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• Math books
  ...https://artofproblemsolving.com/community/user/243060 cargeek9], currently a junior in high school. It covers the basics of algebra, geometry, combinato ...=sr_1_1?ie=UTF8&s=books&qid=1204029652&sr=1-1 Complex Numbers from A to... Z] by [[Titu Andreescu]]
  24 KB (3,177 words) - 12:53, 20 February 2024
• 2005 Alabama ARML TST Problems/Problem 7
  \sum_{n=1}^{\infty} \left(\frac{n^2}{2^n}\right)=z\\ z=1+\frac{3}{2}+\frac{5}{4}+\frac{7}{8}+\frac{9}{16}+\cdots\\
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• Mathematics
  ..., and the formation of turbulence around an aircraft. Mathematics provides a kind of "quality control" for the development of trustworthy theories and e Modern mathematics is built around a system of [[axiom|axioms]], which is a name given to "the rules of the game." Mathematicians then use various meth
  6 KB (902 words) - 12:53, 3 September 2019
• Trivial Inequality
  <cmath>ax^2+bx+c = a(x^2+\frac{b}{a}x+\frac{b^2}{4a^2})+c-\frac{b^2}{4a} = a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a}.</cmath> .../math> is positive and <math>ax^2+bx+c\le c-\frac{b^2}{4a}</math> if <math>a</math> is negative.
  3 KB (560 words) - 22:51, 13 January 2024
• Factoring
  ...rs, is to write the expression (often an [[integer]] or [[polynomial]]) as a product of different terms. This often allows one to find information abou Using the formula for the sum of a [[geometric sequence]], it's easy to derive the general formula for differe
  3 KB (532 words) - 22:00, 13 January 2024
• Relatively prime
  Consecutive positive integers are always relatively prime, since, if a prime <math>p</math> divides both <math>n</math> and <math>n+1</math>, then ...exist some <math>x,y\in \mathbb{Z}</math> such that <math>ax+by=1</math> (a special case of [[Bezout's Lemma]]). The [[Euclidean Algorithm]] can be use
  2 KB (245 words) - 15:51, 25 February 2020
• Discriminant
  ...has two [[nonreal]] roots; and if the discriminant is 0, the equation has a real [[double root]]. The discriminant can tell us something about the roots of a given [[polynomial]] <math>p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0</math> of deg
  4 KB (734 words) - 19:19, 10 October 2023
• Menelaus' theorem
  ...earity]] of points on each of the three sides (extended when necessary) of a [[triangle]]. pair A=(7,6), B=(0,0), C=(10,0), P=(4,0), Q=(6,8), R;
  5 KB (804 words) - 03:01, 12 June 2023
• Fermat's Little Theorem
  ...] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to ...}</math> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>.
  16 KB (2,658 words) - 16:02, 8 May 2024
• Euler's Totient Theorem
  '''Euler's Totient Theorem''' is a theorem closely related to his [[totient function]]. ...is a positive integer [[relatively prime]] to <math>a</math>, then <math>{a}^{\phi (m)}\equiv 1 \pmod {m}</math>.
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• Elementary symmetric sum
  An '''elementary symmetric sum''' is a type of [[summation]]. ...>). For example, if <math>n = 4</math>, and our set of numbers is <math>\{a, b, c, d\}</math>, then:
  2 KB (275 words) - 12:51, 26 July 2023
• Algebra
  ...the [[variable]], an unknown quantity given a name and usually denoted by a letter or symbol. Many contest problems test one's fluency with [[algebraic ...o add to the confusion, "[[algebra (structure) |algebra]]" is the name for a certain kind of structure in modern algebra.
  3 KB (369 words) - 21:18, 18 June 2021
• Number theory/Advanced
  ...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 ...points. This means that given 2 points on the curve, they can be added in a way that satisfies the normal laws of addition, like associativity, commuta
  5 KB (849 words) - 16:14, 18 May 2021
• Real number
  A '''real number''' is a number that falls on the real number line. It can have any value. Some exam ...tion <math>\mathbb{Z}^{+}</math> means positive integers and <math>\mathbb{Z}^{-}</math> means negative integers. The real numbers can also be divided b
  3 KB (496 words) - 23:22, 5 January 2022
• Euclidean algorithm
  ...the most common of which is the [[nonnegative]] [[integer]]s <math>\mathbb{Z}{\geq 0}</math>, without [[factoring]] them. ...c idea is to repeatedly use the fact that <math>\gcd({a,b}) \equiv \gcd({b,a - b})</math>
  6 KB (924 words) - 21:50, 8 May 2022
• Integer
  ...th>a\div b</math> may or may not be an integer, depending on whether <math>a</math> can be divided by <math>b</math> with no remainder. ...ward definition, an integer is a number that is '''not''' a [[decimal]] or a [[fraction]].
  2 KB (296 words) - 15:04, 5 August 2022
• Natural number
  ...non-negative integers, and <math>\mathbb{Z}_{>0}</math> and <math>\mathbb{Z}^+</math> for the set of positive integers.
  1 KB (162 words) - 21:44, 13 March 2022
• Pi
  Pi is the [[ratio]] of the [[circumference]] ([[perimeter]]) of a given [[circle]] to its [[diameter]]. It is approximately equal to 3.14159 ...i</math> is to inscribe a unit circle in a square of side length 2. Using a computer, random points are placed inside the square. Because the area of
  8 KB (1,469 words) - 21:11, 16 September 2022
• Chain Rule
  ...<math>h'(x_0)</math>,<math>f'(g(x_0))</math>, and <math>g'(x_0)</math> is a matrix.) ...as <math>\Delta x</math> approaches <math>0</math>. This can be made into a rigorous proof. (But we do have to worry about the possibility that <math>
  12 KB (2,377 words) - 11:48, 22 July 2009
• Injection
  ...t takes distinct values on distinct inputs. Equivalently, an injection is a function for which every value in the [[range]] is the image of exactly one ...: A function <math>f:A\to B</math> is an injection if for all <math>x,y\in A</math>, if <math>f(x)=f(y)</math> then <math>x=y</math>.
  1 KB (228 words) - 01:01, 17 November 2019
• 2011 AMC 10B Problems/Problem 22
  ...ength <math>1</math> and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the py <math> \textbf{(A)}\ 5\sqrt{2} - 7 \qquad\textbf{(B)}\ 7 - 4\sqrt{3} \qquad\textbf{(C)}\ \fra
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• 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
• Schur's Inequality
  Schur's inequality states that for all non-negative <math>a,b,c \in \mathbb{R}</math> and <math>r>0</math>: <cmath>a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0</cmath>
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• Complex plane
  ...e system) corresponds to the complex number [[zero (constant) | 0]], while a point two units to the right and one unit down from the origin corresponds
  892 bytes (134 words) - 16:52, 3 September 2017
• Complex number
  ...n has no solutions in the real numbers. However, it is possible to define a number, <math> i </math>, such that <math> i = \sqrt{-1} </math>. If we ad ...s the set <math>\mathbb{R}</math> of the [[real number]]s, since <math>a = a + 0i</math>.
  5 KB (860 words) - 15:36, 10 December 2023
• Cis
  ...plex number]]s. For all complex numbers <math>z</math>, we can write <math>z=r\mathrm{cis }(\theta)=r\cos \theta + ir\sin \theta</math>. Notice that <ma
  1 KB (171 words) - 20:59, 11 July 2023
• Polar form
  ...ows us to graph complex numbers given an [[angle]] <math>\theta</math> and a [[radius]] or [[magnitude]] <math>r</math>. ...u do not understand this notation.) This represents a complex number <math>z</math> that is <math>r</math> units away from the origin, and <math>\theta<
  633 bytes (105 words) - 13:35, 1 April 2022
• Roots of unity
  The '''Roots of unity''' are a topic closely related to [[trigonometry]]. Roots of unity come up when we e ...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
• Angle bisector
  ...internal) angle bisector of <math>\angle BAC</math> is the line from <math>A</math> such that the angle between this line and <math>\overline{AB}</math> pair A,B,C,D,E,F;
  3 KB (575 words) - 15:27, 19 March 2023
• Fermat's Last Theorem
  ...> with <math>n \geq 3</math>, there are no solutions to the equation <math>a^n + b^n = c^n</math>. ...ater than the second to be the sum of two like powers. ''I have discovered a truly marvelous demonstration of this proposition that this margin is too n
  3 KB (453 words) - 11:13, 9 June 2023
• 2005 USAMO Problems/Problem 4
  ...math> from the end of leg <math>L_i \; (i = 1,2,3,4)</math> and still have a stable table? ...can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)
  7 KB (1,276 words) - 20:51, 6 January 2024
• Diophantine equation
  A '''Diophantine equation''' is an [[equation]] relating [[integer]] (or some ...closely tied to [[modular arithmetic]] and [[number theory]]. Often, when a Diophantine equation has infinitely many solutions, [[parametric form]] is
  9 KB (1,434 words) - 13:10, 20 February 2024
• Area
  ...size of the region that a two-[[dimension]]al figure occupies. The size of a region in higher dimensions is referred to as [[volume]]. It is often possible to find the area of a region bounded by parts of [[circle]]s and [[line segment]]s through elemen
  6 KB (1,181 words) - 22:37, 22 January 2023
• Exponential form
  ...> as <math>z=re^{i\theta}</math>, which is the general exponential form of a complex number. So <math>z=re^{i\theta}</math> looks like:
  1 KB (238 words) - 22:51, 20 February 2022
• Floor function
  ...]</math>. The action of this function is the same as "rounding down." On a [[positive]] argument, this function is the same as "dropping everything af ...+b\rfloor\ge \lfloor a\rfloor+\lfloor b \rfloor</math> for all real <math>(a,b)</math>.
  3 KB (508 words) - 21:05, 26 February 2024
• Rational approximation
  The main theme of this article is the question how well a given [[real number]] <math>x</math> can be approximated by [[rational numb ...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
• Complex analysis
  ...ctions of a real variable that cannot occur in complex variables. Here are a few spectacular results in complex analysis. ...ly connected]] [[domain]] ''D'', and let <math>\Gamma\subseteq D</math> be a [[simple closed Jordan curve]]. Then for any <math>z_0</math> in the interi
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• Fundamental Theorem of Algebra
  ...tant]] [[polynomial]] with [[complex number|complex]] [[coefficient]]s has a complex [[root]]. In fact, every known proof of this theorem involves some ...s of generality, suppose that <math>P</math> is [[monic]]. Then <math>1/P(z)</math> is an [[entire]] function; we wish to show that it is bounded. It
  5 KB (832 words) - 14:22, 11 January 2024
• Surjection
  ..., the range is a subset of the codomain.) In adjectival form, we say that a function is ''surjective'' or ''onto''. ...defined by <math>f(x) = x+1</math> is not surjective because there exists a [[natural number]] which is not one more than any other natural number.
  794 bytes (131 words) - 22:39, 13 May 2020
• Prime Number Theorem
  constants <math>A</math> and <math>B</math>, <cmath> \pi(x) \sim \frac{x}{A \log x - B} . </cmath>
  10 KB (1,729 words) - 19:52, 21 October 2023
• Infinite
  ...said to be '''infinite''' if there is a [[surjection]] <math>f:S\to\mathbb{Z}</math>. If this is not the case, <math>S</math> is said to be [[finite]]. In simplified language, a set is infinite if it doesn't end, i.e. you can always find another element
  1 KB (186 words) - 23:19, 16 August 2013
• Mock AIME
  A '''Mock AIME''' is a contest that is intended to mimic the [[AIME]] competition. (In more recent ** [https://artofproblemsolving.com/community/c5h2987334 Or(z)IME]
  8 KB (906 words) - 17:30, 26 April 2024
• Commutative property
  An [[operation]] (especially a [[binary operation]]) is said to have the '''commutative property''' or to ...eal number]]s, [[integer]]s, etc.) because <math>\displaystyle a + b = b + a</math>. However, the operation of [[division]] is not commutative over the
  2 KB (301 words) - 17:46, 16 March 2012
• Algebraic topology
  ...of [[topology]] using methods from [[abstract algebra]]. In general, given a [[topological space]], we can associate various algebraic objects, such as ...a,1)=q(a)</math>, and <math>g(0,b)=g(1,b)=x</math>. We call <math>g</math> a [[homotopy]]. Now define <math>\pi_1(X)=L/\sim</math>. That is, we equate a
  3 KB (479 words) - 15:35, 1 December 2015
• Gamma function
  The '''Gamma function''' is a generalization of the notion of a [[factorial]] to [[complex number|complex numbers]]. For <math>\Re(z)>0</math>, we define <cmath>\Gamma(z)=\int_0^\infty e^{-t}t^{z-1}\; dt</cmath>
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• 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
• Subset
  ...r than <math>B</math> itself. In the latter case, <math>A</math> is called a ''proper subset''. The following is a true statement:
  1 KB (217 words) - 09:32, 13 August 2011
• 2006 AIME I Problems/Problem 5
  ...n be written as <math> a\sqrt{2}+b\sqrt{3}+c\sqrt{5}, </math> where <math> a, b, </math> and <math> c </math> are [[positive]] [[integer]]s. Find <math> <cmath> a\sqrt{2}+b\sqrt{3}+c\sqrt{5} = \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2
  3 KB (439 words) - 18:24, 10 March 2015
• 2021 JMC 10
  <math>\textbf{(A) } 3 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 9 \qqu <math>\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2006 AMC 12A Problems
  {{AMC12 Problems|year=2006|ab=A}} <math> \mathrm{(A) \ } 31\qquad \mathrm{(B) \ } 32\qquad \mathrm{(C) \ } 33\qquad \mathrm{(D)
  15 KB (2,223 words) - 13:43, 28 December 2020
• 2005 AMC 12A Problems
  {{AMC12 Problems|year=2005|ab=A}} (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D})
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2004 AMC 12A Problems
  {{AMC12 Problems|year=2004|ab=A}} <math>\text{(A) } 0.0029 \qquad \text{(B) } 0.029 \qquad \text{(C) } 0.29 \qquad \text{(D)
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2000 AMC 12 Problems
  <math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\ <math>\textbf{(A)}\ 2000^{2001} \qquad \textbf{(B)}\ 4000^{2000} \qquad \textbf{(C)}\ 2000^{
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2002 AMC 12B Problems
  ...numbers in the set <math>\{9, 99, 999, 9999, \ldots, 999999999\}</math> is a <math>9</math>-digit number <math>M</math>, all of whose digits are distinc <math>\mathrm{(A)}\ 0
  10 KB (1,547 words) - 04:20, 9 October 2022
• 2004 AMC 12B Problems
  <math>(\mathrm {A}) 3\qquad (\mathrm {B}) 6 \qquad (\mathrm {C}) 9 \qquad (\mathrm {D}) 12 \q In the expression <math>c\cdot a^b-d</math>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are 0, 1, 2, and
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at t \mathrm{(A)}\ 100 \qquad
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12A Problems/Problem 15
  ...os (x+z)=1/2</math>. What is the smallest possible positive value of <math>z</math>? <math> \mathrm{(A) \ } \frac{\pi}{6}\qquad \mathrm{(B) \ } \frac{\pi}{3}\qquad \mathrm{(C) \
  919 bytes (138 words) - 12:45, 4 August 2017
• 2006 AMC 12A Problems/Problem 17
  ...ies on the circle, on the same side of <math>\overline{BE}</math> as <math>A</math>. Segment <math>AF</math> is tangent to the circle, and <math>AF=\sqr label("A", (-s,0), W);
  6 KB (958 words) - 23:29, 28 September 2023
• 2006 AMC 12A Problems/Problem 24
  <cmath>(x+y+z)^{2006}+(x-y-z)^{2006}</cmath> <math> \mathrm{(A) \ } 6018\qquad \mathrm{(B) \ } 671,676\qquad \mathrm{(C) \ } 1,007,514\qqu
  8 KB (1,332 words) - 17:37, 17 September 2023
• 2005 AMC 12A Problems/Problem 20
  (\text {A}) \ 0 \qquad (\text {B}) \ 2005 \qquad (\text {C})\ 4010 \qquad (\text {D}) ...different form(although they'll all satisfy the final equation), so we get a different starting value of <math>x</math>. Every time we have two choices
  3 KB (437 words) - 23:49, 28 September 2022
• 2005 AMC 12A Problems/Problem 25
  ...e]]s <math>(x,y,z)</math>, where <math>x</math>, <math>y</math>, and <math>z</math> are each chosen from the set <math>\{0,1,2\}</math>. How many [[equi <math>(\mathrm {A}) \ 72\qquad (\mathrm {B}) \ 76 \qquad (\mathrm {C})\ 80 \qquad (\mathrm {D
  4 KB (498 words) - 00:46, 4 August 2023
• 2005 AMC 12B Problems/Problem 16
  \mathrm {(A)}\ \sqrt{2} \qquad ...ce, the centers of the spheres are <math>(\pm 1, \pm 1, \pm 1)</math>. For a sphere centered at the origin to contain all eight spheres, its radius must
  2 KB (364 words) - 04:54, 16 January 2023
• 2005 AMC 12B Problems/Problem 22
  A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is define \textbf{(A)}\ 1 \qquad
  4 KB (660 words) - 17:40, 24 January 2021
• 2005 AMC 12B Problems/Problem 23
  Let <math>S</math> be the set of ordered triples <math>(x,y,z)</math> of real numbers for which <cmath>\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.</cmath>
  5 KB (786 words) - 11:36, 19 May 2024
• 2006 AMC 10A Problems
  {{AMC10 Problems|year=2006|ab=A}} <math>\mathrm{(A)}\ 31\qquad\mathrm{(B)}\ 32\qquad\mathrm{(C)}\ 33\qquad\mathrm{(D)}\ 34\qqu
  13 KB (2,028 words) - 16:32, 22 March 2022
• 2006 AMC 10A Problems/Problem 17
  pair A,B,C,D,E,F,G,H,W,X,Y,Z; A=(0,2); B=(1,2); C=(2,2); D=(3,2);
  6 KB (1,066 words) - 00:21, 2 February 2023
• 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 II Problems/Problem 2
  ...e indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the [[probability]] each guest got o ...We need only calculate the probability the first and second person all get a roll of each type, since then the rolls for the third person are determined
  4 KB (628 words) - 11:28, 14 April 2024
• 2005 AIME II Problems/Problem 7
  Like Solution <math>2</math>, let <math>z=\sqrt[16]{5}</math> <math>x=\frac{4}{(z+1)(z^2+1)(z^4+1)(z^8+1)}</math>
  2 KB (279 words) - 12:33, 27 October 2019
• Euler's identity
  ...ers]] and/or [[trigonometry]]. Euler's formula replaces "[[cis]]", and is a superior notation, as it encapsulates several nice properties: A special, and quite fascinating, consequence of Euler's formula is the ident
  3 KB (452 words) - 23:17, 4 January 2021
• 2004 AIME I Problems/Problem 15
  and define a [[sequence]] as follows: <math>x_1=x</math> and <math>x_{n+1}=f(x_n)</math> ...>9</math> operations; however, not all <math>9</math> of them may be <math>A: (-1)</math> otherwise we return back to <math>x_{10} = 1</math>, contradic
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2004 AIME I Problems/Problem 12
  ...e [[floor function|greatest integer]] that is less than or equal to <math> z. </math> Graphing this yields a series of [[rectangle]]s which become smaller as you move toward the [[orig
  2 KB (303 words) - 22:28, 11 September 2020
• 2004 AIME I Problems
  The digits of a positive integer <math> n </math> are four consecutive integers in decreasi ...he absolute value of the difference between the greatest element of <math> A </math> and the greatest element of <math> B </math> is 99. Find <math> m.
