</math> to a plane with equation <math>Ax+By+Cz+D=0</math> is
  4 KB (613 words) - 11:37, 8 September 2022
• 2011 AIME II Problems/Problem 10
  ...14 intersect at point <math>P</math>. The distance between the [[midpoint]]s of the two chords is 12. The quantity <math>OP^2</math> can be represented The line through the midpoint of a chord of a circle and the center of that circle is perpendicular to that chord, so <ma
  11 KB (1,720 words) - 03:12, 18 December 2023
• 1999 AMC 8 Problems
  <math>\text{(A)} \div \qquad \text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \ 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 USAMO Problems/Problem 3
  ...r of opposite sides are parallel. The internal angles satisfy <math>\angle A = 3\angle D</math>, <math>\angle C = 3\angle F</math>, and <math>\angle E = ...B = \beta</math>, <math>AB=DE=p</math>, <math>BC=EF=q</math>, <math>CD=FA=r</math>. Define the vectors: <cmath>\vec{u} = \vec{AB} + \vec{DE}</cmath> <
  11 KB (1,925 words) - 12:07, 31 August 2023
• 2001 AMC 8 Problems/Problem 23
  ...front page--><onlyinclude>Points <math>R</math>, <math>S</math> and <math>T</math> are vertices of an equilateral triangle, and points <math>X</math>, drawn using any three of these six points as vertices?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude>
  3 KB (549 words) - 16:35, 23 July 2023
• 2007 AMC 10B Problems
  Isabella's house has <math>3</math> bedrooms. Each bedroom is <math>12</math> feet lon <math>\textbf{(A) } 678 \qquad\textbf{(B) } 768 \qquad\textbf{(C) } 786 \qquad\textbf{(D) }
  15 KB (2,297 words) - 12:57, 19 February 2020
• Olympiad books
  Here is a list of '''Olympiad Books''' that have Olympiad-level problems used to trai ...om/community/c142h405377 here] about these books or request new books. Let's categorize books into '''Theory''' books, '''Problem''' books, and '''Both'
  17 KB (2,261 words) - 00:30, 22 April 2024
• 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
• 1967 AHSME Problems
  ...umber <math>5b9</math>. If <math>5b9</math> is divisible by 9, then <math>a+b</math> equals <math> \text{(A)}\ 2\qquad\text{(B)}\ 4\qquad\text{(C)}\ 6\qquad\text{(D)}\ 8\qquad\text{(E
  20 KB (3,108 words) - 14:14, 20 February 2020
• 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
• 2005 CEMC Gauss (Grade 7) Problems
  = Part A: Each correct answer is worth 5 points = <math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad
  16 KB (2,317 words) - 03:54, 24 October 2014
• 2009 AMC 8 Problems
  Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. The <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 11\qquad\
  18 KB (2,551 words) - 18:46, 27 February 2024
• 1987 USAMO Problems
  ...sector of <math>\angle A</math>, find all possible values for <math>\angle A</math>. ...<math>X</math>, then <math>x\cdot r(x)</math> and <math>(x + (1 - x) \cdot r(x) )</math> both belong to <math>X</math>.
  2 KB (356 words) - 18:42, 18 July 2016
• 1988 USAMO Problems
  ...t</math>. Show that <math>k=a^2-3b\ge 0</math> and that <math>\sqrt k\le r-t</math>. ...ath>X</math>. Show that for any map <math>f: P\mapsto X</math> we can find a 10-element subset <math>Y</math> of <math>X</math>, such that <math>f(Y-\{k
  2 KB (319 words) - 16:33, 5 February 2018
• 1996 AJHSME Problems/Problem 5
  ...<math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math>, and <math>T</math> represent numbers located on the number line as shown. label("$-3$",(-3,-0.1),S);
  3 KB (506 words) - 00:24, 5 July 2013
• 1996 AJHSME Problems/Problem 17
  ...<math>Q</math> has coordinates (2,2). What are the coordinates for <math>T</math> so that the area of triangle <math>PQT</math> equals the area of squ pair O,P,Q,R,T;
  2 KB (257 words) - 11:20, 22 March 2015
• 2010 PUMaC Problems
  == Division A == ...math> such that <math>4^x = x^4</math>. Find the nearest integer to <math>S</math>.
  22 KB (3,694 words) - 23:58, 3 June 2022
• 2016 AMC 10A Problems/Problem 19
  ...<math>r,s,</math> and <math>t</math> is <math>1.</math> What is <math>r+s+t</math>? <math>\textbf{(A) } 7 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 15
  8 KB (1,386 words) - 15:10, 8 October 2023
• 1993 AJHSME Problems/Problem 23
  ...math> beats <math>R</math>, <math>Q</math> beats <math>S</math>, and <math>T</math> finishes after <math>P</math> and before <math>Q</math>. Who could ...and }S \qquad \text{(D)}\ P\text{ and }T \qquad \text{(E)}\ P,S\text{ and }T</math>
  1 KB (228 words) - 00:12, 5 July 2013
• 2003 AMC 8 Problems/Problem 14
  In this addition problem, each letter stands for a different digit. ...sep}{0.5mm}\begin{array}{cccc}&T & W & O\\ +&T & W & O\\ \hline F& O & U & R\end{array} </math>
  1 KB (177 words) - 19:56, 15 April 2023
• 2003 AMC 8 Problems/Problem 18
  ...east one of her friends. How many classmates will not be invited to Sarah's party? pair a=(102,256), b=(68,131), c=(162,101), d=(134,150);
  2 KB (250 words) - 22:17, 5 January 2024
• 1950 AHSME Problems
  <math> \textbf{(A)}\ 5\frac{1}{3}\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 10\frac{2}{3}\qqua ...>. When <math>S=8</math>, <math>R=16</math>. When <math>S=10</math>, <math>R</math> is equal to:
  22 KB (3,306 words) - 19:50, 3 May 2023
• 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
• 2010 AMC 8 Problems
  ...students in Mr. Newton's class, and <math>9</math> students in Mrs. Young's class taking the AMC 8 this year. How many mathematics students at Euclid M <math> \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qq
  18 KB (2,768 words) - 21:05, 9 January 2024
• 2011 AMC 8 Problems
  ...math> apples at a cost of <math> 50 </math> cents per apple. She paid with a 5-dollar bill. How much change did Margie receive? <math>\textbf{(A) }\ \textdollar 1.50 \qquad \textbf{(B) }\ \textdollar 2.00 \qquad \textbf{
  16 KB (2,371 words) - 17:34, 9 January 2024
• 1985 IMO Problems/Problem 6
  ...ution, one can write <math>x_n=P_n(x_1)</math> , where <math>P_n</math> is a polynomial with non-negative coefficients and zero constant term. Thus, <ma (a) <math>\displaystyle x_n<x_{n+1} \Leftrightarrow x_n>1-\frac{1}{n} \Leftrig
  9 KB (1,788 words) - 00:02, 30 January 2021
• Mathtime Version 1 Issue 2
  ...e the positive integer <math>n</math> such that <math>n^2+20n+40</math> is a perfect square. (Proposed by djmathman) ...umber of ordered integer pairs <math>(a,b)</math> such that <math>a^2+b^2|(a+b)^2</math>. (Proposed by djmathman)
  15 KB (2,444 words) - 21:46, 1 January 2012
• 2012 AMC 10A Problems/Problem 25
  <math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\ ...0</math>. Then the set of points <math>(x,y,z)</math> is a tetrahedron, or a triangular pyramid. The point <math>(x,y,z)</math> distributes uniformly in
  13 KB (2,133 words) - 01:22, 6 February 2024
• 2012 AMC 10A Problems/Problem 18
  ...each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve? <math>\textbf{(A)}\ 2\pi+6\qquad\textbf{(B)}\ 2\pi+4\sqrt{3}\qquad\textbf{(C)}\ 3\pi+4\qquad
  5 KB (775 words) - 22:33, 22 October 2023
• 2012 AMC 12A Problems/Problem 16
  <math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf label("$Z$",Z,S);
  9 KB (1,496 words) - 02:40, 2 October 2022
• 2012 AMC 12A Problems/Problem 22
  ...igcup_{j=1}^{k}p_{j} </math>. The intersection of <math>P</math> and <math>S</math> consists of the union of all segments joining the midpoints of every <math> \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qqua
  5 KB (940 words) - 17:13, 4 April 2020
• 1989 AHSME Problems/Problem 19
  A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of l <math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \frac{18}{\pi^2} } \qquad \mathrm{(C) \frac{9}{\p
  4 KB (649 words) - 10:04, 20 May 2021
• 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 ...e, and they have been traveling for <math>t</math> minutes. Find <math>n + t</math>.
  10 KB (1,617 words) - 14:49, 2 June 2023
• 2012 AIME I Problems/Problem 14
  ...math>b,</math> and <math>c</math> in the complex plane are the vertices of a right triangle with hypotenuse <math>h.</math> Find <math>h^2.</math> ...9}\cdot\frac{x^2+y^2}{4}=\frac{x^2}{9}+\frac{y^2}{9}</math>. Hence, <math>|a|^2+|b|^2+|c|^2=\frac{6x^2+6y^2}{9}=\frac{2x^2+2y^2}{3}=250</math>. Therefor
  12 KB (2,228 words) - 23:59, 20 January 2024
• 2012 AIME II Problems/Problem 15
  ...les <math>\omega</math> and <math>\gamma</math> meet at <math>E</math> and a second point <math>F</math>. Then <math>AF^2 = \frac mn</math>, where <math ...math>. Solving gives <math>x^2=\frac{49}{19}</math>, from which by Ptolemy's we see <math>AF=\frac{30}{\sqrt{19}}</math>. We conclude the answer is <mat
  13 KB (2,298 words) - 12:56, 10 September 2023
• 2017 AMC 8 Problems
  <math>\textbf{(A) }2+0+1+7\qquad\textbf{(B) }2 \times 0 +1+7\qquad\textbf{(C) }2+0 \times 1 Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the vo
  12 KB (1,771 words) - 21:13, 20 January 2024
• Mock AIME I 2012 Problems
  ...<math>q</math> and <math>r</math> are relatively prime. What is <math>p+q+r</math>? A permutation of the numbers <math>\{2</math>, <math>3</math>, <math>4</math>
  7 KB (1,309 words) - 11:13, 8 April 2012
• Mock AIME I 2012 Problems/Problem 8
  ...e expressed in the form <math>\dfrac{r+\sqrt{s}}{t}</math>. Find <math>r+s+t</math>. ...rth power, the desired answer is <math>\dfrac{7+\sqrt{45}}{2} \implies r+s+t=\boxed{054}</math>.