  9 KB (1,434 words) - 13:34, 29 December 2021
• 2004 AIME II Problems/Problem 12
  ...and circles of radius 2 centered at <math> C </math> and <math> D. </math> A circle contained within the trapezoid is [[tangent]] to all four of these c pair A=(0,0), B=(6,0), D=(1, 24^.5), C=(5,D.y), O = (3,(r^2 + 6*r)^.5);
  3 KB (431 words) - 23:21, 4 July 2013
• 2004 AIME II Problems/Problem 10
  ...{40} </math> whose binary expansions have exactly two <math>1</math>'s. If a number is chosen at random from <math> S, </math> the [[probability]] that ...ath>y</math> are <math>0 \leq x < y < 40</math> and <math>x, y \in \mathbb{Z}</math>.
  8 KB (1,283 words) - 19:19, 8 May 2024
• 2004 AIME II Problems/Problem 6
  ...es the rest equally between the other two. Given that each monkey receives a [[whole number]] of bananas whenever the bananas are divided, and the numbe ...> is divisible by <math>72</math> (however, since the denominator contains a <math>27</math>, the factors of <math>3</math> cancel, and it only really n
  6 KB (950 words) - 14:18, 15 January 2024
• 1983 AIME Problems
  ...th> and <math>z</math> all exceed <math>1</math> and let <math>w</math> be a positive number such that <math>\log_xw=24</math>, <math>\log_y w = 40</mat ...hat of <math>BC</math> is <math>2</math> cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B<
  7 KB (1,104 words) - 12:53, 6 July 2022
• 1984 AIME Problems
  A point <math>P</math> is chosen in the interior of <math>\triangle ABC</math pair A=(0,0),B=(12,0),C=(4,5);
  6 KB (933 words) - 01:15, 19 June 2022
• 1990 AIME Problems
  ...neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. ...}</math> be a regular <math>r~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that ea
  6 KB (870 words) - 10:14, 19 June 2021
• 1992 AIME Problems
  A positive integer is called ascending if, in its decimal representation, the ...she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math>0.500</math>. During the weekend, s
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1996 AIME Problems
  ...mn, or diagonal is the same value. The figure shows four of the entries of a magic square. Find <math>x</math>. ...at <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer?
  6 KB (931 words) - 17:49, 21 December 2018
• 1997 AIME Problems
  ...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 ...ny of which could be zero. It is desired to approximate <math>r</math> by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The
  7 KB (1,098 words) - 17:08, 25 June 2020
• 1999 AIME Problems
  ...> <math>(10,114),</math> <math>(28,153),</math> and <math>(28,84).</math> A line through the origin cuts this figure into two congruent polygons. The ...all positive integers <math>n</math> for which <math>n^2-19n+99</math> is a perfect square.
  7 KB (1,094 words) - 13:39, 16 August 2020
• 2000 AIME I Problems
  ...Let <math>A = (u,v)</math>, let <math>B</math> be the reflection of <math>A</math> across the line <math>y = x</math>, let <math>C</math> be the reflec ...icients of <math>x^{2}</math> and <math>x^{3}</math> are equal. Find <math>a + b</math>.
  7 KB (1,204 words) - 03:40, 4 January 2023
• 2001 AIME I Problems
  A finite set <math>\mathcal{S}</math> of distinct real numbers has the follow ...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
• 2000 AIME II Problems
  A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola <math>x^2 - y^ ...h>m/n</math> be the probability that two randomly selected cards also form a pair, where <math>m</math> and <math>n</math> are relatively prime positive
  6 KB (947 words) - 21:11, 19 February 2019
• 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
• 2002 AIME II Problems
  ...math>x</math>; and <math>z=|x-y|</math>. How many distinct values of <math>z</math> are possible? Three [[vertex|vertices]] of a [[cube]] are <math>P=(7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=
  7 KB (1,177 words) - 15:42, 11 August 2023
• 1983 AIME Problems/Problem 1
  ...th> and <math>z</math> all exceed <math>1</math> and let <math>w</math> be a positive number such that <math>\log_x w = 24</math>, <math>\log_y w = 40</ ...verything to a power of <math>120</math>, it will be easy to isolate <math>z</math> and <math>w</math>.
  4 KB (642 words) - 03:14, 17 August 2022
• 1983 AIME Problems/Problem 5
  Hence observe that we can write <math>w=x+y</math> and <math>z=xy</math>. <math>w(7-z)=10</math>.
  4 KB (672 words) - 10:17, 17 March 2023
• 1983 AIME Problems/Problem 9
  We can rewrite the numerator to be a perfect square by adding <math>-\dfrac{12x \sin x}{x \sin x}</math>. Thus, ...use quadratic equation on <math>9z^2 - yz + 4 = 0</math> to see that <math>z = \frac{y\pm\sqrt{y^2-144}}{18}</math>. We know that <math>y</math> must be
  4 KB (722 words) - 20:25, 14 January 2023
• 1983 AIME Problems/Problem 15
  ...r arc <math>AB</math> is a rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <mat pair A=(-0.91,-0.41);
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1984 AIME Problems/Problem 3
  A [[point]] <math>P</math> is chosen in the interior of <math>\triangle ABC</ pair A=(0,0),B=(12,0),C=(4,5);
  4 KB (726 words) - 13:39, 13 August 2023
• 1984 AIME Problems/Problem 8
  The equation <math>z^6+z^3+1=0</math> has complex roots with argument <math>\theta</math> between <m ...ake the equation easier to solve. If <math>r</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 <m
  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
• 1984 AIME Problems/Problem 12
  ...<math>f(7+x)=f(7-x)</math> for all <math>x</math>. If <math>x=0</math> is a root for <math>f(x)=0</math>, what is the least number of roots <math>f(x)= Since <math>0</math> is a root, all multiples of <math>10</math> are roots, and anything congruent to
  3 KB (588 words) - 14:37, 22 July 2020
• 1984 AIME Problems/Problem 15
  Determine <math>x^2+y^2+z^2+w^2</math> if ...6^2-7^2}=1 </math><br /><math> \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 </math></div>
  6 KB (1,051 words) - 04:52, 8 May 2024
• 1984 AIME Problems/Problem 13
  ...ath>\tan(x+y) = \frac{\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}</math>. Let <math>a = \cot^{-1}(3)</math>, <math>b=\cot^{-1}(7)</math>, <math>c=\cot^{-1}(13)</ <center><p><math>\tan(a)=\frac{1}{3},\quad\tan(b)=\frac{1}{7},\quad\tan(c)=\frac{1}{13},\quad\tan(d
  3 KB (473 words) - 12:06, 18 December 2018
• 1995 IMO Problems/Problem 2
  Let <math>a, b, c</math> be positive real numbers such that <math>abc = 1</math>. Prove <cmath> \frac{1}{a^3(b+c)} + \frac{1}{b^3(c+a)} + \frac{1}{c^3(a+b)} \geq \frac{3}{2}. </cmath>
  6 KB (1,122 words) - 12:23, 6 January 2022
• 1985 AIME Problems/Problem 10
  ...rfloor</math> denotes the greatest [[integer]] less than or equal to <math>z</math>? ...oor</math>, then <math>k=20a+b</math> for some non-negative integers <math>a</math>, <math>b</math>, where there are <math>12</math> values for <math>b<
  12 KB (1,859 words) - 18:16, 28 March 2022
• 1986 AIME Problems/Problem 9
  pair C=(0,0),A=(510,0),B=IP(circle(C,450),circle(A,425)); pair Da=IP(Circle(A,289),A--B),E=IP(Circle(C,324),B--C),Ea=IP(Circle(B,270),B--C);
  11 KB (1,850 words) - 18:07, 11 October 2023
• 1986 AIME Problems/Problem 2
  ...,y,z)=\left(\sqrt5,\sqrt6,\sqrt7\right)</math> so that <math>\left(x^2,y^2,z^2\right)=(5,6,7).</math> We rewrite the original expression in terms of <math>x,y,</math> and <math>z,</math> then apply the difference of squares repeatedly:
  3 KB (460 words) - 00:44, 5 February 2022
• 1987 AIME Problems/Problem 15
  ...iangle]]s in the figure are [[similar]] to triangle <math>ABC</math>, it's a good idea to use [[area ratios]]. In the diagram above, <math>\frac {T_1}{T ...ath>, and the ratio between the sides is <math>\sqrt {441} = 21</math>. As a result, <math>AB = 21\sqrt {440} = \sqrt {AC^2 + BC^2}</math>. We now need
  5 KB (838 words) - 18:05, 19 February 2022
• 1987 AIME Problems/Problem 7
  ...r which <math>[a,b] = 1000</math>, <math>[b,c] = 2000</math>, and <math>[c,a] = 2000</math>. It's clear that we must have <math>a = 2^j5^k</math>, <math>b = 2^m 5^n</math> and <math>c = 2^p5^q</math> for s
  3 KB (547 words) - 22:54, 4 April 2016
• 1988 AIME Problems/Problem 11
  ...mplex number]]s. A line <math>L</math> in the [[complex plane]] is called a mean [[line]] for the [[point]]s <math>w_1, w_2, \dots, w_n</math> if <math ...>, <math>w_4 = 1 + 27i</math>, and <math>w_5 = - 14 + 43i</math>, there is a unique mean line with <math>y</math>-intercept 3. Find the [[slope]] of th
  2 KB (422 words) - 00:22, 6 September 2020
• 1988 AIME Problems/Problem 9
  A little bit of checking tells us that the units digit must be 2. Now our cub ...value for the hundreds digit is <math>4</math>, and so <math>442</math> is a valid solution.
  6 KB (893 words) - 08:15, 2 February 2023
• 1988 AIME Problems/Problem 8
  ...ty in terms of <math>x</math> and <math>z,</math> then solve for <math>f(x,z):</math> zf(x,z-x) &= (z-x)f(x,z) \\
  4 KB (538 words) - 13:24, 12 October 2021
• 1989 AIME Problems/Problem 15
  ...th>RST</math>. We'll make use of the following fact: if <math>P</math> is a point in the interior of triangle <math>XYZ</math>, and line <math>XP</math pair X = (1,2), Y = (0,0), Z = (3,0);
  13 KB (2,091 words) - 00:20, 26 October 2023
• 1989 AIME Problems/Problem 8
  ...tic function of <math>k:</math> <cmath>f(k)=ak^2+bk+c,</cmath> where <math>a,b,</math> and <math>c</math> are linear combinations of <math>x_1,x_2,x_3,x f(1)&=\phantom{42}a+b+c&&=1, \\
  8 KB (1,146 words) - 04:15, 20 November 2023
• 1990 AIME Problems/Problem 14
  ...ments <math>\overline{CP}</math> and <math>\overline{DP}</math>, we obtain a [[triangular pyramid]], all four of whose faces are [[isosceles triangle]]s pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6);
  7 KB (1,086 words) - 08:16, 29 July 2023
• 1990 AIME Problems/Problem 13
  ...tive integers n, there's exactly one n-digit power of 9 that does not have a left-most digit 9 ...hat would result in <math>\frac{m}{9}</math> also being an n-digits, hence a contradiction. Therefore, this single n-digit power of 9 must not begin wit
  5 KB (762 words) - 01:18, 10 February 2023
• 1990 AIME Problems/Problem 10
  ...The set <math>C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}</math> is also a set of complex roots of unity. How many distinct elements are in <math>C_{ ...ine <math>n = e^{2\pi i/144}</math>. We can write the numbers of set <math>A</math> as <math>\{n^8, n^{16}, \ldots n^{144}\}</math> and of set <math>B</
  3 KB (564 words) - 04:47, 4 August 2023
• 1991 AIME Problems/Problem 14
  A [[hexagon]] is inscribed in a [[circle]]. Five of the sides have length <math>81</math> and the sixth, de pair A=expi(-pi/2-acos(475/486)), B=expi(-pi/2+acos(475/486)), C=expi(-pi/2+acos(4
  2 KB (284 words) - 03:56, 23 January 2023
• 1992 AIME Problems/Problem 14
  In triangle <math>ABC^{}_{}</math>, <math>A'</math>, <math>B'</math>, and <math>C'</math> are on the sides <math>BC</ma ...the weights of <math>A</math>, <math>B</math>, and <math>C</math> be <math>a</math>, <math>b</math>, and <math>c</math> respectively.
  4 KB (667 words) - 01:26, 16 August 2023
• 1992 AIME Problems/Problem 10
  ...1</math>, inclusive. What is the integer that is nearest the area of <math>A</math>? ...q 1</math> we have the inequality <cmath>0\leq a,b \leq 40</cmath>which is a square of side length <math>40</math>.
  2 KB (323 words) - 12:05, 16 July 2019
• 1992 AIME Problems/Problem 9
  ...}_{}</math>, with <math>AB^{}_{}</math> parallel to <math>CD^{}_{}</math>. A circle with center <math>P^{}_{}</math> on <math>AB^{}_{}</math> is drawn t ...0x=70\cdot 92,</math> hence <math>AP=x=\frac{161}{3}</math>. This gives us a final answer of <math>161+3=\boxed{164}</math>
  5 KB (874 words) - 10:27, 22 August 2021
• 1993 AIME Problems/Problem 6
  ...ying, <math>9a = 10b + 9 = 11c + 19</math>. The relationship between <math>a,\ b</math> suggests that <math>b</math> is divisible by <math>9</math>. Als ...information given, it can be determined that, for positive integers <math>a, \ b, \ c</math>:
  3 KB (524 words) - 18:06, 9 December 2023
• 1994 AIME Problems/Problem 10
  In triangle <math>ABC,\,</math> angle <math>C</math> is a right angle and the altitude from <math>C\,</math> meets <math>\overline{AB ...ath>, <math>29^2 = x^2 - y^2 = (x-y)(x+y)</math>. As <math>x,y \in \mathbb{Z}</math>, the pairs of factors of <math>29^2</math> are <math>(1,29^2)(29,29
  3 KB (534 words) - 16:23, 26 August 2018
• 1994 AIME Problems/Problem 4
  ...x+1}</math> for some <math>x\in\mathbb{Z}</math>, then <math>\lfloor\log_2{a}\rfloor=\log_2{2^{x}}=x</math>. ...th>. So the sum of <math>\lfloor\log_2{a}\rfloor</math> for all such <math>a</math> is <math>x\cdot2^x</math>.
  2 KB (264 words) - 13:33, 11 August 2018
• 1995 AIME Problems/Problem 12
  // projection of point A onto line BC triple projectionofpointontoline(triple A, triple B, triple C)
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1995 AIME Problems/Problem 11
  ...1995,</math> for how many ordered triples <math>(a, b, c)</math> does such a plane exist? ...ath>P'</math> be of length <math>x,y,z</math>, such that <math>x \le y \le z</math>. Then
  2 KB (292 words) - 19:30, 4 July 2013
• 1996 AIME Problems/Problem 14
  A <math>150\times 324\times 375</math> [[rectangle|rectangular]] [[solid]] is ...oordinates of the diagonally opposite corner of the rectangle, where <math>a, b, c \in \mathbb{Z_{+}}</math>.
  5 KB (923 words) - 21:21, 22 September 2023
• 1996 AIME Problems/Problem 13
  ...th> [[bisect]]s <math>\overline{BC}</math>, and <math>\angle ADB</math> is a right angle. The ratio <math>\frac{[ADB]}{[ABC]}</math> can be written in t pair B=(0,0), C=(15^.5, 0), A=IP(CR(B,30^.5),CR(C,6^.5)), E=(B+C)/2, D=foot(B,A,E);
  3 KB (521 words) - 01:18, 25 February 2016
• 1996 AIME Problems/Problem 11
  ...the product of the [[root]]s of <math>z^6+z^4+z^3+z^2+1=0</math> that have a positive [[imaginary]] part, and suppose that <math>\mathrm {P}=r(\cos{\the 0 &=& z^6 - z + z^4 + z^3 + z^2 + z + 1 = z(z^5 - 1) + \frac{z^5-1}{z-1}\\
  6 KB (1,022 words) - 20:23, 17 April 2021
• 1997 AIME Problems/Problem 14
  ...math>w</math> be distinct, randomly chosen [[root]]s of the equation <math>z^{1997}-1=0</math>. Let <math>\frac{m}{n}</math> be the [[probability]] tha :<math>z^{1997}=1=1(\cos 0 + i \sin 0)</math>
  5 KB (874 words) - 22:30, 1 April 2022
• 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
• 1997 AIME Problems/Problem 3
  ...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 ...e trial and error on factors of 1000. If <math>9x - 1 = 100</math>, we get a non-integer. If <math>9x - 1 = 125</math>, we get <math>x=14</math> and <ma
  2 KB (375 words) - 19:34, 4 August 2021
• 1998 AIME Problems/Problem 11
  Three of the edges of a cube are <math>\overline{AB}, \overline{BC},</math> and <math>\overline{CD} triple A=(0,0,0),B=(20,0,0),C=(20,0,20),D=(20,20,20);
  7 KB (1,084 words) - 11:48, 13 August 2023
• 1998 AIME Problems/Problem 5
  ...mber]]), and the [[cosine]] of <math>n\pi</math> where <math>n \in \mathbb{Z}</math> is 1 if <math>n</math> is even and -1 if <math>n</math> is odd. <ma If we group the terms in pairs, we see that we need a formula for <math>-\frac{(n)(n-1)}{2} + \frac{(n+1)(n)}{2} = \left(\frac n2
  1 KB (225 words) - 02:20, 16 September 2017
• 1999 AIME Problems/Problem 15
  ...The vertices of its midpoint triangle are the [[midpoint]]s of its sides. A triangular [[pyramid]] is formed by folding the triangle along the sides of ..."\(D\)",foot(A,B,C),NE);label("\(E\)",foot(B,A,C),SW);label("\(F\)",foot(C,A,B),NW);label("\(P\)",P,NW);label("\(Q\)",Q,NE);label("\(R\)",R,SE);</asy><a
  7 KB (1,169 words) - 15:28, 13 May 2024
• 1999 AIME Problems/Problem 14
  pair A=(0,0),B=(13,0),C=IP(circle(A,15),circle(B,14)); D(MP("A",A)--MP("B",B)--MP("C",C,N)--cycle);
  7 KB (1,184 words) - 13:25, 22 December 2022
• 1999 AIME Problems/Problem 9
  ...is [[equidistant]] from that point and the [[origin]]. Given that <math>|a+bi|=8</math> and that <math>b^2=m/n,</math> where <math>m_{}</math> and <ma ...= (a-b)</math> and <math>y = (a+b)</math>, we get <math>2a = 1 \Rightarrow a = \frac 12</math>.
  6 KB (1,010 words) - 19:01, 24 May 2023
• 1999 AIME Problems/Problem 7
  ...step 1000 has been completed, how many switches will be in position <math>A</math>? ...}{d}</math> must be a multiple of 4 to ensure that a switch is in position A:
  3 KB (475 words) - 13:33, 4 July 2016
• 1999 AIME Problems/Problem 4
  pair W=dir(225), X=dir(315), Y=dir(45), Z=dir(135), O=origin; pair w=dir(alpha)*W, x=dir(alpha)*X, y=dir(alpha)*Y, z=dir(alpha)*Z;
  3 KB (398 words) - 13:27, 12 December 2020
• 1999 AIME Problems/Problem 8
  ...)</math> when exactly two of the following are true: <math>x\ge a, y\ge b, z\ge c.</math> Let <math>\mathcal{S}</math> consist of those triples in <mat This problem just requires a good diagram and strong 3D visualization.
  3 KB (445 words) - 19:40, 4 July 2013
• 2000 AIME I Problems/Problem 9
  \log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ \log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\
  4 KB (623 words) - 15:56, 8 May 2021
• 2000 AIME I Problems/Problem 7
  ...frac {1}{z} = 5,</math> and <math>y + \frac {1}{x} = 29.</math> Then <math>z + \frac {1}{y} = \frac {m}{n},</math> where <math>m</math> and <math>n</mat We can rewrite <math>xyz=1</math> as <math>\frac{1}{z}=xy</math>.
  5 KB (781 words) - 15:02, 20 April 2024
• 2001 AIME I Problems/Problem 13
  .../math>-degree arc, where <math>d < 120.</math> The length of the chord of a <math>3d</math>-degree arc is <math>- m + \sqrt {n}</math> centimeters, whe Note that a cyclic quadrilateral in the form of an isosceles trapezoid can be formed fr
  3 KB (561 words) - 19:25, 27 November 2022
• 2001 AIME I Problems/Problem 12
  A [[sphere]] is inscribed in the [[tetrahedron]] whose vertices are <math>A = (6,0,0), B = (0,4,0), C = (0,0,2),</math> and <math>D = (0,0,0).</math> triple A = (6,0,0), B = (0,4,0), C = (0,0,2), D = (0,0,0);
  6 KB (1,050 words) - 18:44, 27 September 2023
• 2014 USAJMO Problems
  ...frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc. </cmath> ...riangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60^\circ</math>, and let <math>O</math> and <math>H</math> denote the circu
  3 KB (600 words) - 16:42, 5 August 2023
• 2001 AIME I Problems/Problem 10
  ...t satisfy <math>0\le x\le2,</math> <math>0\le y\le3,</math> and <math>0\le z\le4.</math> Two distinct points are randomly chosen from <math>S.</math> The distance between the <math>x</math>, <math>y</math>, and <math>z</math> coordinates must be even so that the midpoint can have integer coord
  8 KB (1,187 words) - 02:40, 28 November 2020
• 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
• 2001 AIME I Problems/Problem 6
  A fair die is rolled four times. The [[probability]] that each of the final t ...n the diagram below, the lowest <math>y</math>-coordinate at each of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> corresponds to t
  11 KB (1,729 words) - 20:50, 28 November 2023
• 2002 AIME I Problems/Problem 15
  ...CD</math> is a square with <math>AB = 12;</math> face <math>ABFG</math> is a trapezoid with <math>\overline{AB}</math> parallel to <math>\overline{GF},< triple A=(-6,-6,0), B = (-6,6,0), C = (6,6,0), D = (6,-6,0), E = (2,0,12), H=(-6+2*s
  7 KB (1,181 words) - 20:32, 8 January 2024
• 2002 AIME I Problems/Problem 12
  ...ath>, where <math>a</math> and <math>b</math> are real numbers, find <math>a+b</math>. F(z) &= \frac{z+i}{z-i}\\
  1 KB (217 words) - 22:39, 21 November 2018
• 2002 AIME I Problems/Problem 11
  ...ht path from the time it leaves point <math>A</math> until it next reaches a vertex of the cube is given by <math>m\sqrt{n}</math>, where <math>m</math> ...isible by 12. Since the light's Y changes by 5 and the X changes by 7 (the Z changes by 12, don't worry about that), and 5 and 7 are relatively prime to
  3 KB (591 words) - 15:11, 21 August 2019
• 2003 AIME II Problems/Problem 14
  ...(b,2)</math> be points on the coordinate plane. Let <math>ABCDEF</math> be a convex equilateral hexagon such that <math>\angle FAB = 120^\circ,</math> < ..., with height <math>8</math> and base <math>\frac{8}{\sqrt{3}}</math>) and a parallelogram (<math>ABDE</math>, with height <math>8</math> and base <math
  9 KB (1,461 words) - 15:09, 18 August 2023
• 2003 AIME II Problems/Problem 9
  So, since <math>z_1</math> is a root, <math>P(z_1)=(z_1)^2-z_1+1</math>. So we just have to find: <math>\sum_{n=1}^{4} z^{4}_n - \sum_{n=1}^{4} z^{3}_n - \sum_{n=1}^{4} z_n</math>.
  3 KB (475 words) - 21:53, 6 May 2024
• 2003 AIME II Problems/Problem 4
  In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to ...</math> on <math>BCD</math>, <math>Y</math> on <math>ACD</math>, and <math>Z</math> on <math>ABD</math>. Clearly, the two regular tetrahedrons are simil
  3 KB (563 words) - 17:36, 30 July 2022
• 2002 AIME II Problems/Problem 1
  &(3)& z=|x-y|. How many distinct values of <math>z</math> are possible?
  971 bytes (150 words) - 21:56, 11 February 2020
• 2001 AIME II Problems/Problem 14
  ...er]]s that satisfy both <math>z^{28} - z^{8} - 1 = 0</math> and <math>\mid z \mid = 1</math>. These numbers have the form <math>z_{m} = \cos\theta_{m} + <math>z</math> can be written in the form <math> \text{cis\,}\theta</math>. Rearran
  2 KB (380 words) - 15:03, 22 July 2018
• 2000 AIME II Problems/Problem 14
  Every positive [[integer]] <math>k</math> has a unique factorial base expansion <math>(f_1,f_2,f_3,\ldots,f_m)</math>, mean Let us consider a base <math>n</math> number system. It’s a well known fact that when we take the difference of two integral powers of
  7 KB (1,131 words) - 14:49, 6 April 2023
• 2000 AIME II Problems/Problem 13
  ...ic (or in this case, nearly symmetric) polynomials is to divide through by a power of <math>x</math> with half of the polynomial's degree (in this case, A slightly different approach using symmetry:
  6 KB (1,060 words) - 17:36, 26 April 2024
• 2000 AIME II Problems/Problem 12
  ...ts <math>A</math>, <math>B</math> and <math>C</math> lie on the surface of a [[sphere]] with center <math>O</math> and radius <math>20</math>. It is giv ...eron's Formula]] the area of <math>\triangle ABC</math> is (alternatively, a <math>13-14-15</math> triangle may be split into <math>9-12-15</math> and <
  3 KB (532 words) - 13:14, 22 August 2020
• 2000 AIME II Problems/Problem 11
  ...of [[isosceles trapezoid]] <math>ABCD</math> are all integers, with <math>A=(20,100)</math> and <math>D=(21,107)</math>. The trapezoid has no horizonta ...on of <math>D</math> across that perpendicular. Then <math>ABCD'</math> is a [[parallelogram]], and <math>\overrightarrow{AB} = \overrightarrow{D'C}</ma
  4 KB (750 words) - 22:55, 5 February 2024
• 2000 AIME II Problems/Problem 9
  .../math>, find the least integer that is greater than <math>z^{2000}+\frac 1{z^{2000}}</math>. ...adratic equation on <math>z^2 - (2 \cos 3 )z + 1 = 0</math>, we have <math>z = \frac{2\cos 3 \pm \sqrt{4\cos^2 3 - 4}}{2} = \cos 3 \pm i\sin 3 = \text{c
  4 KB (675 words) - 13:42, 4 April 2024
• 2005 AMC 10B Problems/Problem 23
  <math>\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \ Mark <math>DC=z</math>, <math>AB=x</math>, and <math>FE=y.</math>
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• 2006 USAMO Problems/Problem 3
  ...efficients such that the sequence <math>\{ p(f(n^2))-2n) \}_{n \in \mathbb{Z} \ge 0}</math> is bounded above. (In particular, this requires <math>f(n^2) Let <math>f(x)</math> be a non-constant polynomial in <math>x</math> of degree <math>d</math> with
  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
• Equation
  An '''equation''' is a [[relation]] which states that two [[expression]]s are equal, identical, or A unique aspect to equations is the ability to modify an original equation by
  5 KB (932 words) - 12:57, 26 July 2023
• Ring
  ...lgebra]], similar to a [[group]] or a [[field]]. A ring <math>R</math> is a [[set]] of elements closed under two [[operation]]s, usually called multipl * <math>(R,\cdot)</math> is a [[monoid]];
  6 KB (994 words) - 06:16, 8 April 2015
• Field
  ...field''' is a structure in [[abstract algebra]], similar to a [[group]] or a [[ring]]. Informally, fields are the general structure in which the usual ...>k</math> stands for Körper, the German word for a mathematical field) is a [[set]] of elements with two [[operation]]s, usually called multiplication
  2 KB (362 words) - 23:24, 31 December 2021
• Holomorphic function
  A '''holomorphic function''' <math>f: \mathbb{C} \to \mathbb{C}</math> is a as in the [[real number|real]] case, <math>f</math> is holomorphic at <math>z</math> if
  9 KB (1,537 words) - 21:04, 26 July 2017
• Multinomial Theorem
  Note that this is a direct generalization of the [[Binomial Theorem]], when <math>k = 2</math> For a positive integer <math>k</math> and a non-negative integer <math>n</math>,
  3 KB (476 words) - 19:37, 4 January 2023
• Platonic solid
  A '''Platonic solid''' is a [[polyhedron]], or 3 dimensional figure, in which all [[face]]s are [[congr void drawFrontFace(int x, int y, int z){
  8 KB (1,168 words) - 22:48, 19 February 2022
• Abelian group
  An '''abelian group''' is a [[group]] in which the group [[operation]] is [[commutative]]. They are nam For a [[group]] to be considered '''abelian''', it must meet several requirements
  2 KB (346 words) - 18:30, 14 June 2020
• Mock AIME 1 2006-2007 Problems/Problem 9
  Let <math>a_{n}</math> be a [[geometric sequence]] of [[complex number]]s with <math>a_{0}=1024</math> Let <math>a_{n}</math> be a geometric sequence for <math>n\in\mathbb{Z}</math> with <math>a_{0}=1024</math> and <math>a_{10}=1</math>. Let <math>S
  5 KB (744 words) - 19:46, 20 October 2020
• Mock AIME 1 2006-2007/Problems
  Let <math>\star (x)</math> be the sum of the digits of a positive integer <math>x</math>. <math>\mathcal{S}</math> is the set of pos ...\sqrt{7}</math>, <math>CA=1</math>, and <math>AB=3</math>. If <math>\angle A=\frac{\pi}{n}</math> where <math>n</math> is an integer, find the remainder
  8 KB (1,355 words) - 14:54, 21 August 2020
• Mock AIME 2 2006-2007 Problems/Problem 5
  ...ac 2z + \frac{3}{z^2}+\frac{4}{z ^3}+\frac{5}{z^4}+\cdots</math> and <math>z=n\pm \sqrt{-i},</math> find <math> \lfloor 100n \rfloor</math>. Multiplying both sides of the equation by <math>z</math>, we get
  912 bytes (145 words) - 10:51, 4 April 2012
• Mock AIME 2 2006-2007 Problems/Problem 11
  <math>x+y+z=3,~x^2+y^2+z^2=3,~x^3+y^3+z^3 =3.</math> The roots are <math>x</math>, <math>y</math>, and <math>z</math>, and we add the squares:
  459 bytes (75 words) - 21:55, 20 October 2019
• Mock AIME 2 2006-2007 Problems
  ...r positive integers <math>a,b,c,</math> and <math>d</math> such that <math>a+4=b-4=4c=d/4.</math> Find the smallest dragon. ...ll positive integers <math>n</math> such that <math>n^2+12n-2007</math> is a perfect square. Find the remainder when <math>S</math> is divided by <math>
  5 KB (848 words) - 23:49, 25 February 2017
• Expected value
  Given an event with a variety of different possible outcomes, the '''expected value''' is what on A video that goes over the type of Expected value, practical examples, and pr
  5 KB (789 words) - 20:56, 10 May 2024
• 2006 Romanian NMO Problems
  ...<math>2 \cdot \frac{CN}{BC} = \frac{AM}{AB}</math>. Let <math>P</math> be a point on the line <math>AC</math>. Prove that the lines <math>MN</math> and ...nimal <math>n</math>, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or
  11 KB (1,779 words) - 14:57, 7 May 2012
• Addition
  ...ign) which combines two quantities. The result of addition is called [[sum|a sum]]. For example, the sum of 3 and 2 is 5 because <math>3+2=5</math>. ...)</math>, where <math>f</math> is a [[function]], is denoted <math>\sum_{i=a}^bf(i)</math>. (See also [[Sigma notation]])
  2 KB (309 words) - 20:34, 4 July 2019
• Congruent (geometry)
  '''Congruency''' is a property of multiple geometric figures. [[Image:reflection.PNG|thumb|right|100px|100px|A collection of isometries.]]