  3 KB (543 words) - 18:51, 7 May 2020
• 2015 IMO Problems/Problem 1
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> ...any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
  4 KB (773 words) - 08:14, 19 July 2016
• 2002 Romanian NMO Problems
  Eight card players are seated around a table. One remarks that at some moment, any player and his two neighbours h Prove that any real number <math>0<x<1</math> can be written as a difference of two positive and less than <math>1</math> irrational numbers.
  10 KB (1,695 words) - 10:03, 10 May 2012
• Install Eclipse
  Note that, first, you need to make sure you have a version of JVM(Java Virtual Machine) installed. See [[Install Java]] page f You can read if you have a 32- or 64-bit system under System Type:
  8 KB (1,222 words) - 17:47, 9 October 2014
• 1988 USAMO Problems/Problem 2
  ...t</math>. Show that <math>k=a^2-3b\ge 0</math> and that <math>\sqrt k\le r-t</math>. By Vieta's Formulas, <math>a=-r-s-t</math>, <math>b=rs+st+rt</math>, and <math>c=-rst</math>.
  3 KB (674 words) - 18:17, 13 June 2017
• 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
  <math> \mathrm{(A)\ } 4x+3y=xy \qquad \mathrm{(B) \ }y=\frac{4x}{6-y} \qquad \mathrm{(C) \ } <math> \mathrm{(A)\ } -\frac{h}{3} \qquad \mathrm{(B) \ }\frac{h}{3} \qquad \mathrm{(C) \ }
  15 KB (2,151 words) - 14:04, 19 February 2020
• 1974 AHSME Problems/Problem 23
  ...t of tangency. If <math> TP=4 </math> and <math> T'Q=9 </math> then <math> r </math> is pair O = (0,0); pair T = dir(90);
  2 KB (337 words) - 12:44, 5 July 2013
• 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
• 2006 SMT/Calculus Problems
  ...equation <math> 4y''+3y'-y=0 </math> and its solution <math> y=e^{\lambda t} </math>, what are the values of <math> \lambda </math>? ...orn is a point on the ground, and that the elven rope is so light it makes a straight line from the unicorn to the tower.
  3 KB (525 words) - 13:59, 27 May 2012
• 2010 AMC 8 Problems/Problem 3
  ....4), M=(1,-4.2), N=(1,-4), P=(1,-3.8), Q=(1,-3.6), R=(1,-3.4), S=(1,-3.2), T=(1,-3), U=(1,-2.8), V=(1,-2.6), W=(1,-2.4), Z=(1,-2.2), E_1=(1.4,-2.6), F_1 ...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
• 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
• 1954 AHSME Problems
  <math>\textbf{(A)}\ y^2-5\sqrt{y^2-25} \qquad \textbf{(B)}\ -y^2 \qquad \textbf{(C)}\ y^2 \\ <math>\textbf{(A)}\ 4 \text{ and }1 \qquad \textbf{(B)}\ \text{only }1 \qquad \textbf{(C)}\
  23 KB (3,535 words) - 16:29, 24 April 2020
• 1955 AHSME Problems
  <math> \textbf{(A)}\ 3.75\times 10^{-7}\qquad\textbf{(B)}\ 3\frac{3}{4}\times 10^{-7}\qquad\t The smaller angle between the hands of a clock at <math>12:25</math> p.m. is:
  22 KB (3,509 words) - 21:29, 31 December 2023
• 1998 USAMO Problems/Problem 6
  ...er <math>k</math> (as a function of <math>n</math>) such that there exists a convex <math>n</math>-gon <math>A_{1}A_{2}\dots A_{n}</math> for which exac pair A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U;
  5 KB (871 words) - 18:59, 10 May 2023
• 1999 USAMO Problems/Problem 1
  (a) every square that does not contain a checker shares a side with one that does; ...ven squares, such that every two consecutive squares of the sequence share a side.
  4 KB (747 words) - 05:23, 13 May 2023
• 2020 Mock Combo AMC 10
  Here are the problems from the 2020 Mock Combo AMC 10 II, a mock contest created by the AoPS user fidgetboss_4000. ...friend, Sara, along with eight other classmates, are randomly seated along a row of ten chairs. What is the probability that Fred sits next to Sara?
  15 KB (2,452 words) - 03:03, 4 July 2020
• 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 10A Problems
  {{AMC10 Problems|year=2013|ab=A}} A taxi ride costs \$1.50 plus \$0.25 per mile traveled. How much does a 5-mile taxi ride cost?
  12 KB (1,894 words) - 15:59, 3 January 2024
• 2013 AIME I Problems/Problem 5
  ...ath>, <math>b</math>, and <math>c</math> are positive integers. Find <math>a+b+c</math>. ...3]{9}+1</math> to rationalize the denominator, and we therefore have <math>r=\frac{\sqrt[3]{81}+\sqrt[3]{9}+1}{8}</math>, and the answer is <math>\boxed
  3 KB (573 words) - 21:11, 27 August 2023
• 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 12
  ..., and <math>c</math> are integers. Find the number of polynomials in <math>S</math> such that each of its roots <math>z</math> satisfies either <math>|z ...real and two nonreal roots which are conjugates. This follows from [[Vieta's formulas]].
  8 KB (1,393 words) - 19:00, 24 May 2023
• 2013 USAMO Problems/Problem 1
  In triangle <math>ABC</math>, points <math>P,Q,R</math> lie on sides <math>BC,CA,AB</math> respectively. Let <math>\omega_A pair A = (-1.0, 3.0);
  14 KB (1,830 words) - 18:22, 10 May 2023
• Mock AIME 6 2006-2007 Problems
  ...integers that do not exceed <math>4</math>. Find the remainder when <math>T</math> is divided by <math>1000</math>. Draw in the diagonals of a regular octagon. What is the sum of all distinct angle measures, in degree
  7 KB (1,173 words) - 21:04, 7 December 2018
• 2014 AIME II Problems/Problem 3
  ...of 24, the hexagon has the same area as the original rectangle. Find <math>a^2</math>. pair A,B,C,D,E,F,R,S,T,X,Y,Z;
  3 KB (454 words) - 22:00, 24 January 2024
• 2014 AIME II Problems/Problem 5
  ...0</math>. Find the sum of all possible values of <math>|b|</math>.<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> ...</math> in <math>p(x)</math> and <math>q(x)</math> are both equal to <math>a</math>, and equating the two coefficients gives
  6 KB (1,069 words) - 15:51, 31 December 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
• 1975 IMO Problems/Problem 6
  ...nteger <math>n</math> and all real <math>t, x, y</math> <cmath>P(tx, ty) = t^nP(x, y)</cmath> (that is, <math>P</math> is homogeneous of degree <math>n< (ii) for all real <math>a, b, c</math>, <cmath>P(b + c, a) + P(c + a, b) + P(a + b, c) = 0,</cmath>
  3 KB (563 words) - 16:21, 29 January 2021
• 1962 AHSME Problems/Problem 13
  ...c37</math>. Find <math>S</math> when <math>R = \sqrt {48}</math> and <math>T = \sqrt {75}</math>. <math> \textbf{(A)}\ 28\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 42\qqu
  969 bytes (143 words) - 22:19, 10 April 2023
• 1962 AHSME Problems/Problem 18
  ...decagon (<math>12</math> sides) is inscribed in a circle with radius <math>r</math> inches. The area of the dodecagon, in square inches, is: ...bf{(B)}\ 2r^2\qquad\textbf{(C)}\ \frac{3r^2\sqrt{3}}{4}\qquad\textbf{(D)}\ r^2\sqrt{3}\qquad\textbf{(E)}\ 3r^2\sqrt{3} </math>
  902 bytes (147 words) - 22:17, 3 October 2014
• 1962 AHSME Problems
  <math>\textbf{(A)}\ \frac{4y-1}{8} \qquad <math>\textbf{(A)}\ \frac{\sqrt{3}}{6} \qquad
  17 KB (2,459 words) - 22:40, 10 April 2023
• 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 11
  ...mill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wed <math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \
  2 KB (323 words) - 17:57, 5 May 2022
• 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
• 2014 AMC 10A Problems/Problem 4
  Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he pa <math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\t
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• 2000 PMWC Problems/Problem T3
  ...th> is a square, <math>P</math>, <math>Q</math>, <math>R</math>, and <math>S</math> are midpoints of the sides <math>AB</math>, <math>BC</math>, <math>C label("A",(0,2),NW);
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• 2018 AIME I Problems/Problem 14
  ...1P_2P_3EP_4P_5</math> be a heptagon. A frog starts jumping at vertex <math>S</math>. From any vertex of the heptagon except <math>E</math>, the frog may ...f left/right (L/R) paths of length <math>\le 11</math> that start at <math>S</math> and end at either <math>P_4</math> or <math>P_3</math>.
  4 KB (649 words) - 17:29, 22 December 2023
• 1960 AHSME Problems
  If <math>2</math> is a solution (root) of <math>x^3+hx+10=0</math>, then <math>h</math> equals: <math>\textbf{(A)}10\qquad
  21 KB (3,242 words) - 21:27, 30 December 2020
• 2014 AMC 12B Problems
  ...d have the same number of pennies and nickels. In cents, how much are Leah's coins worth? <math> \textbf{(A)}\ 33\qquad\textbf{(B)}\ 35\qquad\textbf{(C)}\ 37\qquad\textbf{(D)}\ 39\qqu
  13 KB (2,066 words) - 14:08, 1 November 2022
• 2014 AMC 10B Problems
  ...ld have the same number of pennies as nickels. In cents, how much are Leah's coins worth? <math> \textbf {(A) } 33 \qquad \textbf {(B) } 35 \qquad \textbf {(C) } 37 \qquad \textbf {(D)
  13 KB (2,011 words) - 21:54, 8 November 2022
• 2014 AMC 10B Problems/Problem 23
  A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is <math>\text{(A) } \dfrac32 \quad \text{(B) } \dfrac{1+\sqrt5}2 \quad \text{(C) } \sqrt3 \q
  6 KB (1,060 words) - 19:15, 11 August 2023
• 2014 AMC 12B Problems/Problem 19
  A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is real r=(3+sqrt(5))/2;
  4 KB (703 words) - 16:24, 9 September 2022
• 2014 AIME II Problems
  ...s Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is pai ...has none of the three risk factors given that he does not have risk factor A is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are re
  8 KB (1,410 words) - 00:04, 29 December 2021
• 2014 AIME I Problems/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 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
• 2014 AIME I Problems/Problem 15
  ...is a positive integer not divisible by the square of any prime. Find <math>a+b+c</math>. ...=90^\circ</math>. But <math>DF=FE</math>, so <math>\triangle DEF</math> is a 45-45-90 triangle. Letting <math>DG=3x</math>, we have that <math>EG=4x</ma
  10 KB (1,643 words) - 22:30, 28 January 2024
• 1973 AHSME Problems
  A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length <math> \textbf{(A)}\ 3\sqrt3\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 6\sqrt3\qquad\textbf{(D
  18 KB (2,788 words) - 13:55, 20 February 2020
• 1969 AHSME Problems/Problem 35
  ...n, as <math>m</math> is made arbitrarily close to zero, the value of <math>r</math> is: <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
• 1952 AHSME Problems/Problem 48
  ...if they traveled in the same direction, but would pass each other in <math>t</math> hours if they traveled in opposite directions. The ratio of the spee <math>\text{(A) } \frac {r + t}{r - t} \qquad
  2 KB (370 words) - 01:27, 31 January 2023
• 1986 AHSME Problems
  <math>\textbf{(A)}\ 2y \qquad <math>\textbf{(A)}\ y = \frac{1}{3} x + 8 \qquad
  17 KB (2,512 words) - 18:30, 12 October 2023
• 1958 AHSME Problems/Problem 47
  <math> ABCD</math> is a rectangle (see the accompanying diagram) with <math> P</math> any point on MP("A",(-2,1),NW);MP("B",(2,1),NE);MP("C",(2,-1),SE);MP("D",(-2,-1),SW);
  2 KB (368 words) - 14:07, 28 August 2022
• 1977 AHSME Problems
  <math>\text{(A)}\ x \qquad <math>\textbf{(A)}\ \text{ equiangular}\qquad
  15 KB (2,412 words) - 05:09, 27 November 2020
• 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
• 1969 IMO Problems/Problem 4
  ...Prove that <math>\gamma_1, \gamma_2</math>, and <math>\gamma_3</math> have a second tangent in common. ...ath>K_2</math> is the intersection of the inversion circle <math>(B, r_B = a)</math> with the line <math>AB</math>.