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• 2005 AMC 10A Problems/Problem 25
  <math> \textbf{(A) } \frac{266}{1521}\qquad \textbf{(B) } \frac{19}{75}\qquad \textbf{(C) }\f pair A, B, C, D, E;
  6 KB (854 words) - 20:25, 24 July 2022
• Tetrahedron
  ...ce will be the vertices of a tetrahedron as long as they do not all lie on a single [[plane]]. void drawFrontFace(int x, int y, int z){ draw(P[x] -- P[y] -- P[z] -- cycle, linewidth(0.7));
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• Volume
  The '''volume''' of an object is a [[measure]] of the [[amount]] of [[space]] that it occupies. Note that volu The volume of a [[prism]] of [[height]] <math>h</math> and base of [[area]] <math>b</math>
  3 KB (523 words) - 20:24, 17 August 2023
• Integral
  ...ath>c</math> is a [[real]] [[constant]]. This is because the derivative of a constant is <math>0</math>. *The integral of a function <math>f(x)</math> is written as <math>\int f(x)\,dx</math>, where
  5 KB (909 words) - 14:16, 31 May 2022
• 1970 Canadian MO Problems
  Find all number triples <math>(x,y,z)</math> such that when any of these numbers is added to the product of the ...gram, prove that <math>a+h\ge b+k</math>. Find under what conditions <math>a+h=b+k</math>.
  4 KB (604 words) - 04:32, 8 October 2014
• Multiplicative function
  ...e field of [[number theory]], where an alternate definition is often used: a function from the [[positive integer]]s to the [[complex number]]s is said ...on the [[real numbers]] by <math>f(x) = x^2</math> is a simple example of a multiplicative function.
  3 KB (450 words) - 12:59, 21 July 2009
• Complex conjugate
  ...s a root of the polynomial its conjugate <math>\overline{z}</math> will be a root as well. * <math>\overline{(\overline z)} = z</math>.
  2 KB (307 words) - 18:26, 25 January 2020
• Argument
  Given a [[complex number]] <math>z</math>, the '''argument''' <math>\arg z</math> is the measure of the [[signed angle]] the [[ray]] <math>\overrighta ...r this interpretation, we can write <math>\arg(w \cdot z) = \arg(w) + \arg(z)</math>. The other common solution is to restrict the [[range]] of <math>\
  1 KB (188 words) - 20:59, 31 July 2020
• Metric space
  ...es S \to \mathbb{R}_{\geq 0}</math>. The metric <math>d</math> represents a distance function between pairs of points of <math>S</math> which has the f ...r all points <math>x, y, z \in S</math>, <math>d(x, y) + d(y, z) \geq d(x, z)</math>
  2 KB (324 words) - 01:19, 22 December 2012
• Metric (analysis)
  A '''metric''' <math>d</math> on a [[set]] <math>S</math> is a [[function]] <math>d: S \times S \to \mathbb{R}</math> which obeys the foll ...quality]]: <math>d(x, y) + d(y, z) \geq d(x, z)</math> for all <math>x, y, z \in S</math>.
  1 KB (227 words) - 20:15, 13 October 2019
• 2007 iTest Problems
  A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</ <math>\mathrm{(A)}\, 4</math>
  30 KB (4,794 words) - 23:00, 8 May 2024
• Euler line
  ...] <math>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain fixed orders and distan ...followed by a homothety with scale factor <math>2</math> centered at <math>A</math> brings <math>\triangle ABC \to \triangle O_AO_BO_C</math>. Let us ex
  59 KB (10,203 words) - 04:47, 30 August 2023
• 1961 IMO Problems/Problem 1
  \quad x + y + z \!\!\! &= a \; \, \\ x^2 +y^2+z^2 \!\!\! &=b^2 \\
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• Nine-point circle
  ...s ''Euler's circle'' or ''Feuerbach's circle'') of a given [[triangle]] is a circle which passes through 9 "significant" points: "The nine-point circle is tangent to the incircle, has a radius equal to half the circumradius, and its center is the midpoint of th
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• 2003 AMC 10A Problems/Problem 13
  <math> \mathrm{(A) \ } 28\qquad \mathrm{(B) \ } 40\qquad \mathrm{(C) \ } 100\qquad \mathrm{(D Let the numbers be <math>x</math>, <math>y</math>, and <math>z</math> in that order. The given tells us that
  3 KB (383 words) - 15:45, 19 August 2023
• 2003 AMC 10A Problems/Problem 21
  Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. Th <math> \mathrm{(A) \ } 22\qquad \mathrm{(B) \ } 25\qquad \mathrm{(C) \ } 27\qquad \mathrm{(D)
  7 KB (994 words) - 17:51, 11 April 2024
• 2001 IMO Problems/Problem 1
  ...at <math>\angle C \geq \angle B+30^{\circ}</math>. Prove that <math>\angle A+\angle COP < 90^{\circ}</math>. ...of the circumcircle. But <math>AZ = YP</math> (since <math>AZYP</math> is a rectangle).
  2 KB (417 words) - 17:24, 21 July 2018
• 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
• 1985 AIME Problems
  When a right triangle is rotated about one leg, the volume of the cone produced is ...>b</math>, and <math>c</math> are positive integers which satisfy <math>c=(a + bi)^3 - 107i</math>, where <math>i^2 = -1</math>.
  7 KB (1,071 words) - 19:24, 23 February 2024
• 1987 IMO Problems/Problem 4
  ...ath>f(f(n)) = n + k</math> for all <math>n</math>, where <math>k</math> is a fixed positive integer, then <math>k</math> must be even. If <math>k = 2h</ ...n + kr</math>, <math>f(n + kr) = m + k(r+1)</math>. We show this leads to a contradiction. Suppose <math>m < n</math>, so <math>n = m + ks</math> for s
  5 KB (923 words) - 19:51, 21 January 2024
• 2006 Seniors Pancyprian/2nd grade/Problem 3
  i)Convert <math>\alpha=\sin(x-y)+\sin(y-z)+\sin(z-x)</math> into product. ii)Prove that: If in a triangle <math>\alpha\beta\gamma</math> is true that <math>\alpha \sin \bet
  649 bytes (85 words) - 01:24, 13 March 2020
• 2006 Seniors Pancyprian/2nd grade/Problem 4
  ...math>E</math> and its sides <math>BA</math>, <math>CD</math> meet at <math>Z</math>. ii)the points <math>K , L, M ,N</math> are vertices of a rhombus.
  856 bytes (129 words) - 00:10, 20 February 2020
• 2005 Cyprus Seniors TST/Day 1/Problem 2
  Given a triangle with sides <math>\alpha, \beta, \gamma</math>. Prove that: Let <math>\alpha =y+z, \beta =x+z, \gamma =x+y</math>. Then the inequality becomes
  851 bytes (134 words) - 22:50, 12 October 2022
• De Moivre's Theorem
  ...wers. It states that for <math>x\in\mathbb{R}</math> and <math>n\in\mathbb{Z}</math>, <math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math>. ...la holds true because <math>\cos(0x)+i\sin (0x)=1+i0=1</math>. Since <math>z^0=1</math>, the equation holds true.
  2 KB (363 words) - 01:42, 11 January 2024
• Fundamental Theorem of Arithmetic
  ...'' states that every [[positive integer]] <math>n</math> can be written as a product <math>n = p_1 \cdot p_2 \cdot \cdots \cdot p_k</math> where the <ma ...ath> is a prime number such that <math>p \mid ab</math>, then <math>p \mid a</math> or <math>p \mid b</math>. This proof is not terribly interesting, b
  2 KB (376 words) - 23:28, 4 August 2022
• Asymptote (Vector Graphics Language)
  ...[http://netlib.bell-labs.com/who/hobby/MetaPost.html MetaPost], which was a language used only for two-dimensional graphics. In a sense, Asymptote is the ruler and compass of typesetting.
  2 KB (245 words) - 11:42, 14 January 2024
• Asymptote: Macros and Packages
  ...+r,b+r)</math>. In addition, you want it to simply return the pair <math>(a,b)</math> if no value of <math>r</math> is specified, so you want <math>r</ pair newfunction(pair z, real r=0)
  3 KB (515 words) - 00:43, 24 April 2019
• 2005 IMO Shortlist Problems/A1
  ...splaystyle q(x) </math> such that <math> \displaystyle p(x)q(x) </math> is a polynomial having all coefficients <math> \pm 1 </math>. We note that for <math>|z| \ge 2 </math>, <math>n \ge 2 </math> (<math> n \in \mathbb{N} </math>), we
  2 KB (346 words) - 08:56, 14 May 2024
• 2006 Canadian MO Problems/Problem 4
  ...math> beats <math>Y</math>, <math>Y</math> beats <math>Z</math>, and <math>Z</math> beats <math>X</math>. (a) Find the minimum number of cycle triplets possible.
  604 bytes (100 words) - 18:20, 28 November 2023
• 2006 Canadian MO Problems/Problem 5
  The vertices of a right triangle <math>ABC</math> inscribed in a circle divide the circumference into three arcs. ...ht angle is at <math>A</math>, so that the opposite arc <math>BC</math> is a semicircle while arc <math>AB</math> and arc <math>AC</math> are
  2 KB (347 words) - 17:02, 3 June 2011
• Euclidean domain
  A '''Euclidean domain''' (or '''Euclidean ring''') is a type of [[ring]] in which the [[Euclidean algorithm]] can be used. Formally we say that a ring <math>R</math> is a Euclidean domain if:
  2 KB (357 words) - 15:28, 22 August 2009
• Mock AIME 2 Pre 2005/Problems
  <math>x</math> is a real number with the property that <math>x + \frac{1}{x} = 3</math>. Let < ...l, and a black ball. These balls are randomly drawn out of the box one at a time (without replacement) until two of the same color have been removed.
  6 KB (990 words) - 15:23, 11 November 2009
• 1960 IMO Problems
  ...nteger). Let <math>\alpha</math> be the acute angle subtending, from <math>A</math>, that segment which contains the midpoint of the hypotenuse. Let <ma \tan{\alpha}=\frac{4nh}{(n^2-1)a}.
  3 KB (511 words) - 21:21, 20 August 2020
• 2007 AIME I Problems
  ...ly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the oth ...-1</math>. Given that the imaginary parts of <math>z^{2}</math> and <math>z^{3}</math> are the same, what is <math>b</math> equal to?
  7 KB (1,218 words) - 15:28, 11 July 2022
• 2007 AIME I Problems/Problem 3
  ...-1</math>. Given that the imaginary parts of <math>z^{2}</math> and <math>z^{3}</math> are the same, what is <math>b</math> equal to?<!-- don't remove {{AIME box|year=2007|n=I|num-b=2|num-a=4}}
  1,003 bytes (163 words) - 15:34, 18 February 2017
• 2007 AIME I Problems/Problem 5
  The formula for converting a Fahrenheit temperature <math>F</math> to the corresponding Celsius temperat ...{9}k\right]</math>. Our equation simplifies to <math>k = \left[ \frac{9}{5}z\right]</math>. However, this equation is not possible, as we defined <math>
  7 KB (1,076 words) - 00:10, 29 November 2023
• 2007 AIME I Problems/Problem 13
  ...math> and vertex <math>E</math> has eight edges of length <math>4</math>. A plane passes through the midpoints of <math>AE</math>, <math>BC</math>, and Note first that the intersection is a [[pentagon]].
  7 KB (1,034 words) - 21:56, 22 September 2022
• 2004 USAMO Problems/Problem 5
  Let <math>a </math>, <math>b </math>, and <math>c </math> be positive real numbers. Pr (a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \ge (a+b+c)^3
  4 KB (626 words) - 12:58, 27 August 2023
• 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
• 2002 USA TST Problems/Problem 4
  ...yle S </math> such that <math> \displaystyle f(\{a,b\}), f(\{b,c\}), f(\{c,a\}) </math> are all equal to 0. ...</math> such that for any <math> a, b \in S_k </math>, <math> 2^k \mid f(\{a,b\}) </math>.
  2 KB (392 words) - 14:34, 7 March 2010
• 2005 USAMO Problems
  ...sible to arrange all divisors of <math>n</math> that are greater than 1 in a circle so that no two adjacent divisors are relatively prime. x^3+x^3y+y^2+y+z^9 & = 157^{147}
  4 KB (609 words) - 09:24, 14 May 2021
• 2002 USA TST Problems
  Let <math> \displaystyle ABC </math> be a triangle. Prove that \cos\frac{A-B}{2} + \cos\frac{B-C}{2} + \cos\frac{C-A}{2}.
  3 KB (544 words) - 06:58, 3 August 2017
• 1961 IMO Problems/Problem 2
  Let <math>a</math>, <math>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
• 1989 USAMO Problems
  ...tegers <math>0 < a,\ b,\ c,\ d < 1000000</math> such that <math>T_{1988} = a S_{1989} - b</math> and <math>U_{1988} = c S_{1989} - d</math>. ...laying in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.
  2 KB (326 words) - 18:52, 18 July 2016
• 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
• Nesbitt's Inequality
  ...ely cited, has many instructive proofs. It states that for positive <math>a, b, c </math>, \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} \ge \frac{3}{2}
  7 KB (1,224 words) - 16:21, 24 October 2022
• 2005 IMO Shortlist Problems/A2
  ...t <math>y = \frac{x-z}{f(z)-f(x)} > 0 </math>, so <math>x + yf(x) = z + yf(z) </math>, to obtain <math>f(x)f(y) = 2f(x+yf(x)) = 2f(z+yf(z)) = f(z)f(y) </math>,
  7 KB (1,179 words) - 20:59, 28 March 2010
• LaTeX:Examples
  ...vanced commands and a library of symbols and keywords to memorize, look at a few LaTeX documents prepared by Art of Problem Solving Community members. S ...ples/mathfanBasics.tex Click here for the source file]. This file contains a collection of problems and solutions. Our beginning and newly intermediate-
  5 KB (687 words) - 02:03, 4 February 2020
• 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
• Imaginary number
  An '''imaginary number''' is a [[complex number]] whose [[real part]] is equal to 0. In the [[complex pla ...r <math>z</math> is imaginary if and only if <math>z = i \cdot \textrm{Im}(z)</math>, where <math>i = \sqrt{-1}</math> is the [[imaginary unit]] and <ma
  2 KB (283 words) - 00:18, 8 March 2012
• Chicken McNugget Theorem
  ...ritten in the form <math>am + bn</math> for [[nonnegative]] integers <math>a, b</math> is <math>mn-m-n</math>. A consequence of the theorem is that there are exactly <math>\frac{(m - 1)(n
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• 2005 AMC 12A Problems/Problem 23
  Two distinct [[number]]s a and b are chosen randomly from the [[set]] <math>\{2, 2^2, 2^3, ..., 2^{25} <math>\mathrm{(A)}\ \frac{2}{25}\qquad \mathrm{(B)}\ \frac{31}{300}\qquad \mathrm{(C)}\ \fra
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• 2007 AMC 12A Problems/Problem 9
  <math>\mathrm{(A)}\ \frac 23\qquad \mathrm{(B)}\ \frac 34\qquad \mathrm{(C)}\ \frac 45\qquad ...r}</math>. If Yan walks directly to the stadium, then assuming he walks at a rate of <math>1</math>, it will take him <math>x-r</math> units of time. Si
  5 KB (804 words) - 14:55, 21 August 2022
• Proportion
  ...ists between them. There are several types of proportions, each defined by a separate class of function. Direct proportion is a proportion in which one number is a multiple of the other. Direct proportion between two numbers <math>x</math>
  2 KB (325 words) - 16:34, 1 June 2022
• 2006 AIME II Problems/Problem 15
  Given that <math> x, y, </math> and <math>z</math> are real numbers that satisfy: x &= \sqrt{y^2-\frac{1}{16}}+\sqrt{z^2-\frac{1}{16}}, \\
  4 KB (725 words) - 17:18, 27 June 2021
• 2006 AIME II Problems
  ...vex hexagon <math>ABCDEF</math>, all six sides are congruent, <math>\angle A</math> and <math>\angle D</math> are right angles, and <math>\angle B, \ang ...\log_{10} 75</math>, and <math>\log_{10} n</math>, where <math>n</math> is a positive integer. Find the number of possible values for <math>n</math>.