  5 KB (904 words) - 13:42, 29 January 2021
• 2007 iTest Problems/Problem 21
  ...1's in a row and randomly writes + or - between each pair of consecutive 1's. <math>\text{(A) }0\qquad
  2 KB (247 words) - 03:34, 14 June 2018
• 2007 iTest Problems/Problem 22
  <math>\text{(A) } 0 \quad \text{(R) } 17 \quad
  929 bytes (137 words) - 22:05, 10 January 2019
• 2007 iTest Problems/Problem 25
  Ted's favorite number is equal to Find the remainder when Ted’s favorite number is divided by 25.
  2 KB (270 words) - 14:35, 29 July 2018
• 2007 iTest Problems/Problem 23
  <math>\text{(A) } 0\quad \text{(R) } -8\qquad
  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
• 2007 IMO Problems/Problem 6
  Let <math>n</math> be a positive integer. Consider <cmath>S=\{(x,y,z)~:~x,y,z\in \{0,1,\ldots,n \},~x+y+z>0\}</cmath>
  3 KB (506 words) - 13:38, 23 March 2024
• 2014 IMO Problems/Problem 3
  ...ar from <math>A</math> to <math>BD</math>. Points <math>S</math> and <math>T</math> lie on sides <math>AB</math> and <math>AD</math>, respectively, such pair A = (0,1), C=(0,-1);
  6 KB (1,071 words) - 03:58, 8 September 2018
• 1964 AHSME Problems
  <math>\textbf{(A)}\ \log_{10}50 \qquad <math>\textbf{(A)}\ \text{a parabola} \qquad
  19 KB (2,907 words) - 14:16, 20 February 2020
• 1963 AHSME Problems
  <math>\textbf{(A)}\ (0,0)\qquad <math>\textbf{(A)}\ -14 \qquad
  20 KB (3,122 words) - 14:17, 20 February 2020
• 1961 AHSME Problems
  <math>\textbf{(A)}\ \frac{1}{25} \qquad An automobile travels <math>a/6</math> feet in <math>r</math> seconds. If this rate is maintained for <math>3</math> minutes, how
  18 KB (2,905 words) - 18:33, 5 April 2023
• 1957 AHSME Problems
  ...lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is: <math>\textbf{(A)}\ 9\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 5\qqua
  26 KB (3,950 words) - 21:09, 31 August 2020
• 2013 UNCO Math Contest II Problems
  ...equal to <math>4^3</math>, so <math>64</math> is both a perfect square and a perfect cube. (a) Find the smallest positive integer multiple of <math>12</math> that is a perfect square.
  11 KB (1,648 words) - 09:55, 20 December 2021
• 2011 UNCO Math Contest II Problems
  pair P,R; R=unit(C1-D1)+D1;
  5 KB (846 words) - 22:10, 7 November 2014
• 2010 UNCO Math Contest II Problems
  Find a <math>3</math>-digit integer less than <math>200</math> where each digit is pair A,B,C,D;
  5 KB (826 words) - 22:12, 7 November 2014
• 2010 UNCO Math Contest II Problems/Problem 3
  Suppose <math>r, s</math>, and <math>t</math> are three different positive integers and that their product is <mat What is the smallest possible value of the sum <math>r+s+t</math>?
  550 bytes (83 words) - 02:46, 13 January 2019
• 2007 UNCO Math Contest II Problems/Problem 7
  ...sum <math>S= 1+ \frac{1}{3}+\frac{1}{3^2}+ \frac{1}{3^3}+ \cdots</math> as a reduced fraction. (b) Express the infinite sum <math>T=\frac{1}{5}+ \frac{1}{25}+ \frac{2}{125}+ \frac{3}{625}+ \frac{5}{3125}+ \c
  1 KB (156 words) - 05:26, 12 January 2019
• 2005 CEMC Gauss (Grade 7) Problems/Problem 15
  <math>\text{(A)}\ 18 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 16 \qquad \text{(D)}\ 6 \qqu label("$S$",(0,0),SW);
  2 KB (349 words) - 02:41, 23 October 2014
• 1986 AHSME Problems/Problem 28
  ...P, AQ</math> and <math>AR</math> are the perpendiculars dropped from <math>A</math> onto <math>CD, CB</math> extended and <math>DE</math> extended, ...62), P=(C+D)/2, Q=C+3.10*dir(C--B), R=D+3.10*dir(D--E), S=C+4.0*dir(C--B), T=D+4.0*dir(D--E);
  4 KB (702 words) - 17:13, 17 April 2020
• 2015 AMC 12A Problems
  {{AMC12 Problems|year=2015|ab=A}} <math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf
  13 KB (2,117 words) - 12:33, 24 August 2023
• 2015 AMC 12A Problems/Problem 18
  ...ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>? <math>\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \q
  7 KB (1,193 words) - 14:25, 25 July 2022
• 2015 AMC 12B Problems/Problem 24
  ...Furthermore, <math>AB = CD = 39</math> and <math>PQ = 48</math>. Let <math>R</math> be the midpoint of <math>\overline{PQ}</math>. What is <math>\overli <math>\textbf{(A)}\; 180 \qquad\textbf{(B)}\; 184 \qquad\textbf{(C)}\; 188 \qquad\textbf{(D)
  7 KB (1,188 words) - 19:18, 19 October 2023
• 2015 AIME I Problems/Problem 8
  ...math>n</math>. Find the smallest positive integer satisfying <math>s(n) = s(n+864) = 20</math>. ...it has to have 3 digits, because if it had only two, the maximum of <math>s(n)</math> is 18.
  10 KB (1,797 words) - 14:06, 21 January 2024
• 2015 AIME I Problems/Problem 10
  Let <math>f(x)</math> be a third-degree polynomial with real coefficients satisfying Since <math>f(x)</math> is a third degree polynomial, it can have at most two bends in it where it goes
  8 KB (1,474 words) - 10:00, 10 November 2023
• 2015 AIME II Problems/Problem 7
  ...math> on <math>\overline{AC}</math>, and vertices <math>R</math> and <math>S</math> on <math>\overline{BC}</math>. In terms of the side length <math>PQ ...ath> is half the area of the triangle. Using Heron's formula, since <math>s = \frac{12 + 17 + 25}{2} = 27</math>,
  7 KB (1,180 words) - 14:08, 14 February 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
• 2016 AMC 10A Problems/Problem 25
  <math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qqu ...,2),(1,2),(0,2)</math>. Thus our answer is <math>5\times 3=\boxed{\textbf{(A) }15}.</math>
  7 KB (1,135 words) - 09:28, 5 September 2022
• 2015 USAMO Problems/Problem 2
  ...n segment <math>\overline{PQ}</math>, show that <math>M</math> moves along a circle. <cmath>A=(1,0) P=(1-a,b), Q=(1-a,-b)</cmath>,
  5 KB (902 words) - 09:58, 20 August 2021
• 2015 USAJMO Problems/Problem 3
  ...n segment <math>\overline{PQ}</math>, show that <math>M</math> moves along a circle. pair A=(1,0);
  4 KB (760 words) - 16:45, 29 April 2020
• Aops font
  The AoPS font is a font that can be used to make certain special symbols in the forum. You enc | <span class="aops-font">a</span>
  6 KB (924 words) - 21:31, 22 February 2024
• 2017 AIME I Problems
  ...points are designated on <math>\triangle ABC</math>: the 3 vertices <math>A</math>, <math>B</math>, and <math>C</math>; <math>3</math> other points on ...der is always the positive integer <math>s \neq r</math>. Find <math>m+n+r+s</math>.
  7 KB (1,163 words) - 16:43, 2 June 2022
• 2016 AIME II Problems/Problem 1
  ...ex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats <math>5</math> of his peanuts, Betty eats ...solving for <math>r</math> gives <math>r=\dfrac{4}{3}</math>, since <math>r>1</math>, so Alex had <math>\dfrac{3}{4} \cdot 144=\boxed{108}</math> peanu
  5 KB (788 words) - 02:50, 1 March 2024
• 2016 AMC 12A Problems/Problem 24
  ...mial <math>x^3-ax^2+bx-a</math> are real. In fact, for this value of <math>a</math> the value of <math>b</math> is unique. What is this value of <math>b <math>\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad
  3 KB (568 words) - 17:14, 20 April 2024
• 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
• 2017 UNM-PNM Statewide High School Mathematics Contest II Problems
  ...<math>\sqrt{STARS} = SAT</math>, what are <math>A, R, S</math>, and <math>T</math>? (a) Determine <math>f\circ g(x)</math> and <math>g\circ f(x)</math>.