  8 KB (1,350 words) - 12:00, 4 December 2022
• 1998 PMWC Problems
  int triangle(pair z, int n){ dot((z.x+j -1 + i/2 ,z.y + i*sqrt(3)/2));
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2001 PMWC Problems
  pair A,B,C,D,E,F,G,O; A=origin;
  4 KB (641 words) - 21:24, 21 April 2014
• British Flag Theorem
  ...British flag theorem due to the similarities between the British flag and a diagram of the points (shown below): pair A,B,C,D,P;
  3 KB (405 words) - 01:14, 3 December 2023
• 1989 USAMO Problems/Problem 3
  ...</math> and <math>b</math> such that <math>P(a + bi) = 0</math> and <math>(a^2 + b^2 + 1)^2 < 4 b^2 + 1</math>. <cmath> P(z) = \prod_{j=1}^n (z- z_j) . </cmath>
  2 KB (340 words) - 19:11, 18 July 2016
• 1961 IMO Problems
  \quad x + y + z \!\!\! &= a \; \, \\ x^2 +y^2+z^2 \!\!\! &=b^2 \\
  3 KB (425 words) - 21:18, 20 August 2020
• 2004 AMC 10A Problems/Problem 20
  pair A, B, C, D, E, F; A = (0,0);
  4 KB (710 words) - 02:47, 18 April 2024
• 1960 IMO Problems/Problem 5
  ...ABCDA'B'C'D'</math> (with face <math>ABCD</math> directly above face <math>A'B'C'D'</math>). a) Find the locus of the midpoints of the segments <math>XY</math>, where <ma
  2 KB (402 words) - 23:28, 18 July 2016
• Excircle
  ...is a [[circle]] [[Tangent line|tangent]] to the extensions of two sides of a [[triangle]] and the third side. pair excenter(pair A, pair B, pair C){
  5 KB (843 words) - 03:02, 1 July 2020
• 2001 USAMO Problems/Problem 2
  ...t <math>AD_2</math> at two points, the closer of which to the vertex <math>A</math> is denoted by <math>Q</math>. Prove that <math>AQ = D_2P</math>. It is well known that the excircle opposite <math>A</math> is tangent to <math>\overline{BC}</math> at the point <math>D_2</mat
  11 KB (2,091 words) - 08:35, 16 November 2017
• L'Hôpital's Rule
  '''L'Hopital's Rule''' is a theorem dealing with [[limit]]s that is very important to [[calculus]]. The definition of a derivative is <math>f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}</ma
  2 KB (475 words) - 15:04, 24 March 2022
• Magnitude
  ...gin, a measure of the size of a complex number. The magnitude is generally a nonnegative real number. ...orm of the [[Triangle Inequality]] which states that <math>|a| + |b| \geq |a + b|</math>.
  2 KB (246 words) - 18:27, 2 March 2023
• Zorn's Lemma
  '''Zorn's Lemma''' is a [[set theory | set theoretic]] result which is equivalent to the [[Axiom of Let <math>A</math> be a [[partially ordered set]].
  9 KB (1,669 words) - 19:02, 1 August 2018
• Root (operation)
  ...denoted by <math>\sqrt[n]{x}</math>, is a common operation on numbers and a partial [[inverse]] to [[exponentiation]]. (The proper inverse is the [[log A known method to compute all the roots of <math>\sqrt[n]{x}</math> is by the
  3 KB (532 words) - 16:52, 20 May 2020
• Ideal
  In [[ring theory]], an '''ideal''' is a special kind of [[subset]] of a [[ring]]. Two-sided ideals in rings are the [[kernel]]s of ring [[homomorp ...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
• 2004 AMC 12A Problems/Problem 14
  ...ath>20</math> is added to the third term, the three resulting numbers form a [[geometric progression]]. What is the smallest possible value for the thir <math>\text {(A)}\ 1 \qquad \text {(B)}\ 4 \qquad \text {(C)}\ 36 \qquad \text {(D)}\ 49 \q
  4 KB (689 words) - 03:35, 16 January 2023
• 2004 AMC 12A Problems/Problem 6
  ...>, <math>X=2\cdot 2004^{2004}</math>, <math>Y=2004^{2004}</math> and <math>Z=2004^{2003}</math>. Which of the following is the largest? ...W \qquad \mathrm {(C)} W-X \qquad \mathrm {(D)} X-Y \qquad \mathrm {(E)} Y-Z \qquad</math>
  1 KB (139 words) - 02:10, 30 December 2020
• 1992 USAMO Problems/Problem 2
  ...O=(0,0), a=expi(0), b=expi(1/6), c=expi(2/6), d=expi(3/6), y=expi(32/30), z= expi(34/30); pair A=a, B=b/b.x, C= c/c.x, D=d/d.x, Y=y/y.x, Z=z/z.x, E=(D+Y)/2;
  4 KB (628 words) - 07:41, 19 July 2016
• 1976 USAMO Problems/Problem 5
  prove that <math>x-1</math> is a factor of <math>P(x)</math>. ...ath>, and then note that since <math>P(1)= 0</math>, <math>(x-1)</math> is a factor of <math>P(x)</math>.
  3 KB (572 words) - 17:14, 16 August 2015
• 2000 AMC 12 Problems/Problem 10
  \text {(A) } (1,7, - 3) \qquad \text {(B) } ( - 1,7, - 3) \qquad \text {(C) } ( - 1, Step 1: Reflect in the <math>xy</math>-plane. Replace <math>z</math> with its additive inverse: <math>(1,2,-3)</math>
  1 KB (173 words) - 08:56, 21 August 2023
• 2000 AMC 12 Problems/Problem 15
  ...rac{x}{3}\right) = x^2 + x + 1</math>. Find the sum of all values of <math>z</math> for which <math>f(3z) = 7</math>. <cmath>\text {(A)}\ -1/3 \qquad \text {(B)}\ -1/9 \qquad \text {(C)}\ 0 \qquad \text {(D)}\
  3 KB (441 words) - 21:11, 29 April 2023
• 2000 AMC 12 Problems/Problem 20
  If <math>x,y,</math> and <math>z</math> are positive numbers satisfying ...h>x + \frac{1}{y} = 4,\qquad y + \frac{1}{z} = 1, \qquad \text{and} \qquad z + \frac{1}{x} = \frac{7}{3}</cmath>
  2 KB (244 words) - 07:34, 4 November 2022
• 1997 PMWC Problems
  ...represent a single digit. Different letters represent different digits but a box can represent any digit. What does the five-digit number <math>\mathrm{ ...&\Box\\ \hline &\Box &\Box & 9 &\Box\\ \Box &\Box &\Box & 7 &\\ \hline H & A & P & P & Y\end{array} </cmath>
  15 KB (2,057 words) - 19:13, 10 March 2015
• 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
• 1959 AHSME Problems
  Each edge of a cube is increased by <math>50</math>%. The percent of increase of the surfa <math>\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300
  22 KB (3,345 words) - 20:12, 15 February 2023
• 2002 AMC 12B Problems/Problem 5
  ...hat <math>v < w < x < y < z</math> and <math>v, w, x, y, </math> and <math>z</math> form an [[arithmetic sequence]]. Find the value of <math>x</math>. <math>\mathrm{(A)}\ 72
  2 KB (263 words) - 05:58, 8 August 2023
• 1991 IMO Problems/Problem 1
  ...isectors of the angles <math>A,B,C</math> meet the opposite sides in <math>A^\prime,B^\prime,C^\prime</math> respectively. Prove that ...>\frac{1}{8}\prod\frac{x+2y+z}{x+y+z}=\frac{1}{8}\prod\left(1+\frac{y}{x+y+z}\right)</math>. To prove the right side of the inequality, we simply apply
  2 KB (311 words) - 09:40, 19 November 2023
• 2001 AMC 12 Problems/Problem 4
  <math>\textbf{(A)}\ 5\qquad \textbf{(B)}\ 20\qquad \textbf{(C)}\ 25\qquad \textbf{(D)}\ 30\q Say the three numbers are <math>x</math>, <math>y</math> and <math>z</math>. When we arrange them in ascending order then we can assume <math>y<
  2 KB (311 words) - 21:53, 10 February 2024
• 2007 AMC 10A Problems/Problem 18
  ...f its sides has length <math>4</math>, and each two consecutive sides form a right angle. Suppose that <math>\overline{AG}</math> and <math>\overline{CH pair A=(1,3), B=(2,3), C=(2,2), D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1
  6 KB (867 words) - 00:17, 20 May 2023
• 1974 USAMO Problems/Problem 1
  ...s impossible that <math>P(a)=b</math>, <math>P(b)=c</math>, and <math>P(c)=a</math>. ...P</math> is a polynomial with integral coefficients, then <cmath>a - b | P(a) - P(b).</cmath> (Why?)
  7 KB (1,291 words) - 20:30, 27 April 2020
• 2004 AMC 12B Problems/Problem 16
  ...alues of <math>z</math> satisfy both <math>|z| = 5</math> and <math>f(z) = z</math>? <math>\mathrm{(A)}\ 0
  1 KB (240 words) - 12:59, 30 March 2023
• 2002 AMC 10A Problems/Problem 25
  pair A,B,C,D; A=(0,0);
  8 KB (1,308 words) - 07:05, 19 December 2022
• Isogonal conjugate
  ...nal conjugates''' are pairs of [[point]]s in the [[plane]] with respect to a certain [[triangle]]. ...ll</math> and a point <math>O</math> lying on <math>\ell</math> be given. A pair of lines symmetric with respect to <math>\ell</math> and containing t
  54 KB (9,416 words) - 08:40, 18 April 2024
• 1976 IMO Problems/Problem 3
  A box whose shape is a parallelepiped can be completely filled with cubes of side <math>1.</math> We name a,b,c the sides of the parallelepiped, which are positive integers. We also p
  4 KB (605 words) - 16:27, 29 January 2021
• 2008 AMC 12A Problems
  {{AMC12 Problems|year=2008|ab=A}} A bakery owner turns on his doughnut machine at <math>\text{8:30}\ {\small\te
  13 KB (2,025 words) - 13:56, 2 February 2021
• 2008 AMC 12A Problems/Problem 18
  ...is, one on the positive <math>y</math>-axis, and one on the positive <math>z</math>-axis. Let <math>O</math> be the [[origin]]. What is the volume of [[ <math>\mathrm{(A)}\ \sqrt{85}\qquad\mathrm{(B)}\ \sqrt{90}\qquad\mathrm{(C)}\ \sqrt{95}\qqua
  2 KB (302 words) - 04:51, 16 January 2023
• 2008 AMC 12A Problems/Problem 23
  ...ns of the equation <math>z^4+4z^3i-6z^2-4zi-i=0</math> are the vertices of a convex polygon in the complex plane. What is the area of the polygon? <math>\mathrm{(A)}\ 2^{\frac{5}{8}}\qquad\mathrm{(B)}\ 2^{\frac{3}{4}}\qquad\mathrm{(C)}\ 2\
  1 KB (199 words) - 01:38, 10 November 2019
• 1992 USAMO Problems/Problem 3
  ...integer <math>n \le 1500</math>, there is a subset <math>S</math> of <math>A</math> for which <math>\sigma(S) = n</math>. What is the smallest possible ...th>x\le p</math>, <math>x\in \mathbb{Z}^+</math>, <math>\exists Y\subseteq Z</math>, <math>\sigma(Y)=x</math>, <math>\nexists \sigma(Y)=p+1</math>.
  5 KB (858 words) - 07:52, 19 July 2016
• 2007 AMC 12B Problems/Problem 23
  <math>\mathrm {(A)} 6\qquad \mathrm {(B)} 7\qquad \mathrm {(C)} 8\qquad \mathrm {(D)} 10\qqua Let <math>a</math> and <math>b</math> be the two legs of the triangle.
  4 KB (725 words) - 19:59, 4 January 2024
• 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
• 2006 Alabama ARML TST Problems
  River draws four cards from a standard 52 card deck of playing cards. Exactly 3 of them are 2’s. Find t There exist positive integers <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math> with no common f
  5 KB (769 words) - 20:56, 24 March 2015
• 2008 AMC 12B Problems/Problem 19
  ...(z) = (4 + i) z^2 + \alpha z + \gamma</math> for all complex numbers <math>z</math>, where <math>\alpha</math> and <math>\gamma</math> are complex numbe <math>\textbf{(A)} \; 1 \qquad \textbf{(B)} \; \sqrt {2} \qquad \textbf{(C)} \; 2 \qquad \te
  2 KB (307 words) - 19:28, 26 September 2023
• 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
• 1985 AJHSME Problems
  <math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 49 \qquad \textbf{(D)} \fr <math>\textbf{(A)}\ 845 \qquad \textbf{(B)}\ 945 \qquad \textbf{(C)}\ 1005 \qquad \textbf{(D
  12 KB (1,670 words) - 17:42, 24 November 2021
• 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 3
  ...of <math>x,y,z</math> is odd. We see from our second expression that <math>z</math> must be odd, because <math>91</math> is also odd and <math>2y</math> ...readdressing the first equation, we see that if <math>2x+3y</math> will be a multiple of <math>6</math>, <math>4z \equiv 2 \pmod{6} = 5</math>, we get t
  3 KB (564 words) - 22:15, 28 November 2023
• 2008 AIME I Problems/Problem 9
  a + b + c &= 10\end{align*}</cmath> ...>. The last doesn't work, obviously. This gives the three solutions <math>(a,b,c) = (5,2,3),(3,5,2),(1,8,1)</math>. In terms of choosing which goes wher
  6 KB (909 words) - 15:39, 8 August 2022
• 2008 AIME I Problems/Problem 13
  ...d <math>c</math> are relatively prime, and <math>c > 1</math>. Find <math>a + b + c</math>. ...the original polynomial <math>p</math>, we find that at <math>\left(\frac{a}{c}, \frac{b}{c}\right)</math>,
  8 KB (1,218 words) - 00:07, 11 April 2024
• 2008 AIME I Problems/Problem 15
  .... When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of t triple A=(0,0,0),B=(1,0,0),C=(1,1,0),D=(0,1,0);
  6 KB (1,041 words) - 00:54, 1 February 2024
• Vornicu-Schur Inequality
  The '''Vornicu-Schur Inequality''' is a generalization of [[Schur's Inequality]] discovered by the Romanian mathema ...ger]] and let <math>f:\mathbb{R} \rightarrow \mathbb{R}_{\geq 0}</math> be a [[function]] from the reals to the nonnegative reals that is either [[conve
  1 KB (228 words) - 16:58, 29 December 2021
• Cauchy Functional Equation
  ...lutions are only the [[linear function]]s <math>f(x)=ax</math>, with <math>a\in\mathbb Q</math>. ...possible to write down a formula for any such function, and the [[graph of a function | graph]] of any such function is [[dense]] in the plane.)
  2 KB (295 words) - 13:57, 22 July 2009
• 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
• 2008 AIME II Problems/Problem 8
  Let <math>a = \pi/2008</math>. Find the smallest positive integer <math>n</math> such t 2[\cos(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)]
  4 KB (621 words) - 18:26, 16 January 2023
• 2008 AIME II Problems/Problem 9
  ...ckwise rotation of <math>\pi/4</math> radians about the origin followed by a translation of <math>10</math> units in the positive <math>x</math>-directi ...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
  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 D(A--B--C--cycle); D(Y--Q--P--X); D(Q--M); D(P--C,dashed);
  6 KB (1,065 words) - 20:12, 9 August 2022
• Filter
  A '''filter''' on a [[set]] <math>X</math> is a structure of [[subset]]s of <math>X</math>. ...set of subsets of <math>X</math>. We say that <math>\mathcal{F}</math> is a filter on <math>X</math> if and only if each of the following conditions ho
  2 KB (368 words) - 21:14, 13 October 2019
• 2005 USAMO Problems/Problem 3
  ...th>. Prove that points <math>B_1, C_1,P</math>, and <math>Q</math> lie on a circle. pair A=(2,5), B=(-1,0), C=(5,0);
  6 KB (1,117 words) - 01:17, 11 October 2021
• 2005 USAMO Problems/Problem 2
  x^3 + x^3y + y^2 + y + z^9 &= 157^{147}\end{align*} has no solutions in integers <math>x</math>, <math>y</math>, and <math>z</math>.
  7 KB (1,053 words) - 10:38, 12 August 2015
• 2006 Alabama ARML TST Problems/Problem 6
  ...a</math> and let <math>\{a\}=a-\lfloor a \rfloor</math>. Find <math>10(x+y+z)</math> given that x+\lfloor y \rfloor +\{z\}=14.2,\\
  862 bytes (139 words) - 12:13, 20 January 2018
• 1997 PMWC Problems/Problem T7
  ...Take out all the smaller cubes which have at least one red surface and fix a cuboid, <b>keeping the surfaces of the cuboid red</b>. Now what is the maxi ...math>. This is because the total length of the edges is simply <math>4(x+y+z)</math>, and both have <math>8</math> corners, so simply divide by <math>4<
  2 KB (283 words) - 21:47, 13 July 2023
• 2008 AIME II Problems/Problem 13
  ...where <math>a</math> and <math>b</math> are positive integers. Find <math>a + b</math>. ...its conjugate but we can do that in the Cartesian plane too (just reflect a point in the Cartesian plane over the x-axis)! If you're familiar with inve
  6 KB (894 words) - 18:56, 25 December 2022
• 0.999...
  ...r. But how do we know that there are no 'gaps' so it could possibly not be a real number. But lets prove it. ...f real numbers has a upper bound, then it has a least upper bound (that is a real number).
  3 KB (577 words) - 20:04, 4 February 2023
• 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 2
  ...ath>F</math>, inside of triangle <math>ABC</math>. Prove that points <math>A</math>, <math>N</math>, <math>F</math>, and <math>P</math> all lie on one c ...\angle OFA = 90^{\circ}</math>, as this would immediately prove that <math>A,P,O,F,N</math> are concyclic.
  20 KB (3,565 words) - 11:54, 1 May 2024
• 2008 USAMO Problems/Problem 4
  ...- 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles tria ..., P_2, \ldots, P_n</math>. Consider a diagonal <math>d = \overline{P_a\,P_{a+k}},\,k \le n/2</math> in the triangulation. We show that <math>k</math> mu
  8 KB (1,318 words) - 12:37, 20 April 2022
• 2008 USAMO Problems/Problem 6
  ...mber of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form <math>2^k</math> for some positive integ ...e [[vector space]] of all such functions. Define the linear operator <math>A : V \to V</math> as
  13 KB (2,414 words) - 14:37, 11 July 2016
• Normal subgroup
  ...mod <math>{\rm H}</math>"). (Hence the notation <math>\mathbb{Z}/n\mathbb{Z}</math> for the integers mod <math>n</math>.) <math>N</math> is said to be a normal subgroup of a group <math>G</math> if <math>aNa^{-1}=N</math>.Note that this means <math>
  15 KB (2,840 words) - 12:22, 9 April 2019
• Stabilizer
  ...zer''' is a part of a [[monoid]] (or [[group]]) [[group action|acting]] on a set. ...er words, the stabilizer of <math>A</math> is the [[transporter]] of <math>A</math> to itself.
  3 KB (531 words) - 18:47, 9 September 2008
• Order (group theory)
  The order of a group <math>G</math>, sometimes denoted <math>\text{ord(G)}</math>, is the ...multiplicative group of non-[[zero divisor]]s in <math>\mathbb{Z}/n\mathbb{Z}</math>.
  1 KB (183 words) - 10:43, 11 April 2020
• Conjugate (group theory)
  Let <math>G</math> be a [[group]] operating on a [[set]] <math>S</math>. An element <math>y\in S</math> ''conjugate'' to an ...s, the orbit of a complex number <math>z</math> is the set <math>\{z, \bar{z}\}</math>.
  1 KB (214 words) - 22:20, 21 May 2008
• 2008 Mock ARML 1 Problems/Problem 8
  For positive real numbers <math>a,b,c,d</math>, <cmath>\begin{align*}2a^2 + \sqrt {(a^2 + b^2)(a^2 + c^2)} &= 2bc\\
  2 KB (267 words) - 20:17, 4 December 2016
• 2003 AMC 12A Problems/Problem 17
  ...h>M</math> intersects a circle with radius <math>4</math> and center <math>A</math> at points <math>P</math> and <math>D</math>. What is the distance fr pair A,B,C,D,M,P;
  6 KB (1,026 words) - 22:35, 29 March 2023
• P-group
  ...a [[finite]] [[group]] whose [[order (group theory) |order]] is a power of a [[prime]] <math>p</math>. '''Lemma.''' Let <math>G</math> be a <math>p</math>-group acting on a finite set <math>S</math>; let <math>S^G</math> denote the set of fixed poi
  4 KB (814 words) - 22:50, 3 November 2023
• Sylow Theorems
  ...a collection of results in the theory of [[finite]] [[group]]s. They give a partial converse to [[Lagrange's Theorem]], and are one of the most importa :'''Theorem.''' Every finite group contains a [[Sylow p-subgroup |Sylow <math>p</math>-subgroup]].
  11 KB (2,071 words) - 12:25, 9 April 2019
• Statistics
  ...st data sets from statistics are from [[sample (statistics)|samples]] from a much larger [[population (statistics)|population]] size. ""Inferential stat ...the sample data set came from a population with known [[parameters]], like a [[normal distribution]], while others are [[non-parametric tests]]
  2 KB (212 words) - 22:42, 13 February 2016
• Division Algorithm
  ...re exist unique integers <math>q</math> and <math>r</math> such that <math>a = qb + r</math> and <math>0 \leq r < b</math>. ...to be the smallest positive value that can be expressed in the form <math>a-nb</math>, but <math>r-b</math> is smaller, positive, and can also be expre
  2 KB (311 words) - 02:25, 22 June 2009
• 1999 AHSME Problems/Problem 30
  <math> \mathrm{(A) \ }2 \qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 33\qquad \mathrm{(D) \ ...+ y^3 + z^3 - 3xyz = \frac12\cdot(x + y + z)\cdot((x - y)^2 + (y - z)^2 + (z - x)^2)</cmath>
  881 bytes (134 words) - 14:36, 5 July 2013
• 2008 IMO Problems
  ...ter of an acute-angled triangle <math>ABC</math>. The circle <math>\Gamma_{A}</math> centered at the midpoint of <math>BC</math> and passing through <ma Let <math>x, y, z\neq 1</math> be three real numbers, such that <math>xyz = 1</math>
  4 KB (613 words) - 00:30, 18 February 2021
• 1998 AHSME Problems
  ...7, x = 5, y = 4, z = 8</math>. In rectangle E, <math>w = 9, x = 2, y = 7, z = 0</math>. These five rectangles are placed, without rotating or reflectin ...,(0,6),(-1,0));label("$x$",(1.5,7),(0,1));label("$y$",(3,6),(1,0));label("$z$",(1.5,5),(0,-1));
  15 KB (2,222 words) - 10:40, 11 August 2020
• 2001 IMO Shortlist Problems/C1
  ...math> and <math>a_k = a_j + 1</math>. Considering all such sequences <math>A</math> find the greatest value of <math>m</math>.'' ...ue, so does it unstrictly increasing version. Therefore, we can look at an A such that it is unstrictly increasing and attains the maximal number of suc
  4 KB (703 words) - 12:45, 27 November 2017
• 2000 USAMO Problems/Problem 2
  ...>r = \frac {A}{s} = \frac{\sqrt{(x+y+z)xyz}}{x+y+z} = \sqrt{\frac{xyz}{x+y+z}}</math>. Thus, <cmath>6\sqrt{\frac{x+y+z}{xyz}} = \frac{2yz + 5xy + 5xz}{xyz}</cmath>
  4 KB (703 words) - 18:40, 3 January 2019
• 2001 IMO Shortlist Problems/C4
  ...\}</math> with <math>x < y < z</math> is called <i>historic</i> if <math>\{z - y,y - x\} = \{1776,2001\}</math>. Show that the set of all nonnegative i ...ition in its historic set (e.g. in <math>\{1,2,4\}</math>, 4 is in column z, 2 in column y, and 1 in column x).
  2 KB (494 words) - 17:34, 5 July 2020
• 2001 IMO Shortlist Problems/G6
  Let <math>ABC</math> be a triangle and <math>P</math> an exterior point in the plane of the triangle. So <math>A</math> is <math>(1,0,0)</math>, <math>B</math> is <math>(0,1,0)</math>, <ma
  3 KB (565 words) - 23:25, 2 April 2012
• 2001 IMO Shortlist Problems/N2
  ...that <math>m \leq x/y</math> for any positive integer solution <math>(x,y,z,u)</math> of the system, with <math>x \geq y</math>. ..., we discard the second inequality and have that <math>3+2\sqrt2</math> is a lower bound for <math>r</math>
  2 KB (267 words) - 14:38, 1 September 2017
• 2001 IMO Shortlist Problems/N4
  ...\leq p - 2</math> such that neither <math>a^{p - 1} - 1</math> nor <math>(a + 1)^{p - 1} - 1</math> is divisible by <math>p^2</math>. Let us work in the group <math>(\mathbb{Z}/p^2\mathbb{Z})^{\times}</math>.
  3 KB (454 words) - 22:27, 16 October 2020
• 2008 IMO Problems/Problem 4
  ...ions <math>f: (0, \infty) \mapsto (0, \infty)</math> (so <math>f</math> is a function from the positive real numbers) such that ...ht)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}</math>
  4 KB (661 words) - 01:10, 19 November 2023
• 1997 PMWC Problems/Problem T1
  ...<math>V</math>, <math>W</math>, <math>X</math>, <math>Y</math>, and <math>Z</math> divide the sides into lengths of one unit. Find the ratio of the are label("$Z$",(1/6,sqrt(3)/3),NE);
  1 KB (188 words) - 14:38, 20 April 2014
• 1985 IMO Problems/Problem 4
  ...e divisor greater than <math>23</math>, prove that <math>M</math> contains a subset of <math>4</math> elements whose product is the <math>4</math>th po ...have <math>xy=z^2</math> for some <math>z</math> which implies <math>abcd=z^4</math> and the claim is proved.
  1 KB (261 words) - 23:56, 29 January 2021
• Carnot's Theorem
  '''Carnot's Theorem''' states that in a [[triangle]] <math>ABC</math>, the signed sum of [[perpendicular]] distance pair a,b,c,O,i,d,f,g;
  4 KB (723 words) - 01:45, 18 February 2021
• 1995 IMO Problems/Problem 1
  ...Let <math>P</math> be a point on the line <math>XY</math> other than <math>Z</math>. The line <math>CP</math> intersects the circle with diameter <math> Think about Radical Axis, Power of a Point and Radical Center.
  5 KB (847 words) - 19:03, 12 October 2021
• 2008 IMO Problems/Problem 2
  '''(i)''' If <math>x</math>, <math>y</math> and <math>z</math> are three real numbers, all different from <math>1</math>, such that ...\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1</math>.
  3 KB (593 words) - 01:09, 19 November 2023
• Integral domain
  An '''integral domain''' is a [[commutative]] [[Domain (Ring theory)|domain]]. More explicitly a [[ring]], <math>R</math>, is an integral domain if:
  826 bytes (125 words) - 17:43, 16 March 2012
• Field of fractions
  ...>\mathbb{Z}</math>, and can be viewed as a way turning <math>R</math> into a [[field]]. ...division in <math>R</math> is undefined, so an expression like <math>\frac{a}{b}</math> is meaningless.
  2 KB (439 words) - 14:09, 4 March 2022
• 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
• Gauss's Lemma (polynomial)
  '''Gauss's Lemma for Polynomials''' is a result in [[abstract algebra | algebra]]. ...al statement concerns [[polynomial]]s with [[integer]] coefficients. Such a polynomial is called ''primitive'' if the [[greatest common divisor]] of it
  3 KB (483 words) - 12:23, 30 May 2019
• 1989 IMO Problems/Problem 2
  <math>ABC</math> is a triangle, the bisector of angle <math>A</math> meets the circumcircle of triangle <math>ABC</math> in <math>A_1</ma ...the hexagon is equal to <math>[ABC]+\sum\frac{1}{4}a^2\left(\frac{2r}{b+c-a}\right)</math>.
  3 KB (568 words) - 11:50, 30 January 2021
• Fermat's Two Squares Theorem
  ...' states that that a [[prime number]] <math>p</math> can be represented as a sum of two nonzero [[perfect squares |squares]] if and only if <math>p = 2< The theorem is a starting point in the investigation of [[binary quadratic forms]]. Histori
  4 KB (612 words) - 12:10, 30 May 2019
• Boolean
  ...the capitalization) are constants representing the two possible values of a boolean. (x and not y) or (z and not y)
  2 KB (314 words) - 16:29, 25 March 2011
• 1995 AHSME Problems/Problem 30
  A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube a <math> \mathrm{(A) \ 16 } \qquad \mathrm{(B) \ 17 } \qquad \mathrm{(C) \ 18 } \qquad \mathrm{
  5 KB (828 words) - 05:52, 30 September 2023
• 2007 AMC 12B Problems/Problem 25
  Points <math>A,B,C,D</math> and <math>E</math> are located in 3-dimensional space with <ma <math>\mathrm {(A)} \sqrt{2}\qquad \mathrm {(B)} \sqrt{3}\qquad \mathrm {(C)} 2\qquad \mathrm
  3 KB (470 words) - 19:46, 17 July 2023
• 1973 USAMO Problems/Problem 4
  <center><math>x+y+z=3</math>, <math>x^2+y^2+z^2=3</math>,
  5 KB (888 words) - 08:18, 22 April 2024
• Spiral similarity
  ...milarity is a plane transformation composed of a rotation of the plane and a dilation of the plane having the common center. The order in which the com Any two directly similar figures are related either by a translation or by a spiral similarity (directly similar figures are similar and have the same o
  28 KB (4,863 words) - 00:29, 16 December 2023
• Van Aubel's Theorem
  ...: <math>ABA'B'</math>, <math>BCB'C'</math>, <math>CDC'D'</math>, <math>DAD'A'</math>; yielding centers <math>P_{AB}, P_{BC}, P_{CD}, P_{DA}</math>. Van pair A, B, C, D, O, P, Q, R, SS;
  2 KB (410 words) - 14:01, 4 March 2023
• Well Ordering Principle
  ...ement. While this theorem is mostly brushed off as common sense, there is a bit of formalism required to actually prove it sufficiently. We will do th ...<math>\mathbb{R}</math>. Then the positive integers denoted <math>\mathbb{Z}^{+}</math> are defined by the equation
  2 KB (408 words) - 22:20, 12 May 2022
• 2004 AMC 10B Problems
  ...math> seats. Rows <math>12</math> through <math>22</math> are reserved for a youth club. How many seats are reserved for this club? <math> \mathrm{(A) \ } 297 \qquad \mathrm{(B) \ } 330\qquad \mathrm{(C) \ } 363\qquad \mathrm
  13 KB (1,988 words) - 20:19, 15 May 2024
• 1999 AHSME Problems/Problem 23
  <math> \mathrm{(A) \ } \frac {15}2\sqrt{3} \qquad \mathrm{(B) \ }9\sqrt{3} \qquad \mathrm{(C) label("$A$",O,SW);
  6 KB (949 words) - 17:48, 19 March 2020
• 1973 USAMO Problems
  Two points <math>P</math> and <math>Q</math> lie in the interior of a regular tetrahedron <math>ABCD</math>. Prove that angle <math>PAQ<60^o</mat Three distinct vertices are chosen at random from the vertices of a given regular polygon of <math>(2n+1)</math> sides. If all such choices are
  2 KB (273 words) - 18:53, 3 July 2013
• 1972 USAMO Problems/Problem 1
  ...or and least common multiple, respectively, of the positive integers <math>a,b,\ldots, g</math>. For example, <math>(3,6,18)=3</math> and <math>[6,15]=3 <center><math>\frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}</math>.</center>
  5 KB (1,018 words) - 11:14, 6 October 2023
• 1974 USAMO Problems/Problem 2
  Prove that if <math>a</math>, <math>b</math>, and <math>c</math> are positive real numbers, then <center><math>a^ab^bc^c\ge (abc)^{(a+b+c)/3}</math></center>
  9 KB (1,726 words) - 14:01, 1 February 2024
• 1974 USAMO Problems/Problem 4
  A father, mother and son hold a family tournament, playing a two person board game with no ties. The tournament rules are: ...ath>L</math> denote a game the father loses, and let <math>x</math> denote a game the father does not participate in. Then, with the knowledge that the
  6 KB (1,037 words) - 23:42, 11 May 2018
• 1975 USAMO Problems/Problem 2
  ...denote four points in space and <math>AB</math> the distance between <math>A</math> and <math>B</math>, and so on. Show that pair A=(2,4), B=(0,0), C=(4,0), D=(4,2);
  6 KB (1,096 words) - 23:07, 26 August 2017
• 1960 IMO Problems/Problem 7
  An isosceles trapezoid with bases <math>a</math> and <math>c</math> and altitude <math>h</math> is given. a) On the axis of symmetry of this trapezoid, find all points <math>P</math>
  2 KB (410 words) - 15:25, 23 March 2020
• 1962 IMO Problems/Problem 4
  First, note that we can write the left hand side as a cubic function of <math>\cos^2 x</math>. So there are at most <math>3</math ...c{(2k+1)\pi}4,\, \frac{(6k+1)\pi}6,\, \frac{(6k+5)\pi}6 \quad \forall k\in Z</math>
  1 KB (187 words) - 00:37, 4 September 2023
• 2001 AMC 10 Problems
  <math>\textbf{(A) } 4 \qquad\textbf{(B) } 6 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 10 \qq A number <math>x</math> is <math>2</math> more than the product of its recipr
  14 KB (1,983 words) - 16:25, 2 June 2022
• 2000 AMC 10 Problems
  <math>\textbf{(A)}\ 23 \qquad\textbf{(B)}\ 55 \qquad\textbf{(C)}\ 99 \qquad\textbf{(D)}\ 111 <math>\textbf{(A)}\ {2000}^{2001} \qquad \textbf{(B)}\ {4000}^{2000} \qquad \textbf{(C)}\ {2
  14 KB (2,035 words) - 21:57, 2 May 2024
• 1998 AHSME Problems/Problem 30
  <math> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \ The number <math>a_n</math> is a product of <math>10</math> consecutive integers. Out of these, <math>5</mat
  3 KB (486 words) - 10:37, 11 August 2020
• 1999 AHSME Problems/Problem 18
  <math> \mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) ...< 1</math> we have <math>-\infty < \log x < 0</math>, and the logarithm is a strictly increasing function on this interval.