  5 KB (846 words) - 03:36, 19 January 2019
• 2016 AMC 12B Problems/Problem 14
  ...the series is <math>1</math>. What is the smallest possible value of <math>S?</math> <math>\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad
  7 KB (1,264 words) - 14:57, 5 July 2022
• 2016 AIME I Problems/Problem 10
  A strictly increasing sequence of positive integers <math>a_1</math>, <math>a We first create a similar sequence where <math>a_1=1</math> and <math>a_2=2</math>. Continuin
  5 KB (883 words) - 15:24, 6 January 2024
• 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 9
  Since all the terms of the sequences are integers, and 100 isn't very big, we should just try out the possibilities for <math>b_2</math>. Wh We have <math>a_k=r^{k-1}</math> and <math>b_k=(k-1)d</math>. First, <math>b_{k-1}<c_{k-1}=100<
  6 KB (983 words) - 01:18, 2 February 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
• 1954 AHSME Problems/Problem 37
  ...ath>, <math>PQ</math> extended to <math>D</math> and <math>\angle n</math> a right angle, then: path anglemark2(pair A, pair B, pair C, real t=8, bool flip=false)
  3 KB (560 words) - 10:11, 3 May 2020
• 2016 USAMO Problems/Problem 4
  Find all functions <math>f:\mathbb{R}\rightarrow \mathbb{R}</math> such that for all real numbers <math>x</math> and <math>y</math>, < ...that <math>f(t) = 0 \implies f(-t) = 0</math> for all <math>t \in \mathbb{R}.</math>
  8 KB (1,391 words) - 00:13, 28 February 2020
• 2016 USAMO Problems/Problem 3
  ...</math> are colinear. This can be proved by the Trigonometric Form of Ceva's Theorem for <math>\triangle I_AI_BI_C.</math> We find point <math>T</math> on line <math>YZ,</math> we prove that <math>TY \perp OI_A</math> an
  6 KB (998 words) - 21:36, 17 October 2022
• 2013 APMO Problems/Problem 5
  ...and <math>\omega</math>. Prove that <math>B</math>, <math>E</math>, <math>R</math> are collinear. ...th>ABCD</math> is harmonic, hence the tangents at <math>C</math> and <math>A</math> concur on <math>BD</math> at <math>X</math>, say.
  5 KB (927 words) - 21:28, 28 May 2016
• 1990 IMO Problems/Problem 3
  Assume that <math> P\ne\{3\}</math> and let's try to determine the second smallest element <math> q = \min (P\setminus\{3 ...> n</math>, a non-sense). Therefore, <math> n = 3^t n'</math> where <math> t = 0</math> or <math> 1</math> and <math> n'</math> does not have prime divi
  6 KB (999 words) - 18:15, 7 August 2021
• 1987 USAMO Problems/Problem 3
  ...<math>X</math>, then <math>x\cdot r(x)</math> and <math>(x + (1 - x) \cdot r(x) )</math> both belong to <math>X</math>. ...>s(x)</math> are distinct elements of <math>X</math>, then <math>r(x) \neq s(x)</math> for any <math>0 < x < 1</math>.
  2 KB (357 words) - 16:49, 18 May 2018
• 1994 USAMO Problems/Problem 5
  ...\sigma(U)}{|S|} = \pi(S) </cmath> for all integers <math>\, m \geq \sigma(S)</math>. ...<math>\left|\bigcup_{k \in U}A_k\right| = \sum_{\emptyset \neq U \subseteq S} (-1)^{|U|+1}\left|\bigcap_{k \in U}A_k\right|</math>.
  3 KB (628 words) - 10:42, 4 August 2023
• 2016 AMC 8 Problems/Problem 24
  ...h>3</math>, <math>4</math>, and <math>5</math> are each used once to write a five-digit number <math>PQRST</math>. The three-digit number <math>PQR</mat <math>\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \text
  5 KB (781 words) - 22:06, 21 January 2024
• 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
  {{AMC10 Problems|year=2017|ab=A}} <math>\textbf{(A)}\ 70\qquad\textbf{(B)}\ 97\qquad\textbf{(C)}\ 127\qquad\textbf{(D)}\ 159\q
  15 KB (2,285 words) - 18:02, 28 October 2023
• 2017 AMC 10A Problems/Problem 21
  ...ertex of the square coincides with the right-angle vertex of the triangle. A square with side length <math>y</math> is inscribed in another right triang <math>\textbf{(A) } \dfrac{12}{13} \qquad \textbf{(B) } \dfrac{35}{37} \qquad \textbf{(C) }
  3 KB (461 words) - 22:24, 8 January 2023
• 2017 AMC 10A Problems/Problem 17
  Distinct points <math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math> lie on the circle <math>x^{2}+y^{2}=25</math> and have integer coord <math>\textbf{(A) } 3 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 3\sqrt{5} \qquad \textbf{(
  5 KB (864 words) - 01:42, 17 August 2023
• 2017 AMC 12A Problems/Problem 25
  The vertices <math>V</math> of a centrally symmetric hexagon in the complex plane are given by <math>\textbf{(A) } \dfrac{5\cdot11}{3^{10}} \qquad \textbf{(B) } \dfrac{5^2\cdot11}{2\cdot3
  18 KB (2,878 words) - 01:47, 16 December 2023
• 2017 AMC 12A Problems/Problem 19
  ...ertex of the square coincides with the right-angle vertex of the triangle. A square with side length <math>y</math> is inscribed in another right triang <math> \textbf{(A)}\ \frac{12}{13}
  2 KB (351 words) - 14:55, 10 June 2023
• 2017 AMC 12A Problems/Problem 16
  ...ly tangent to each other and internally tangent to the largest semicircle. A circle centered at <math>P</math> is drawn externally tangent to the two sm <math> \textbf{(A)}\ \frac{3}{4}
  13 KB (1,982 words) - 17:12, 20 December 2022
• 2017 AMC 10B Problems/Problem 18
  ...reen. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different pair A, B, C, D, E, F;
  7 KB (1,057 words) - 23:27, 27 August 2022
• 2017 AMC 12B Problems/Problem 22
  ...laced in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one c <math>\textbf{(A)}\quad \dfrac{7}{576} \qquad \qquad \textbf{(B)}\quad \dfrac{5}{192} \qquad
  12 KB (1,935 words) - 11:37, 16 December 2023
• 2017 AIME I Problems/Problem 15
  pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); real t = .385, s = 3.5*t-1;
  22 KB (3,622 words) - 17:11, 6 January 2024
• 2017 AIME II Problems/Problem 3
  ...he triangle is closer to vertex <math>B</math> than to either vertex <math>A</math> or vertex <math>C</math> can be written as <math>\frac{p}{q}</math>, pair A,B,C,D,X,Z,P;
  9 KB (1,411 words) - 22:18, 29 January 2024
• 2017 USAJMO Problems/Problem 6
  ...unterclockwise, and writing the color of the points in that order - either R or B. For example, possible sequences include <math>RB</math>, <math>RBBR</ ...h> from my sequence when I choose them. I define this to be a move. Hence, a possible move sequence of <math>BBRRRB</math> is:
  8 KB (1,465 words) - 15:30, 12 June 2020
• 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
• 2017 AMC 8 Problems/Problem 25
  ...th>\overarc{TR}</math> and <math>\overarc{SR}</math> are each one-sixth of a circle with radius 2. What is the area of the region shown? ...abel("$S$", (1,1.732), W); label("$T$", (3,1.732), E); label("$R$", (2,0), S);</asy>
  4 KB (711 words) - 21:00, 16 January 2024
• MIE 96/97
  <math>\begin{cases}x^y=y^x\\y=ax\end{cases}~\mbox{where }a\neq1\mbox{ and }a>0</math> ...> to <math>A</math> and from <math>P</math> to <math>B</math> are given by a constant <math>k</math>. Justify your answer discussing every possibility f
  2 KB (377 words) - 14:52, 7 January 2018
• 2018 AMC 10A Problems/Problem 22
  ...0<\gcd(d, a)<100</math>. Which of the following must be a divisor of <math>a</math>? <math>\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \q
  10 KB (1,599 words) - 04:51, 6 August 2023
• 2018 AMC 12B Problems/Problem 16
  ...o form a convex regular polygon, three of whose vertices are labeled <math>A,B,</math> and <math>C</math>. What is the least possible area of <math>\tri <math>\textbf{(A) } \frac{1}{6}\sqrt{6} \qquad \textbf{(B) } \frac{3}{2}\sqrt{2}-\frac{3}{2}
  11 KB (1,708 words) - 12:01, 18 March 2023
• 2018 AIME I Problems/Problem 1
  ...t) linear factors with integer coefficients. Find the remainder when <math>S</math> is divided by <math>1000</math>. Then, <math>a=c+d</math> and <math>b=cd</math>.
  10 KB (1,670 words) - 16:38, 15 January 2024
• 2016 JBMO Problems/Problem 4
  ...y once in every <math>2 \times 2</math> subtable.The sum of all numbers of a regular table is called the total sum of the table. With any four numbers, In a regular table either each row contains exactly two of the numbers or each c
  5 KB (799 words) - 01:20, 12 January 2023
• 2018 AIME I Problems/Problem 13
  ...gn*}bc\sin\frac A2\sin\frac B2\sin\frac C2&=bc\cdot\dfrac{(a-b+c)(b-c+a)(c-a+b)}{8abc}\\&=\dfrac{(30-32+34)(32-34+30)(34-30+32)}{8\cdot 32}=\boxed{126}. ...minimize the area for <math>[A I_1 I_2]</math> because <math>b, c, \angle A, \angle B, \angle C</math> are all constants while only <math>\sin \alpha</
  13 KB (2,200 words) - 21:36, 6 January 2024
• 2018 AIME I Problems/Problem 12
  ...t random among all subsets of <math>U</math>, the probability that <math>s(T)</math> is divisible by <math>3</math> is <math>\frac{m}{n}</math>, where < The question asks us for the probability that a randomly chosen subset of the set of the first 18 positive integers has the
  26 KB (4,044 words) - 13:58, 24 January 2024
• 2007 iTest Problems/Ultimate Question
  ...tions are from the "Ultimate Question" of the [[2007 iTest]], but with the T-values substituted. This is for people who want to work on the problems wi Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesi
  4 KB (636 words) - 00:41, 25 June 2018
• 2017 IMO Problems
  ...math> such that there exists a number <math>A</math> such that <math>a_n = A</math> for infinitely many values of <math>n</math>. Let <math>\mathbb{R}</math> be the set of real numbers , determine all functions
  4 KB (720 words) - 12:12, 5 August 2021
• 2008 iTest Problems/Problem 33
  One night, over dinner Jerry poses a challenge to his younger children: "Suppose we travel <math>50</math> miles ...the beautiful landscape your mother likes to photograph from the passenger's seat, we
  2 KB (421 words) - 20:19, 6 August 2023
• 1973 AHSME Problems/Problem 35
  ...hords <math>MP</math>, <math>PQ</math>, and <math>NR</math> are each <math>s</math> units long and chord <math>MN</math> is <math>d</math> units long. label("1", (5,0), S);
  7 KB (1,181 words) - 20:41, 17 April 2022
• 2006 iTest Problems/Problem 27
  Line <math>\ell</math> passes through <math>A</math> and into the interior of the equilateral triangle <math>ABC</math>. real r = 31, t = -10;
  3 KB (488 words) - 12:54, 7 December 2018
• 2008 iTest Problems/Problem 96
  ...square of any prime, and <math>\gcd(t,r,p)=1</math>. Compute <math>p+q+r+s+t</math>. Using the definition of a cosine, we have <math>XC = 8\sqrt{3} \cos(x)</math>. By the [[Law of Sines]
  3 KB (558 words) - 20:13, 4 January 2019
• 2008 iTest Problems/Problem 98
  ...<math>AB=7</math>, <math>BC=9</math>, <math>CD=15</math>, and there exists a circle, lying inside the quadrilateral and having center <math>I</math>, th ...pposite sides of <math>QMPN</math> have equal length, <math>QMPN</math> is a [[parallelogram]]. Thus, the diagonals bisect each other, and <math>QI = IP
  3 KB (586 words) - 23:47, 8 January 2019
• 2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 2
  ...<math>\sqrt{STARS} = SAT</math>, what are <math>A, R, S</math>, and <math>T</math>? <math>A = 3, T = 9, S = 1</math> and <math>R = 2</math>
  2 KB (385 words) - 12:27, 21 August 2023
• 2019 AMC 10B Problems/Problem 22
  ...lvia, and Ted play the following game. Each starts with <math> \$1</math>. A bell rings every <math>15</math> seconds, at which time each of the players <math>\textbf{(A) } \frac{1}{7} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}
  9 KB (1,568 words) - 11:15, 18 October 2023
• 2019 AMC 12A Problems/Problem 17
  ...h> for <math>k = 2</math>, <math>3</math>, <math>....</math> What is <math>a+b+c</math>? <math>\textbf{(A)} \; -6 \qquad \textbf{(B)} \; 0 \qquad \textbf{(C)} \; 6 \qquad \textbf{(D
  4 KB (731 words) - 22:18, 12 March 2023
• 2019 AMC 10B 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?
  15 KB (2,458 words) - 23:52, 12 November 2023
• 2019 AIME I Problems/Problem 11
  ...ath>\triangle ABC</math>. An ''excircle'' of <math>\triangle ABC</math> is a circle in the exterior of <math>\triangle ABC</math> that is tangent to one ...However, the sum of the radius of <math>\omega</math> and <math>\frac{rs}{s-b}</math> is equivalent to the distance between the incenter and the the <m
  21 KB (3,915 words) - 19:55, 10 October 2023
• 2019 AIME I Problems/Problem 12
  ...(z))</math> are the vertices of a right triangle in the complex plane with a right angle at <math>f(z)</math>. There are positive integers <math>m</math ...\frac{f(f(z))-f(z)}{z-f(z)}\in i\mathbb R .</cmath>However, <math>f(t)-t=t(t-20)</math>, so
  8 KB (1,534 words) - 22:17, 28 December 2023
• 2017 IMO Problems/Problem 4
  ...point of <math>\Gamma</math> and <math>\ell</math> that is closer to <math>R</math>. Line <math>AJ</math> meets <math>\Omega</math> again at <math>K</ma We construct circle <math>\omega</math> centered at <math>R</math> which maps <math>\Gamma</math> into <math>\Gamma.</math>
  4 KB (681 words) - 01:41, 19 November 2023
• Pascal's Triangle and Pythagorean Theorem
  = Pascal's Identity = Pascal's Identity states that
  15 KB (2,425 words) - 09:25, 13 February 2020
• What is the definition of Pure Mathematics?
  ...is welcomed to edit, BUT ONLY IN GOOD WAYS! The AoPS Secret Governemtn has a backup of this page. ...ng it. For example, can a butterfly that flaps its wings in Brazil trigger a tornado in Texas?
  35 KB (5,882 words) - 18:08, 28 June 2021
• 2006 iTest Problems/Problem 20
  <math>\textbf{(A) }2\qquad \textbf{(R) }26\qquad
  4 KB (334 words) - 12:49, 15 January 2020
• 2020 AIME I Problems/Problem 8
  ...On the first day, it starts at point <math>O</math>, faces east, and walks a distance of <math>5</math> units due east. Each night the bug rotates <math pair O, A, B, C, D, F, G, H, I, P, X;
  9 KB (1,380 words) - 16:12, 2 January 2024
• 2020 AIME I Problems/Problem 11
  ...and <math>g(x)=x^2+cx+d.</math> Find the number of ordered triples <math>(a,b,c)</math> of integers with absolute values not exceeding <math>10</math> ...math>f(2),f(4)</math> are the roots of <math>g</math>. This means by Vieta's, that:
  6 KB (964 words) - 22:33, 18 January 2024
• 2020 AIME I Problems/Problem 13
  ...ath>p</math> are relatively prime positive integers, and <math>n</math> is a positive integer not divisible by the square of any prime. Find <math>m+n+p label("$A$", (-6.8002301023571095,8.267690318323321), NE * labelscalefactor);
  35 KB (5,215 words) - 23:08, 29 October 2023
• 2020 AIME I Problems/Problem 14
  ...ons, <math>x=3,4,a,b.</math> Find the sum of all possible values of <math>(a+b)^2.</math> ...ain. Now, it's easy to find that <math>a = -2.5, b = -3.5</math>, yielding a value of <math>36</math>. Finally, we add <math>49 + 36 = \boxed{085}</math
  10 KB (1,819 words) - 19:34, 17 September 2023
• 2020 AIME I Problems/Problem 15
  The following is a power of a point solution to this menace of a problem: var phi=75.5, theta=130, r=4.8;
  16 KB (2,678 words) - 22:45, 27 November 2023
• 2023 AIME I Problems/Problem 12
  pair A, B, C, D, E, F, P; A = 55*sqrt(3)/3 * dir(90);
  16 KB (2,592 words) - 15:40, 13 April 2024
• Factoring Quadratics
  ...oring a quadratic is to turn the quadratic into a product of 2 [[binomial]]s. ...the roots of the quadratic, and where <math>r+s=-b</math> and <math>r\cdot s=c</math>.
  2 KB (367 words) - 12:15, 15 July 2021
• GMAAS
  4. The Infinity Numeral, PI, is the second deity of the Almighty Gmaas's heaven. 5. The Infinite Logarithm, E, is the third deity of the Almighty Gmaas's heaven.
  85 KB (13,971 words) - 18:02, 16 May 2024
• 2006 iTest Problems/Problem U7
  ...ly. Find the perimeter of the triangle ABC which minimizes <math>\frac{s}{r}</math>. <cmath>r = \frac{x+696-y}{2} = 348 - a</cmath>
  4 KB (674 words) - 16:03, 25 February 2021
• 2019 AMC 8 Problems
  Ike and Mike go into a sandwich shop with a total of <math>\$30.00</math> to spend. Sandwiches cost <math>\$4.50</math> <math>\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textb
  17 KB (2,585 words) - 15:27, 4 May 2024
• 2020 AMC 10A Problems/Problem 22
  <math>\textbf{(A) } 22 \qquad\textbf{(B) } 23 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 25 ...th>3 \left\lfloor \frac{999}{n} \right\rfloor \pm 1</math>, so it is never a multiple of <math>3</math>. Thus, we are looking for all <math>n \neq 1</ma
  12 KB (1,978 words) - 22:13, 29 September 2023
• 2020 AMC 10A Problems/Problem 23
  ...by another reflection across the <math>x</math>-axis will not return <math>T</math> to its original position.) <math>\textbf{(A) } 12 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 17 \qquad \textbf{(D) }
  8 KB (1,247 words) - 15:45, 23 September 2023
• 2020 AMC 10A Problems/Problem 24
  <math>\textbf{(A) } 12 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 21\ ...start bashing by plugging in numbers that satisfy these conditions: <math>a=4</math> is the first number that works so we get <math>\ell = 33</math>, <
  17 KB (2,544 words) - 12:09, 1 September 2023
• 2020 AMC 12A Problems/Problem 24
  ...P=1</math>, <math>BP=\sqrt{3}</math>, and <math>CP=2</math>. What is <math>s</math>? <math>\textbf{(A) } 1+\sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3
  16 KB (2,509 words) - 17:49, 8 February 2024
• 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 10B Problems/Problem 18
  ...peration four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to <math>\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad
  11 KB (1,928 words) - 22:40, 12 November 2023
• 2020 AMC 10B Problems/Problem 14
  As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles <math>\textbf{(A) } 6\sqrt3 - 3\pi \qquad \textbf{(B) } \frac{9\sqrt3}{2} - 2\pi \qquad \tex
  17 KB (2,392 words) - 12:36, 24 December 2023
• 2020 AMC 12B Problems/Problem 12
  ...th> be a chord in the circle that intersects <math>\overline{AB}</math> at a point <math>E</math> such that <math>BE=2\sqrt5</math> and <math>\angle AEC <math>\textbf{(A)}\ 96 \qquad\textbf{(B)}\ 98 \qquad\textbf{(C)}\ 44\sqrt5 \qquad\textbf{(D
  10 KB (1,599 words) - 10:05, 6 October 2023
• 2021 AMC 10A Problems
  {{AMC10 Problems|year=2021|ab=A}} What is the value of<cmath>(2^2-2)-(3^2-3)+(4^2-4)?</cmath><math>\textbf{(A)} ~1 \qquad\textbf{(B)} ~2 \qquad\textbf{(C)} ~5 \qquad\textbf{(D)} ~8 \qqu
  15 KB (2,302 words) - 23:41, 14 April 2024
• 2024 AIME II Problems
  ...n a garden spade. In addition, each of the <math>900</math> residents owns a bag of candy hearts. There are <math>437</math> residents who own exactly t A list of positive integers has the following properties:
  8 KB (1,236 words) - 23:11, 12 March 2024
• 2022 AMC 10A Problems
  {{AMC10 Problems|year=2022|ab=A}} <math>\textbf{(A)}\ \frac{31}{10}\qquad\textbf{(B)}\ \frac{49}{15}\qquad\textbf{(C)}\ \frac{
  18 KB (2,662 words) - 02:08, 9 March 2024
• AMC 12C 2020
  <math>\mathrm{(A) \ } -10\qquad \mathrm{(B) \ } -3\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) ...th> plane in the shape of a circle, centered at <math>(5, 6)</math> with a a radius of <math>8</math>. The area that is in the field but above the line
  9 KB (1,450 words) - 18:33, 21 April 2020
• 1976 AHSME Problems/Problem 6
  ...ber and the negative of one of the solutions of <math>x^2-3x+c=0</math> is a solution of <math>x^2+3x-c=0</math>, then the solutions of <math>x^2-3x+c=0 <math>\textbf{(A) }1,~2\qquad \textbf{(B) }-1,~-2\qquad \textbf{(C) }0,~3\qquad \textbf{(D)
  807 bytes (143 words) - 20:11, 12 July 2020
• 2023 AIME I Problems/Problem 11
  Using casework on where the consecutive element pair is, there is a unique consecutive element pair that satisfies the conditions. It is easy t ...by <math>N_1 \left( m \right)</math> the number of subsets of a set <math>S</math> that consists of <math>m</math> consecutive integers, such that each
  15 KB (2,414 words) - 06:57, 26 November 2023
• 2020 AIME II Problems/Problem 7
  ...<math>r</math> lies within both cones. The maximum possible value of <math>r^2</math> is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</mat ...and thus must be at the intersection of their axes of symmetry. Let <math>A</math> be the point in the cross section where the bases of the cones meet,
  7 KB (951 words) - 10:11, 26 December 2023
• 2020 AIME II Problems/Problem 11
  ...hese three common roots are distinct. If <math>Q(0) = 2</math>, then <math>R(0) = \frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relati Let <math>Q(x) = x^2 + ax + 2</math> and <math>R(x) = x^2 + bx + c</math>. We can write the following:
  6 KB (1,015 words) - 21:16, 28 December 2023
• 2020 AIME II Problems/Problem 13
  ...h>, and <math>EA=7</math>. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of <math>ABCDE< ...the area of triangle <math>ANM</math> is 108. Triangle <math>END</math> is a triangle with side lengths 6, 8, 10, so the area of two of them is 48, so t
  13 KB (2,197 words) - 23:00, 8 January 2024
• Superagh's Olympiad Notes
  SINCE MY COMPUTER WON'T LOAD THIS FOR SOME REASON, I'LL BE UPDATING THIS AS I GO THOUGH :) ...o this thing is almost the exact same format as Lcz's :P ). (Ok, actually, a LOT of credits to Lcz)
  4 KB (671 words) - 13:59, 22 July 2020
• 1954 AHSME Problems/Problem 47
  ...dius equal to <math>\frac{1}{2}AB</math>, meeting <math>AB</math> at <math>T</math>. Then <math>AT</math> and <math>TB</math> are the roots of: <math>\textbf{(A)}\ x^2+px+q^2=0\\ \textbf{(B)}\ x^2-px+q^2=0\\ \textbf{(C)}\ x^2+px-q^2=0\\
  971 bytes (170 words) - 16:43, 7 July 2020
• 2021 AIME I Problems/Problem 13
  ...pectively, intersect at distinct points <math>A</math> and <math>B</math>. A third circle <math>\omega</math> is externally tangent to both <math>\omega <font color=red><b>Claim.</b></font color> <math>DPEQ</math> is a harmonic quadrilateral.