  2 KB (270 words) - 14:35, 5 July 2013
• 1985 AJHSME Problems/Problem 11
  ...d as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled <math>\text{X}</math label("Z",(1.5,-1.7),N);
  2 KB (279 words) - 09:25, 10 January 2023
• 1986 AJHSME Problems
  <math>\text{(A)}\ \frac{366}{31\times 24} \qquad \text{(B)}\ \frac{366\times 31}{24}\qquad <math>\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{2}{5} \qquad \text{(C)}\ 1 \qquad \
  14 KB (2,054 words) - 15:41, 8 August 2020
• 1986 AJHSME Problems/Problem 23
  pair A=(-2,0), O=origin, C=(2,0); path X=Arc(O,2,0,180), Y=Arc((-1,0),1,180,0), Z=Arc((1,0),1,180,0), M=X..Y..Z..cycle;
  2 KB (277 words) - 21:32, 3 July 2013
• 2004 AMC 12B Problems/Problem 10
  ...be the radius of the larger circle that contains <math>Z</math>. Let <math>a=XZ</math>, <math>d=YZ</math>, and <math>e=XY</math>. What is the area of th pair Y=(0,r2), Z=(0,r1), X=intersectionpoint( Z--(Z+(10,0)), outer );
  2 KB (340 words) - 14:35, 23 April 2023
• 2004 AMC 12B Problems/Problem 14
  pair C=(0,0), A=(0,5), B=(12,0), M=(0,4), N=(4,0); pair J=intersectionpoint(A--B, M--(M+rotate(90)*(B-A)) );
  7 KB (1,083 words) - 22:41, 23 November 2020
• 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 ...<math> n - z + 1</math> instead of <math> n - z</math> until we get <math> z = n - S</math>, what makes the sum become smaller. So we have<cmath> T \ge
  6 KB (1,192 words) - 14:14, 29 January 2021
• 2009 AMC 10A Problems/Problem 19
  ...e <math>A</math> as it rolls once around the circumference of circle <math>A</math>. The two circles have the same points of tangency at the beginning a \mathrm{(A)}\ 4\
  2 KB (276 words) - 09:57, 8 June 2021
• Fractal
  A fractal is defined as a figure that does not become simpler under any level of magnification. Probably the most well-known example of a fractal, the Mandelbrot set is the set of all points <math>c</math> in the
  915 bytes (163 words) - 18:28, 8 January 2012
• 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
• 2007 AMC 10B Problems/Problem 25
  ...ny pairs of positive integers <math>(a,b)</math> are there such that <math>a</math> and <math>b</math> have no common factors greater than <math>1</math <cmath>\frac{a}{b} + \frac{14b}{9a}</cmath>
  8 KB (1,425 words) - 23:05, 21 June 2023
• 2009 AMC 12B 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>\textbf{(A) } 1\qquad\textbf{(B) } 2\qquad\textbf{(C) } 3\qquad\textbf{(D) } 4\qquad\t
  13 KB (2,030 words) - 03:04, 5 September 2021
• 2009 AMC 12B Problems/Problem 22
  ...<math>k > 1</math>, respectively. How many such parallelograms are there? (A lattice point is any point whose coordinates are both integers.)<!-- don't <math>\textbf{(A)}\ 49\qquad \textbf{(B)}\ 720\qquad \textbf{(C)}\ 784\qquad \textbf{(D)}\ 2
  6 KB (1,071 words) - 22:25, 9 October 2021
• 2009 AMC 12B Problems/Problem 23
  A region <math>S</math> in the complex plane is defined by ...</math>. What is the probability that <math>\left(\frac34 + \frac34i\right)z</math> is also in <math>S</math>?
  2 KB (422 words) - 13:25, 20 January 2020
• 2009 AMC 12B Problems/Problem 10
  ...henever it is supposed to display a <math>1</math>, it mistakenly displays a <math>9</math>. For example, when it is 1:16 PM the clock incorrectly shows <math>\mathrm{(A)}\ \frac 12\qquad
  2 KB (372 words) - 17:36, 28 June 2021
• 2009 AMC 10B Problems/Problem 24
  \text{(A) } 100 label("$Z$",rotate(20)*(R,0),ENE);
  3 KB (546 words) - 04:26, 16 January 2023
• 2009 AIME I Problems
  ...s <math>3</math> distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest g ...s a complex number <math>z</math> with imaginary part <math>164</math> and a positive integer <math>n</math> such that
  7 KB (1,152 words) - 02:24, 23 July 2021
• 2009 AIME I Problems/Problem 2
  ...s a complex number <math>z</math> with imaginary part <math>164</math> and a positive integer <math>n</math> such that <cmath>\frac {z}{z + n} = 4i.</cmath>
  5 KB (850 words) - 12:53, 23 December 2023
• 2009 AIME I Problems/Problem 14
  ...total of <math>350</math> <math>a</math>s, we know that <math>w + x + y + z = 350</math>. We also know that <math>w + 2x + 3y + 4z = 513</math> and <ma <cmath>w = 350 - x - y - z</cmath>
  2 KB (326 words) - 14:12, 16 August 2020
• 2009 AIME I Problems/Problem 9
  ...the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were <math>1, 1, 1, 1, 3, 3, 3</math>. Fin ...u are supposed to find the number of all possible tuples of prices, <math>(A, B, C)</math>, that could have been on that day.]
  2 KB (286 words) - 02:11, 31 August 2022
• 2009 AIME I Problems/Problem 11
  ...y = 2009</math>. Find the number of such distinct triangles whose area is a positive integer. ...</math> is an integer. So, <math>b = 58\sqrt2 \times z</math>, where <math>z</math> is also an integer.
  8 KB (1,319 words) - 15:01, 16 August 2020
• 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 ...s very complex but we can take a guess here. Assume that <math>x</math> is a rational number
  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
• Liouville's Theorem (complex analysis)
  In [[complex analysis]], '''Liouville's Theorem''' states that a [[Picard's Little Theorem]] is a stronger result.
  2 KB (412 words) - 20:30, 16 January 2024
• 2009 AIME II Problems/Problem 2
  ...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_ <cmath> a^{(\log_3 7)^2} + b^{(\log_7 11)^2} + c^{(\log_{11} 25)^2}. </cmath>
  3 KB (396 words) - 15:20, 10 May 2024
• 2009 AIME II Problems/Problem 9
  ...')</math> such that <math>2x + 3y' < 1006</math> we have exactly one <math>z</math> such that the equality holds. Hence we need to count the pairs <math For a fixed <math>y'</math>, <math>x</math> can be at most <math>\left\lfloor \df
  5 KB (845 words) - 15:45, 28 December 2020
• Artinian
  We say that a [[ring]] or [[module]] is '''Artinian''' diverse a category as Noetherian rings. However, this
  2 KB (263 words) - 21:00, 19 April 2012
• 1984 IMO Problems/Problem 1
  ...th>y</math>, <math>z</math> be nonnegative real numbers with <math>x + y + z = 1</math>. Show that Note that this inequality is symmetric with x,y and z.
  2 KB (321 words) - 10:51, 29 June 2022
• 2004 USAMO Problems/Problem 2
  ...th> are integers whose greatest common divisor is 1. Let <math>S</math> be a set of integers with the following properties: (a) For <math>i = 1, \dots, n</math>, <math>a_i \in S</math>.
  3 KB (508 words) - 14:23, 17 July 2014
• KGS math club/solution 5 1
  ...nce P has nonzero degree, n>0. The Taylor polynomial of P about the point a is <math>F(x)=P(a)+P'(a)(x-a)+\dotsb +\dfrac{P^{(n)}(a)(x-a)^n}{n!}=P(x)</math>.
  2 KB (305 words) - 19:24, 21 December 2015
• 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
• 1964 IMO Problems/Problem 2
  Suppose <math>a, b, c</math> are the sides of a triangle. Prove that <cmath>a^2(b+c-a)+b^2(c+a-b)+c^2(a+b-c)\le{3abc}.</cmath>
  2 KB (387 words) - 23:59, 10 December 2022
• 1964 IMO Problems/Problem 3
  ...Find the sum of the areas of all four inscribed circles (in terms of <math>a,b,c</math>). ...tively and let the points of contact of MH,FG,DE with the in circle be X,Y,Z respectively. Then perimeter of BHM = BH+HX+XM+MB=BH+HP+MR+BM=BP+BQ=2(s-b)
  2 KB (414 words) - 12:48, 29 January 2021
• 1965 IMO Problems/Problem 3
  ...</math> whose edges <math>AB</math> and <math>CD</math> have lengths <math>a</math> and <math>b</math> respectively. The distance between the skew lines Let the plane meet AD at X, BD at Y, BC at Z and AC at W. Take plane parallel to BCD through WX and let it meet AB in P.
  2 KB (309 words) - 10:52, 30 September 2022
• 2009 IMO Problems/Problem 3
  Suppose that <math>s_1,s_2,s_3,\ldots</math> is a strictly increasing sequence of positive integers such that the subsequence put S(3) = b, S(2) = a, S(1) = k
  1 KB (184 words) - 01:16, 19 November 2023
• Simple module
  ...module]] that is simple as a [[group with operators]]&mdash;that is, it is a module with no [[submodule]]s other than itself and the zero module, and it If <math>R</math> is a [[commutative ring]], then every simple module over <math>R</math> is
  1 KB (253 words) - 11:02, 29 September 2012
• Zero ring
  A '''zero ring''' is a ring with one element, 0 (equal to 1), with the ...<math>R</math> is a ring in which <math>0=1</math>, then <math>R</math> is a trivial
  1 KB (221 words) - 18:44, 31 January 2022
• 2010 AIME II Problems/Problem 5
  ...468</math>. Find <math>\sqrt {(\log_{10}x)^2 + (\log_{10}y)^2 + (\log_{10}z)^2}</math>. ...log_{10}z)= 468</math> by using the fact that <math>\log_{10}ab = \log_{10}a + \log_{10}b</math>.
  2 KB (392 words) - 23:49, 5 February 2022
• Constructible number
  ...can be constructed with a [[straight edge]] and [[compass]] starting with a segment of length <math>1</math>. ...h> is constructible as a real number if and only if it is constructible as a complex number, i.e., our two definitions coincide in this case.
  8 KB (1,305 words) - 08:39, 21 August 2009
• Irreducible element
  In [[ring theory]] a element <math>r</math> of a [[ring]] <math>R</math> is said to be '''irreducible''' if: * <math>r</math> is not a [[unit (ring theory)|unit]].
  1 KB (284 words) - 20:42, 23 August 2009
• Unique factorization domain
  ...ization domain if for any nonzero element <math>r\in R</math> which is not a [[unit (ring theory)|unit]]: ...of <math>\{1,2\ldots,n\}</math> such that for each <math>k</math> there is a unit <math>u_k</math> such that <math>p_k = u_kq_{\sigma(k)}</math>.
  6 KB (1,217 words) - 23:05, 23 August 2009
• Fundamental group
  ...[0,1]\times[0,1]\to X</math> with <math>F(a,0)=f(a)</math>, <math>F(a,1)=g(a)</math>, and <math>F(0,b)=F(1,b)=x_0</math>). Now let <math>\pi_1(X,x_0)=\O Now define a [[binary operation]] <math>\cdot</math> (called ''concatenation'') on <math
  8 KB (1,518 words) - 20:11, 23 January 2017
• Schonemann's criterion
  If <math>f(x)\in \mathbb{Z}[x]</math> * <math>g(x), h(x)\in \mathbb{Z}[x]</math>, a prime <math>p</math> and an integer <math>n</math> such that <math>f(x)=g(x
  1 KB (279 words) - 17:03, 31 October 2023
• 2010 AMC 12A Problems/Problem 17
  <math>\textbf{(A)}\ \frac{4\sqrt{3}}{3} \qquad \textbf{(B)} \frac{10}{3} \qquad \textbf{(C)} ...t at <math>Y</math>, and <math>DE</math> and <math>FA</math> meet at <math>Z</math>, we find that hexagon <math>ABCDEF</math> is formed by taking equila
  4 KB (690 words) - 10:13, 14 October 2022
• 2010 AMC 12A Problems/Problem 22
  <math>\textbf{(A)}\ 49 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ ...f(x)</math> occurs where the absolute value of the sum of the slopes is at a minimum <math>\ge 0</math>, since it is easy to see that the value will be
  6 KB (994 words) - 23:22, 24 July 2023
• 2010 AMC 12B Problems
  <math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ A big <math>L</math> is formed as shown. What is its area?
  12 KB (1,845 words) - 13:00, 19 February 2020
• Closed set
  ...set''' is a set which contains all of its [[limit point]]s. Equivalently, a [[set]] in some [[topological space]] (including, for example, any [[metric ...the union of any two closed sets (or any finite number of closed sets) is a closed set. Note, however, that the union of an arbitrary collection of clo
  1 KB (232 words) - 18:33, 1 March 2010
• Derivative/Definition
  ...he rate of change between two points on the curve easily since we can draw a line through them. We can move the second point closer to the first one to find a more accurate value of the derivative. Thus, taking the [[limit]] as <math
  5 KB (994 words) - 16:21, 3 January 2012
• Mock AIME 1 2010 Problems
  Let <math>AB</math> be a line segment of length <math>20 \sqrt{2}</math>, and let <math>S</math> be ...ment in which every player plays every other player exactly once. There is a round robin tournament with 2010 people. In each match, the winner scores o
  7 KB (1,297 words) - 01:29, 25 November 2016
• Z-test
  A z-test is a significance test in statistics that one conducts when the population stand
  488 bytes (76 words) - 22:52, 13 February 2016
• 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 ...math>x^y = y^x</math>. The quantity <math>x + y</math> can be expressed as a rational number <math>\frac {r}{s}</math>, where <math>r</math> and <math>s
  8 KB (1,243 words) - 21:58, 10 August 2020
• 2010 AIME I Problems/Problem 7
  ...er of minimally intersecting ordered triples of sets for which each set is a subset of <math>\{1,2,3,4,5,6,7\}</math>. Find the remainder when <math>N</ ...0</math> ways to choose values of <math>x</math>, <math>y</math> and <math>z</math>. There are <math>4</math> unpicked numbers, and each number can eith
  2 KB (255 words) - 17:03, 9 August 2018
• 2010 AIME I Problems/Problem 9
  ...</math>, <math>z^3 - xyz = 20</math>. The greatest possible value of <math>a^3 + b^3 + c^3</math> can be written in the form <math>\frac {m}{n}</math>, ...adratic]] to get <math>p = - 4, - \frac {15}{7}</math>. To maximize <math>a^3 + b^3 + c^3</math> choose <math>p = - \frac {15}{7}</math> and so the sum
  3 KB (483 words) - 07:35, 4 November 2022
• 2010 AIME I Problems/Problem 13
  [[Rectangle]] <math>ABCD</math> and a [[semicircle]] with diameter <math>AB</math> are coplanar and have nonoverl label("$A$",(-16.43287,-9.3374),NE/2);
  10 KB (1,418 words) - 23:05, 20 October 2021
• 2010 AIME I Problems/Problem 15
  ...th>, <math>BC = 13</math>, and <math>AC = 15</math>, let <math>M</math> be a point on <math>\overline{AC}</math> such that the [[incircle]]s of <math>\t label("$A$",(-1.04408,-0.60958),NE);
  14 KB (2,210 words) - 13:14, 11 January 2024
• Mock AIME 2 2010 Problems
  ...where <math>a, b</math> are relatively prime positive integers. Find <math>a+b</math>. ...ath> and <math>b</math> are relatively prime positive integers, find <math>a + b</math>.