  17 KB (2,852 words) - 03:59, 7 February 2024
• 2021 AIME I Problems/Problem 14
  ...1</math> is divisible by <math>2021</math> for all positive integers <math>a</math>. Find the sum of the prime factors in the prime factorization of <ma .... By Dirichlet's Theorem (Refer to the <b>Remark</b> section.), such <math>a</math> always exists.
  13 KB (2,185 words) - 02:28, 13 November 2023
• 2021 AIME I Problems/Problem 15
  ...m of the least element of <math>S</math> and the greatest element of <math>S</math>. ...with the vertex at <math>(0,-k),</math> and <math>x=2(y-20)^2-k</math> is a rightward-opening parabola with the vertex at <math>(-k,20).</math> We cons
  10 KB (1,742 words) - 02:31, 13 November 2023
• 2021 AMC 12A Problems
  {{AMC12 Problems|year=2021|ab=A}} <math>\textbf{(A) }0 \qquad \textbf{(B) }50 \qquad \textbf{(C) }52 \qquad \textbf{(D) }54 \q
  15 KB (2,383 words) - 09:49, 25 June 2023
• 2022 AIME I Problems/Problem 11
  ...th>ABCD</math> be a parallelogram with <math>\angle BAD < 90^\circ.</math> A circle tangent to sides <math>\overline{DA},</math> <math>\overline{AB},</m pair A,B,C,D,P,Q;
  16 KB (2,517 words) - 20:22, 31 January 2024
• 2022 AIME I Problems/Problem 12
  S_n = \sum | A \cap B | , ...ubsets of <math>\left\{ 1 , 2 , 3, \cdots , n \right\}</math> with <math>|A| = |B|</math>.
  9 KB (1,471 words) - 16:41, 1 February 2024
• 2022 AIME I Problems/Problem 14
  ...h> into two polygons of equal perimeter. Let <math>\triangle ABC</math> be a triangle where <math>BC = 219</math> and <math>AB</math> and <math>AC</math ...mple angle chasing reveals the condition is now equivalent to <math>\angle A=120^\circ</math>.
  16 KB (2,730 words) - 02:56, 4 January 2023
• 2022 AMC 12B Problems
  <math> \textbf{(A)}\ {-}2 \qquad real r = 3, s = 5, t = sqrt(r*r+s*s);
  15 KB (2,233 words) - 13:02, 10 November 2023
• 2021 Fall AMC 10B Problems
  <math>(\textbf{A})\: 10{,}000\qquad(\textbf{B}) \: 10{,}010\qquad(\textbf{C}) \: 10{,}110\qq pen s = linewidth(0.8)+fontsize(8);
  16 KB (2,450 words) - 00:13, 12 November 2023
• 2021 Fall AMC 10A Problems
  {{AMC10 Problems|year=2021 Fall|ab=A}} <math>\textbf{(A) } 7 \qquad\textbf{(B) } 21 \qquad\textbf{(C) } 49 \qquad\textbf{(D) } 64 \
  14 KB (2,162 words) - 21:33, 2 November 2023
• 2021 Fall AMC 10B Problems/Problem 25
  ...rger square as shown. The sum of all possible values for the area of <math>R</math> can be written in the form <math>\tfrac mn</math>, where <math>m</ma label("$3$",(11/2,9/2),S);
  16 KB (2,274 words) - 09:02, 10 December 2022
• 2021 Fall AMC 12B Problems/Problem 20
  A cube is constructed from <math>4</math> white unit cubes and <math>4</math> <math>\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\
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• 1978 IMO Problems/Problem 4
  In a triangle <math>ABC</math> we have <math>AB = AC.</math> A circle which is internally tangent with the circumscribed circle of the tri ...{s - a}{b} \cdot \frac{2R}{h}</math> Denote <math>h' = AL</math> the <math>A</math>-altitude of the triangle <math>\triangle APQ</math>. Then
  3 KB (541 words) - 17:04, 29 January 2021
• 1978 IMO Problems/Problem 3
  ...positive<math>.</math> The <math>n-th</math> positive integer which doesn't belong to the sequence is <math>f(f(n))+1.</math> Find <math>f(240).</math> ...h>-th number missing is <math>f(f(n))+1</math> and <math>f(f(n))</math> is a member of the sequence, it results that there are exactly <math>n-1</math>
  2 KB (448 words) - 17:01, 29 January 2021
• 2021 AMC 12A Problems/Problem 15
  ...ts are that the difference between the number of tenors and basses must be a multiple of <math>4</math>, and the group must have at least one singer. Le <math>\textbf{(A) } 47\qquad\textbf{(B) } 48\qquad\textbf{(C) } 83\qquad\textbf{(D) } 95\qqu
  7 KB (1,152 words) - 14:13, 29 February 2024
• 2021 AMC 12A Problems/Problem 21
  <math>\textbf{(A) }7 \qquad \textbf{(B) }9 \qquad \textbf{(C) }11 \qquad \textbf{(D) }13\qqu ...triangle formed by the points <math>P_1 = (r,0)</math>, <math>P_2 = \left(-r,\sqrt 3\right)</math>, and <math>P_3 = \left(-2r,\sqrt 2\right)</math> lies
  11 KB (1,858 words) - 18:33, 1 August 2022
• 2021 AMC 12A Problems/Problem 22
  <math>\textbf{(A) }{-}\frac{3}{49} \qquad \textbf{(B) }{-}\frac{1}{28} \qquad \textbf{(C) }\ ==Solution 1 (Complex Numbers: Vieta's Formulas)==
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• 2021 AMC 12A Problems/Problem 24
  ...a positive integer not divisible by the square of any prime. What is <math>a+b+c</math>? <math>\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }1
  16 KB (2,539 words) - 07:30, 30 December 2023
• 2020 AMC 8 Problems/Problem 4
  pair[] A; for (int i=0; i<7; ++i) { A[i] = rotate(60*i)*(1,0);}
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• 2020 AMC 8 Problems
  Luka is making lemonade to sell at a school fundraiser. His recipe requires <math>4</math> times as much water a <math>\textbf{(A) } 6\qquad\textbf{(B) } 8\qquad\textbf{(C) } 12\qquad\textbf{(D) } 18\qquad
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• 2020 INMO Problems/Problem 5
  ...r <math>n \geqslant 3</math> is called frameable if it is possible to draw a regular polygon with <math>n</math> sides all whose vertices lie on these l (a) Show that <math>3, 4, 6</math> are frameable.
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• The Devil's Triangle
  ==Generalized Wooga Looga Theorem (The Devil's Triangle)== ...C]}=1-\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}=\frac{rst+1}{(r+1)(s+1)(t+1)}</math>.
  3 KB (582 words) - 18:05, 23 March 2024
• 2020 AMC 8 Problems/Problem 21
  ...p paths are there from <math>P</math> to <math>Q?</math> (The figure shows a sample path.) <math>\textbf{(A) }28 \qquad \textbf{(B) }30 \qquad \textbf{(C) }32 \qquad \textbf{(D) }33 \
  7 KB (1,170 words) - 22:50, 19 January 2024
• Quartic Equation
  A '''quartic equation''' is an algebraic [[equation]] of the form ...ver, there are very clever methods for solving them by bringing it down to a [[Cubic Equation|cubic]]. I am going to list the simplest of the five. Also
  13 KB (2,376 words) - 16:56, 19 February 2024
• 2013 Mock AIME I Problems
  ...ommon tangent that passes through <math>P</math>. If this tangent is also a common tangent to <math>C_2</math> and <math>C_1</math>, find the radius of ...math>b</math>, <math>b+1</math> evenly divides <math>c</math>, and <math>c-a=10</math>.
  7 KB (1,149 words) - 17:16, 15 December 2020
• 1982 AHSME Problems/Problem 22
  In a narrow alley of width <math>w</math> a ladder of length <math>a</math> is placed with its foot at point <math>P</math> between the walls. ...wall at <math>Q</math>, the distance <math>k</math> above the ground makes a <math>45^\circ</math> angle with the ground.
  2 KB (383 words) - 22:47, 13 September 2021
• Functional equation for the zeta function
  The '''functional equation for Riemann zeta function''' is a result due to analytic continuation of [[Riemann zeta function]]: \zeta(s)=2^s\pi^{s-1}\sin\left(\pi s\over2\right)\Gamma(1-s)\zeta(1-s)
  4 KB (682 words) - 03:56, 13 January 2021
• 1971 AHSME Problems/Problem 20
  <math>\textbf{(A) }-1\qquad We can rewrite the equation as <math>x^2 + 2hx - 3 = 0.</math> By Vieta's Formulas, the sum of the roots is <math>-2h</math> and the product of the r
  828 bytes (143 words) - 20:09, 28 January 2021
• 2021 AMC 12B Problems/Problem 17
  ...and <math>\overline{CD}</math> with <math>AB>CD.</math> Line segments from a point inside <math>ABCD</math> to the vertices divide the trapezoid into fo pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75)
  3 KB (484 words) - 12:58, 10 September 2023
• 2021 AMC 12B Problems/Problem 20
  ...nd the degree of <math>R</math> is less than <math>2.</math> What is <math>R(z)?</math> <math>\textbf{(A) }{-}z \qquad \textbf{(B) }{-}1 \qquad \textbf{(C) }2021\qquad \textbf{(D)
  8 KB (1,315 words) - 11:43, 24 October 2023
• 2021 AMC 12B Problems/Problem 16
  ...h> where <math>1<a<b<c.</math> What is <math>g(1)</math> in terms of <math>a,b,</math> and <math>c?</math> ...2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}</math>
  4 KB (720 words) - 10:02, 1 October 2023
• 2009 Polish Mathematical Olympiad Third Round
  Each of the vertices of a convex hexagon is a center of a circle of radius equal to the not longer side of the hexagon which includes ...elements <math>A,B</math> of that subset there exists a point <math>C \in S</math> such that the area of triangle <math>ABC</math> is equal to <math>k<
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• 2021 AIME II Problems/Problem 9
  ...gcd(r,s)=1,}</math> then <math>\boldsymbol{\gcd(r,t)\cdot\gcd(s,t)=\gcd(rs,t).}</math></b> As <math>r</math> and <math>s</math> are relatively prime (have no prime divisors in common), this proper
  7 KB (1,212 words) - 15:54, 15 April 2024
• 2021 AIME II Problems/Problem 10
  triple O1, O2, O3, T1, T2, T3, A, L1, L2; A = (0,0,-36*sqrt(1105)/23);
  13 KB (2,124 words) - 01:00, 14 January 2023
• 2021 AIME II Problems/Problem 13
  ...st positive integer <math>n</math> for which <math>2^n + 5^n - n</math> is a multiple of <math>1000</math>. ...on to be true, <math>2^n \equiv n \pmod{5}</math> is necessary; it shouldn't take too long for us to go through the <math>20</math> possible LHS-RHS com
  16 KB (2,240 words) - 23:16, 26 January 2024
• G285 2021 MC10B
  <math>\textbf{(A)}\ 42\qquad\textbf{(B)}\ 43\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 45\qqu <math>\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\t
  4 KB (675 words) - 14:01, 28 May 2021
• G285 2021 MC10A
  <math>\textbf{(A)}\ -2\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 1\qquad ...h> with equal chances. What is the probability that the number selected is a perfect square, given <math>n=4</math>?
  8 KB (1,385 words) - 12:55, 23 June 2021
• 2021 GMC 12B
  When <math>30\%</math> of <math>x</math> is a positive perfect square integer, what is <math>min(x)</math> such that <mat <math>\textbf{(A)} ~9 \qquad\textbf{(B)} ~12 \qquad\textbf{(C)} ~30 \qquad\textbf{(D)} ~36 \
  11 KB (1,691 words) - 18:56, 25 April 2022
• 2021 Fall AMC 12B Problems/Problem 21
  <math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ ...> have magnitude of <math>1</math>, so the answer is <math>\boxed{\textbf{(A)}\ 0 }</math> ~lopkiloinm
  6 KB (1,078 words) - 17:50, 19 May 2024
• 2022 AIME II Problems/Problem 3
  ...th side length <math>6.</math> The five vertices of the pyramid all lie on a sphere with radius <math>\frac mn</math>, where <math>m</math> and <math>n< triple A = (-3,-3,0);
  4 KB (649 words) - 00:33, 29 January 2024
• 2022 AIME II Problems/Problem 11
  Let <math>ABCD</math> be a convex quadrilateral with <math>AB=2, AD=7,</math> and <math>CD=3</math> su pair A,B,C,D,M,H; real xb=71, xd=121;
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• 1955 AHSME Problems/Problem 29
  ...h>PB</math> is tangent to semicircle <math>RBT</math>; <math>SRT</math> is a straight line; pair O1=(0,0), O2=(3,0), Sp=(-2,0), R=(2,0), T=(4,0);
  2 KB (336 words) - 02:02, 14 October 2021
• 2022 AMC 10B Problems
  <math> \textbf{(A)}\ {-}2 \qquad real r = 3, s = 5, t = sqrt(r*r+s*s);
  15 KB (2,224 words) - 13:10, 20 February 2024
• 2023 AIME I Problems/Problem 9
  ...tegers in <math>\{-20,-19,-18,\ldots,18,19,20\},</math> such that there is a unique integer <math>m \not= 2</math> with <math>p(m) = p(2).</math> ...what we just need to do is to just find the number of ordered pairs <math>(a, b)</math> that work, and multiply it by <math>41.</math>
  5 KB (960 words) - 20:40, 27 December 2023
• 2021 Fall AMC 12A Problems/Problem 23
  ...real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial <math>\tilde{p}(x)</math> for which the sum of the r <math>\textbf{(A) } \dfrac{5}{16} \qquad\textbf{(B) } \dfrac{1}{2} \qquad\textbf{(C) } \dfra
  26 KB (4,697 words) - 22:19, 2 January 2024
• 2021 Fall AMC 10B Problems/Problem 11
  A regular hexagon of side length <math>1</math> is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the re
  6 KB (1,021 words) - 19:40, 1 November 2023
• 2021 Fall AMC 10B Problems/Problem 15
  ...}</math> and <math>\overline{CQ}</math> intersect at right angles at <math>R</math>, with <math>BR = 6</math> and <math>PR = 7</math>. What is the area real r = 3.5;
  11 KB (1,733 words) - 23:06, 13 May 2024
• 2021 Fall AMC 12B Problems
  <math>\textbf{(A)}\: 10{,}000\qquad\textbf{(B)} \: 10{,}010\qquad\textbf{(C)} \: 10{,}110\qq pen s = linewidth(0.8)+fontsize(8);
  14 KB (2,191 words) - 19:57, 12 November 2023
• 2022 IMO Problems/Problem 4
  ...be a convex pentagon such that <math>BC = DE</math>. Assume that there is a point <math>T</math> inside <math>ABCDE</math> with <math>TB = TD</math>, <math>TC = TE</
  2 KB (307 words) - 01:55, 19 November 2023
• 2022 USAMO Problems
  ...amber cells and <math>b</math> bronze cells such that no two of the <math>a+b</math> chosen cells lie in the same row or column. ...>2b</math>-gon <math>B</math> is formed by translating the black rods, and a convex <math>2w</math>-gon <math>W</math> is formed by translating the whit
  4 KB (762 words) - 13:45, 22 November 2023
• Newton's method
  ...guess can cause the recursion to cycle or diverge instead of converging to a root. ...exactly one root, as long as <math>f'(x_i) \neq 0</math> we can construct a [[Taylor polynomial#Tangent-line approximation|tangent-line approximation]]
  13 KB (2,298 words) - 23:34, 28 May 2023
• 2022 USAMO Problems/Problem 5
  ...eq t</math> are real numbers such that <math>f(s)\neq 0</math> and <math>f(t)\neq 0</math>. {{USAMO newbox|year=2022|num-b=4|num-a=6}}
  671 bytes (113 words) - 15:00, 27 March 2022
• Rank Nullity Theorem
  ...n of the linear transformation's image, and nullity is the dimension of it's kernel (the word nullity comes from the term "null space" which is used to <center><math>\text{nullity}(\text{T})+\text{rank}(\text{T})=\dim(\text{T}).</math></center>
  5 KB (893 words) - 22:41, 28 May 2022
• AoPSWiki:Sandbox
  A focused asymptote reference with examples and short code.<p style="margin-l <i>Well, it isn't the perfect graphics package, but we do think it is getting there asympto
  28 KB (4,808 words) - 19:43, 3 June 2022
• 2023 AMC 8 Problems
  <math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 18 A square piece of paper is folded twice into four equal quarters, as shown be
  29 KB (4,048 words) - 17:25, 21 February 2024
• 2022 AMC 10A Problems/Problem 25
  ...s <math>\frac{1}{4}</math> of the lattice points contained in <math>R \cup S.</math> See the figure (not drawn to scale). label(scale(1.3)*"$R$", (55/2,55/2));
  8 KB (1,433 words) - 12:02, 30 April 2023
• 2022 AMC 10B Problems/Problem 21
  Let <math>P(x)</math> be a polynomial with rational coefficients such that when <math>P(x)</math> is d is <math>2x+1</math>. There is a unique polynomial of least degree with these two properties. What is the su
  7 KB (1,219 words) - 23:07, 19 September 2023
• 2022 AMC 10B Problems/Problem 2
  real r = 3, s = 5, t = sqrt(r*r+s*s); pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0);
  3 KB (498 words) - 17:14, 29 June 2023
• 2022 AMC 12B Problems/Problem 11
  <math> \textbf{(A)}\ -2 \qquad ...e base of the exponents are the roots of the characteristic equation <math>r^3-1=0</math>. So we have
  5 KB (866 words) - 22:17, 27 October 2023
• Kimberling’s point X(26)
  ...t lines to the reference triangle's circumcircle at the reference triangle's vertices). ...le ABC.</math> Let <math>\Omega'</math> be circumcircle of <math>\triangle A'B'C'.</math>
  2 KB (297 words) - 14:14, 18 November 2022
• Kimberling’s point X(21)
  ...th> and <math>\triangle ABC</math> are concurrent at Schiffler point <math>S = X(21), \frac {OS}{SG} = \frac {3R}{2r}</math>. ...ler line <math>OG</math> of <math>\triangle ABC</math> at such point <math>S,</math> that <math>\frac {OS}{SG} = \frac {3R}{2r}</math>.
  3 KB (576 words) - 16:41, 23 November 2022
• 2023 AMC 8 Problems/Problem 15
  ...oday, after walking <math>5</math> blocks, Viswam discovers he has to make a detour, walking <math>3</math> blocks of equal length instead of <math>1</m real d,s; // gap and side
  6 KB (1,028 words) - 17:51, 11 May 2024
• 2024 AMC 8 Problems
  <math>\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\t <math>\textbf{(A) } 6.4\qquad\textbf{(B) } 6.504\qquad\textbf{(C) } 6.54\qquad\textbf{(D) }
  21 KB (2,856 words) - 17:51, 8 May 2024
• 2023 AMC 8 Problems/Problem 16
  ...y <math>\text{P}</math>s, <math>\text{Q}</math>s, and <math>\text{R}</math>s will appear in the completed table? label(scale(.9)*"\textsf{R}", (.5,2.5));
  13 KB (1,813 words) - 03:45, 24 January 2024
• 2023 AIME I Problems/Problem 8
  Rhombus <math>ABCD</math> has <math>\angle BAD < 90^\circ.</math> There is a point <math>P</math> on the incircle of the rhombus such that the distances pair A, B, C, D, O, P, R, S, T;
  17 KB (2,612 words) - 14:54, 3 July 2023
• 2023 AIME II Problems/Problem 8
  Because the answer must be a positive integer, it is just equal to the modulus of the product. Define <m ...roduct (two at a time, where <math>1</math> is treated as the identity) as a series of arrays:
  9 KB (1,284 words) - 23:37, 31 January 2024
• Bézout's Lemma
  ...result in number theory that implies many other theorems, such as [[Euclid's Lemma]] and the [[Chinese Remainder Theorem]]. ...ers <math>x</math> and <math>y</math> such that <math>15x + 6y = 3</math>. A solution <math>(x, y)</math> of this equation is <math>(1, -2)</math>.
  3 KB (562 words) - 11:19, 5 March 2023
• 2013 TSTST Problems
  ...ath>\ell_a</math> passes through the feet of the perpendiculars from <math>A</math> to <math>DB</math> and <math>DC</math>. Line <math>m_a</math> passes ...<math>b = a_k</math>. Find the maximum possible number of forced terms in a regular sequence with <math>1000</math> terms.
  5 KB (873 words) - 08:45, 24 February 2023
• 2023 USAJMO Problems
  ...epeatedly sliding dominoes. Find the maximum value of <math>k(C)</math> as a function of <math>n</math>. ...<math>B</math> decides to end the game. At this point, <math>B</math> gets a score, given by the number of unit squares in the largest (in terms of area
  4 KB (681 words) - 21:38, 19 March 2024
• 2023 USAMO Problems
  ...\rightarrow\mathbb{R}^{+}</math> such that, for all <math>x, y \in \mathbb{R}^{+}</math>, ...eatedly sliding dominoes. Find all possible values of <math>k(C)</math> as a function of <math>n</math>.
  4 KB (664 words) - 22:57, 9 March 2024
• 2023 USAMO Problems/Problem 3
  ...epeatedly sliding dominoes. Find the maximum value of <math>k(C)</math> as a function of <math>n</math>. ...is even but its column is odd, a tile B if its row and column is even, and a tile Y if its row is odd but its column is even.
  5 KB (867 words) - 19:18, 6 October 2023
• 2022 SSMO Accuracy Round Problems
  ...distinct element in the range) <math>f:S\rightarrow S,</math> where <math>S=\{1, 2, 3, 4, 5\}</math>. What is the average of <math>f(1) + f(2) + f(3)</ ...and a point <math>P</math> in the same plane as the base of the cone, but a distance of <math>10</math> from the center of the base of the cone. We rot
  5 KB (830 words) - 13:04, 14 December 2023
• 2022 IMO Problems
  ...r. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer <math>k\le 2n</math>, Marianne repeatedly performs t ...<math>x \in \mathbb{R}^+</math>, there is exactly one <math>y \in \mathbb{R}^+</math> satisfying
  3 KB (605 words) - 15:19, 3 June 2023
• 2024 AIME I Problems
  ...ometers per hour, how many minutes will the walk take, including the <math>t</math> minutes at the coffee shop? ...t there is a strategy that guarantees that Bob wins, regardless of Alice’s moves.
  8 KB (1,307 words) - 20:00, 6 February 2024
• 2023 IMO Problems/Problem 2
  ...ath> at <math>D</math> and meets <math>\Omega</math> again at <math>E \neq A</math>. The line through <math>D</math> parallel to <math>BC</math> meets l ...e the point diametrically opposite to a point <math>S</math> through <math>S' \implies AS'</math> is the internal angle bisector of <math>\angle BAC</ma
  7 KB (1,318 words) - 06:38, 1 May 2024
• 1992 IMO Problems/Problem 4
  ...with the following property: there exists two points <math>Q</math>, <math>R</math> on <math>l</math> such that <math>M</math> is the midpoint of <math> ...have been shortened in this solution. The solution in the video provides a much faster solution,
  6 KB (1,052 words) - 00:42, 17 November 2023
• 2023 AMC 10A Problems/Problem 19
  ...ath>B'(4, 3)</math> about the point <math>P(r, s)</math>. What is <math>|r-s|</math>? <math>\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } \frac{
  5 KB (817 words) - 11:51, 19 May 2024
• 2023 AMC 10A Problems/Problem 20
  Each square in a <math>3\times3</math> grid of squares is colored red, white, blue, or green label("R", (19.5,1.5));
  8 KB (1,261 words) - 15:23, 6 February 2024
• 2023 AMC 12A Problems/Problem 13
  In a table tennis tournament every participant played every other participant ex <math>\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\t
  5 KB (810 words) - 22:56, 10 May 2024
• 2023 AMC 12A Problems/Problem 15
  ...th>\theta = \angle PAB=\angle QPC=\angle RQB=\cdots</math> will produce in a length that is <math>120</math> meters? (This figure is not drawn to scale. label("$A$",(-58,-15));
  7 KB (1,074 words) - 20:58, 19 May 2024
• 2023 AMC 12A Problems/Problem 18
  pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.7 <math>\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \fra
  4 KB (527 words) - 23:54, 20 April 2024
• 2023 AMC 12B Problems/Problem 25
  ...tices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? <math>\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qqu
  19 KB (2,967 words) - 16:56, 24 February 2024
• Polya’s method for extremums
  ==Polya’s method== Natural scientists have found a number of empirical laws and methods that describe nature and therefore opt
  12 KB (2,104 words) - 14:11, 24 February 2024
• 2024 AMC 8 Problems/Problem 20
  ...How many of these triangles are equilateral and contain <math>P</math> as a vertex? pair P,Q,R,S,T,U,V,W;
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• 2023 AMC 12B Problems/Problem 23
  <math>\textbf{(A)}~11\qquad\textbf{(B)}~6\qquad\textbf{(C)}~8\qquad\textbf{(D)}~10\qquad\tex We start by trying to prove a function of <math>n</math>, and then we can apply the function and equate i
  13 KB (2,194 words) - 19:10, 18 December 2023
• 2024 AIME I Problems/Problem 7
  ...part of <cmath>(75+117i)z+\frac{96+144i}{z}</cmath>where <math>z</math> is a complex number with <math>|z|=4</math>. ==Video Solution: Cauchy's Inequality==
  12 KB (1,957 words) - 09:04, 18 May 2024
• 2024 AIME I Problems/Problem 15
  ...ar boxes that are elements of <math>\mathcal{B}</math>. The value of <math>r^2</math> can be written as <math>\frac{p}{q}</math>, where <math>p</math> a ...make the length longer.) Thus, let the width and height be of length <math>a</math> and the length be <math>L</math>.
  10 KB (1,554 words) - 22:26, 13 April 2024
• 1987 OIM Problems/Problem 5
  If <math>r</math>, <math>s</math>, and <math>t</math> are all the roots of the equation: (a) Prove that <math>r</math>, <math>s</math>, and <math>t</math> are all postive
  581 bytes (98 words) - 13:27, 13 December 2023
• 1992 OIM Problems/Problem 3
  ...th>G</math>, the sum of the squares of its distances to the vertices <math>A</math>, <math>B</math> and <math>C</math> is 5. ...r every point <math>P</math> in <math>G</math> it is possible to construct a triangle whose sides have the lengths of the segments <math>AP</math>, <mat
  4 KB (814 words) - 09:42, 23 December 2023
• 1995 OIM Problems/Problem 3
  ...be their common perpendicular, where <math>A \in r</math>, and <math>B \in s</math> '''(*)'''. ...e condition that <math>MN</math> is tangent to the sphere at a point <math>T</math>.
  767 bytes (136 words) - 14:48, 13 December 2023
• 1997 OIM Problems/Problem 2
  ...ath>F</math> and <math>R</math> (<math>F</math> being the closest to <math>A</math>). ...section of the diagonals of the quadrilateral <math>EQFR</math>. Let <math>T</math> be the point of intersection of the diagonals of the <math>FRDP</mat
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• 2022 SSMO Accuracy Round Problems/Problem 4
  ...th>r,s,t.</math> A monic polynomial <math>g</math> has roots <math>r^3,s^3,t^3.</math> Given that the minimum possible value of <math>\frac{g(1)}{f(1)}<
  362 bytes (64 words) - 13:03, 14 December 2023
• 2023 SSMO Team Round Problems
  ...2, 3)</math>. Find the largest possible value of <math>a^b + b^c + c^d + d^a</math> ...nd increases in altitude at an angle of <math>30^{\circ}.</math> Let <math>s</math> the speed in feet per second that the plane must fly at to move nort
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• 2024 AIME I Problems/Problem 5
  ...<math>D,E,C,F</math> are collinear. Also, <math>A,D,H,G</math> all lie on a circle. If <math>BC=16</math>,<math>AB=107</math>,<math>FG=17</math>, and < pair A = (0,0);pair B = (70,0);pair C = (70,16);pair D = (0,16);pair E = (3,16);pa
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• 2002 AMC 12P Problems/Problem 24
  ...m <math>E</math> to the edges <math>AB, BC, CA.</math> Then <math>\frac{s}{S}</math> equals \text{(A) }\sqrt{2}
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• 2024 AIME II Problems/Problem 15
  ...h>12</math>-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangle ...l (don't forget to multiply by 3 at the end of Case 1). Then the rectangle's sides coincide with these segments as shown in the diagram.
  8 KB (1,395 words) - 17:26, 9 February 2024
• DVI exam
  ...State University named after M.V. Lomonosov. The first four problems have a standard level. ...le of its upper base touches each of the faces which contains vertex <math>S.</math>
  32 KB (5,375 words) - 11:43, 5 May 2024
• Problems Collection
  This is a page where you can share the problems you made (try not to use past exams). <cmath>\frac{ab+bc+ac}{(a+b+c)^2}</cmath>
  43 KB (7,006 words) - 14:24, 19 February 2024