  7 KB (1,150 words) - 09:10, 8 October 2018
• 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
• 2007 IMO Problems
  (a) Prove that, for any real numbers <math>x_1\le x_2\le \cdots\le x_n</math>, ...h> such that <math>ABCD</math> is a parallelogram and <math>BCED</math> is a cyclic quadrilateral.
  3 KB (505 words) - 09:24, 10 September 2020
• 2010 AIME II Problems
  A point <math>P</math> is chosen at random in the interior of a unit square <math>S</math>. Let <math>d(P)</math> denote the distance from ...where <math>a</math> and <math>b</math> are integers satisfying <math>1\le a < b \le 20</math>. Find the greatest positive integer <math>n</math> such t
  8 KB (1,246 words) - 21:58, 10 August 2020
• 1994 AJHSME Problems
  <math>\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{1}{4} \qquad \text{(C)}\ \dfrac{3 <math>\text{(A)}\ 4\dfrac{1}{2} \qquad \text{(B)}\ 6.4 \qquad \text{(C)}\ 9 \qquad \text{(
  16 KB (2,292 words) - 13:36, 19 February 2020
• 2010 AIME II Problems/Problem 7
  ...w+9i</math>, and <math>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 Since <math>a,b,c\in{R}</math>, the imaginary part of <math>a,b,c</math> must be <math>0</math>.
  4 KB (735 words) - 02:41, 27 June 2022
• 2010 AIME II Problems/Problem 10
  ...ath>r</math> and <math>s</math> are indistinguishable. In one case, <math>(a,r,s) = (2010,1,1)</math>, we have <math>r=s</math>. The other <math>80</mat ...s. For the <math>40</math> cases where <math>|r|\neq |s|</math>, there are a total of four possibilities, For the case <math>|r|=|s|=1</math>, there are
  6 KB (972 words) - 16:01, 16 September 2023
• LaTeX:LaTeX on AoPS
  LaTeX uses a special "math mode" to display mathematics. There are two types of this "ma Besides displaying in-line vs. displaying centered and on a new line, the two modes render differently in other ways. Note that <code>$
  8 KB (1,356 words) - 22:35, 26 June 2020
• 1992 USAMO Problems
  Find, as a function of <math>\, n, \,</math> the sum of the digits of ...integer <math>n \le 1500</math>, there is a subset <math>S</math> of <math>A</math> for which <math>\sigma(S) = n</math>. What is the smallest possible
  2 KB (310 words) - 20:54, 3 July 2013
• 1992 USAMO Problems/Problem 5
  ...<math>\, P(z) \,</math> divides the polynomial <math>\left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991} \right)^2 - a_{1992}</math>. ...> of the polynomial <math>P(z)</math> are distinct, the polynomial <math>P(z)</math> divides the polynomial
  5 KB (825 words) - 02:21, 2 September 2016
• 1993 USAMO Problems/Problem 3
  ...right)^n\right)\le\left(\frac{1}{2}\right)^n</math> for <math>n\in \mathbb{Z}, n\ge0</math> ...ht)^{n+1}<x\le\left(\frac{1}{2}\right)^n\le1</math> and <math>n\in \mathbb{Z}</math>, <math>f(x)\le\left(\frac{1}{2}\right)^n</math>.
  3 KB (523 words) - 17:03, 13 August 2023
• 1993 USAMO Problems/Problem 4
  ...ve integers. Define the sequence <math>(f_n)</math> by putting <math>f_1 = a</math>, value as a function of <math>a</math> and <math>b</math>.
  3 KB (593 words) - 07:56, 19 July 2016
• 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 USAMO Problems/Problem 4
  Let <math>ABC</math> be a triangle with <math>\angle A = 90^{\circ}</math>. Points <math>D</math> pair A = origin; ldot(A, "$A$", plain.SW);
  7 KB (1,230 words) - 19:47, 31 January 2024
• 2010 USAMO Problems/Problem 1
  Let <math>AXYZB</math> be a convex pentagon inscribed in a semicircle of diameter Since <math>XY</math> is a chord of the circle with diameter <math>AB</math>,
  13 KB (2,178 words) - 14:14, 11 September 2021
• 2010 AIME II Problems/Problem 15
  ...ACE = \angle ECB</math>. Let <math>P</math> be the point, other than <math>A</math>, of intersection of the circumcircles of <math>\triangle AMN</math> pair A,B,C,D,E,M,N,P,Q;
  9 KB (1,523 words) - 15:24, 21 November 2023
• 1995 AJHSME Problems
  <math>\text{(A)}\ 4\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 4 <math>\text{(A)}\ 8 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 14 \qquad \text{(D)}\ 16 \qqu
  14 KB (2,096 words) - 18:29, 2 January 2023
• 2011 AMC 10B Problems/Problem 25
  ...BC</math>, and <math>AC,</math> respectively, then <math>T_{n+1}</math> is a triangle with side lengths <math>AD, BE,</math> and <math>CF,</math> if it <math> \textbf{(A)}\ \frac{1509}{8} \qquad\textbf{(B)}\ \frac{1509}{32} \qquad\textbf{(C)}\ \
  4 KB (620 words) - 03:15, 3 November 2022
• 2010 AMC 12B Problems/Problem 15
  ...> are there exactly two distinct elements in the set <math>\{i^x, (1+i)^y, z\}</math>, where <math>i=\sqrt{-1}</math>? <math>\textbf{(A)}\ 149 \qquad \textbf{(B)}\ 205 \qquad \textbf{(C)}\ 215 \qquad \textbf{(D)
  2 KB (360 words) - 17:29, 26 May 2023
• 2010 IMO Problems/Problem 1
  ...t\lfloor a\right\rfloor </math> is greatest integer not greater than <math>a.</math> ...{R}</math> or <math>f(x)=a, \ \forall x \in \mathbb{R}</math>, where <math>a \in [1,2)</math>.
  7 KB (1,128 words) - 12:54, 3 April 2024
• KGS math club/solution 10 1
  where <math> z = x^2 + y^2 + x y </math> So we want to find a pair (x, y) such that
  1 KB (186 words) - 15:11, 11 August 2010
• KGS math club/solutio prince
  Consider a board with at least two rows and columns and label the squares in the upper ...e it only with a move to the south. That is, the short path A -> B -> C is a forced part of the circuit.
  1 KB (242 words) - 15:46, 9 January 2017
• 1969 IMO Problems
  ...ers <math>a</math> with the following property: the number <math>z = n^4 + a</math> is not prime for any natural number <math>n</math>. Let <math>a_1, a_2, \cdots, a_n</math> be real constants, <math>x</math> a real variable, and
  3 KB (428 words) - 13:34, 29 January 2021
• 1971 IMO Problems
  ...; let <math>P_i</math> be the polyhedron obtained from <math>P_1</math> by a translation that moves vertex <math>A_1</math> to <math>A_i (i = 2, 3, \cdo ...math> distinct from <math>A</math> and <math>B</math>; similarly, <math>Y, Z, T</math> are interior points of edges <math>BC, CD, DA</math>, respectivel
  3 KB (495 words) - 13:52, 29 January 2021
• 2006 AMC 8 Problems
  <math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qqu <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 19\qquad
  16 KB (2,215 words) - 19:18, 10 April 2024
• 1998 AJHSME Problems
  <math>\text{(A)}\ \dfrac{6}{x} \qquad \text{(B)}\ \dfrac{6}{x+1} \qquad \text{(C)}\ \dfrac If <math>\begin{tabular}{r|l}a&b \\ \hline c&d\end{tabular} = \text{a}\cdot \text{d} - \text{b}\cdot \text{c}</math>, what is the value of <math>
  14 KB (1,920 words) - 19:31, 31 January 2024
• 2010 AMC 10B Problems/Problem 16
  A square of side length <math>1</math> and a circle of radius <math>\dfrac{\sqrt{3}}{3}</math> share the same center. Wh <math>\textbf{(A)}\ \dfrac{\pi}{3}-1 \qquad \textbf{(B)}\ \dfrac{2\pi}{9}-\dfrac{\sqrt{3}}{3
  3 KB (544 words) - 20:54, 24 March 2024
• 1969 IMO Problems/Problem 1
  ...ers <math>a</math> with the following property: the number <math>z = n^4 + a</math> is not prime for any natural number <math>n</math>. ...at <math>a = 4k^4</math> for some <math>a</math>. We will prove that <math>a</math> satisfies the property outlined above.
  1 KB (170 words) - 18:13, 7 December 2022
• 2011 AMC 12A Problems
  {{AMC12 Problems|year=2011|ab=A}} A cell phone plan costs <math>\$20</math> dollars each month, plus <math>5</m
  13 KB (1,994 words) - 13:52, 3 July 2021
• 2011 AMC 12A Problems/Problem 20
  Let <math>f(x)=ax^2+bx+c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are integers. Suppose that <math \textbf{(A)}\ 1 \qquad
  4 KB (724 words) - 17:36, 9 January 2024
• 2011 AMC 12A Problems/Problem 23
  ...ath> and <math>g(g(z))=z</math> for all <math>z</math> for which <math>g(g(z))</math> is defined. What is the difference between the largest and smalles \textbf{(A)}\ 0 \qquad
  3 KB (519 words) - 15:49, 5 November 2023
• 1958 AHSME Problems
  <math> \textbf{(A)}\ 5\qquad If <math> \frac {1}{x} - \frac {1}{y} = \frac {1}{z}</math>, then <math> z</math> equals:
  25 KB (3,872 words) - 14:21, 20 February 2020
• 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
• 2021 AIME II Problems/Problem 4
  ...ath> is a root of <math>x^3 + cx^2 + d.</math> These two polynomials share a complex root <math>m + \sqrt{n} \cdot i,</math> where <math>m</math> and <m ...</math> and the roots of <math>x^3+cx^2+d</math> are <math>-21,z,\overline{z}.</math>
  8 KB (1,361 words) - 16:35, 12 January 2024
• 2021 AIME II Problems/Problem 6
  ...\{1,2,3,4,5\}</math> that satisfy <cmath>|A| \cdot |B| = |A \cap B| \cdot |A \cup B|</cmath> ...een the other way around. Now we need to subtract the overlaps where <math>A=B</math>, and this case has <math>2^{5}=32</math> ways that could happen. I
  8 KB (1,364 words) - 01:02, 29 January 2024
• KGS math club/solution 11 2
  Given n, k, and a k-member subset K of the n-member set N, we form the (n-k)-member subset it ...cyclically order its members using them. Remove from N a member of K with a non-member of K that immediately follows it; and repeat until there are no
  1 KB (188 words) - 18:08, 21 February 2011
• 2011 AMC 12B Problems
  <math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \ <math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\
  13 KB (1,978 words) - 16:28, 12 July 2020
• 2011 AMC 12B Problems/Problem 24
  ...ons in the complex plane whose vertices are precisely the zeros of <math>P(z)</math>? <math>\textbf{(A)}\ 4\sqrt{3} + 4 \qquad \textbf{(B)}\ 8\sqrt{2} \qquad \textbf{(C)}\ 3\sqr
  2 KB (344 words) - 18:12, 22 August 2021
• 2001 AMC 10 Problems/Problem 10
  .../math>, <math>xz = 48</math>, and <math>yz = 72</math>, then <math>x + y + z</math> is <math>\textbf{(A) }18\qquad\textbf{(B) }19\qquad\textbf{(C) }20\qquad\textbf{(D) }22\qquad\t
  2 KB (257 words) - 23:42, 9 April 2024
• 2001 AMC 10 Problems/Problem 22
  ...<math> x </math>, <math> y </math>, and <math> z </math>. Find <math> y + z </math>. label("$z$",(1.5,0.5));
  5 KB (721 words) - 16:44, 9 August 2022
• 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 6
  ...<math>a + b + c</math> is an integer. The minimum possible value of <math>a</math> can be written in the form <math>\frac{p}{q}</math>, where <math>p</ .../math>, the equation of the parabola can be expressed in the form <cmath>y=a\left(x-\frac{1}{4}\right)^2-\frac{9}{8}.</cmath>
  4 KB (661 words) - 01:18, 11 December 2023
• 2002 AMC 8 Problems
  A [[circle]] and two distinct [[Line|lines]] are drawn on a sheet of paper. What is the largest possible number of points of intersecti <math>\text {(A)}\ 2 \qquad \text {(B)}\ 3 \qquad {(C)}\ 4 \qquad {(D)}\ 5 \qquad {(E)}\ 6<
  15 KB (2,102 words) - 09:58, 5 May 2024
• Bézout's Identity
  ...ath> such that <math>x\alpha+y\beta=g</math>. In other words, there exists a linear combination of <math>x</math> and <math>y</math> equal to <math>g</m ...d in this form, i.e. <math>g = \min\{x\alpha+y\beta|\alpha,\beta\in\mathbb Z, x\alpha+y\beta > 0\}</math>.
  4 KB (768 words) - 16:50, 6 September 2023
• 2011 AIME I Problems/Problem 13
  ...at heights 10, 11, and 12 above the plane. The distance from vertex <math>A</math> to the plane can be expressed as <math> \frac{r-\sqrt{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 I Problems/Problem 14
  Let <math>A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8</math> be a regular octagon. Let <math>M_1</math>, <math>M_3</math>, <math>M_5</math>, pair A,B,C,D,E,F,G,H,M,N,O,O2,P,W,X,Y,Z;
  8 KB (1,344 words) - 18:39, 9 February 2023
• 2011 AIME I Problems/Problem 8
  ...math>n</math> are relatively prime positive integers and <math>m</math> is a positive integer that is not divisible by the square of any prime. Find <ma pair[] A, B, C, U, V, W, X, Y, Z;
  6 KB (1,077 words) - 21:47, 12 April 2022
• 2011 AIME II Problems/Problem 6
  ..., d)</math> as interesting if <math>1 \le a<b<c<d \le 10</math>, and <math>a+d>b+c</math>. How many interesting ordered quadruples are there? ...th>. Let <math>e = 11</math>, then <math>(a, b-a, c-b, d-c, e-d)</math> is a partition of 11 into 5 positive integers or equivalently:
  9 KB (1,535 words) - 01:28, 16 January 2023
• 2011 AIME II Problems/Problem 8
  ...h>z_1,z_2,z_3,\dots,z_{12}</math> be the 12 zeroes of the polynomial <math>z^{12}-2^{36}</math>. For each <math>j</math>, let <math>w_j</math> be one of ...ch the real part of <math>iz</math> is greater than the real part of <math>z</math>. Now, the sum of the real parts of the blue dots is easily seen to b
  5 KB (805 words) - 18:46, 27 January 2024
• 2011 AIME II Problems
  Gary purchased a large beverage, but only drank <math>m/n</math> of it, where <math>m</math> The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer
  8 KB (1,301 words) - 08:43, 11 October 2020
• 2011 AIME II Problems/Problem 10
  A [[circle]] with center <math>O</math> has radius 25. [[Chord]] <math>\overl 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
• 2011 USAJMO Problems/Problem 3
  For a point <math>P = (a, a^2)</math> in the coordinate plane, let <math>\ell(P)</math> denote the line ...th>a_n^3-\frac{3ka_n^2}2-\frac{3a_n}4+\frac k8=0.</cmath>Note that this is a cubic, and the roots are <math>a_1,a_2</math> and <math>a_3</math> which ar
  15 KB (2,593 words) - 13:37, 29 January 2021
• 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 = ...}+\vec{v}+\vec{w}=\vec{0}</math>, we can arrange the three vectors to form a triangle, so the triangle with sides of lengths <math>2p \cos \gamma</math>
  11 KB (1,925 words) - 12:07, 31 August 2023
• 2011 AMC 10B Problems
  <math> \textbf{(A)}\ -1\qquad\textbf{(B)}\ \frac{5}{36}\qquad\textbf{(C)}\ \frac{7}{12}\qquad <math> \textbf{(A)}\ 80 \qquad\textbf{(B)}\ 82 \qquad\textbf{(C)}\ 85 \qquad\textbf{(D)}\ 90
  13 KB (2,090 words) - 18:05, 7 January 2021
• 2011 AMC 10B Problems/Problem 15
  ...(x + y) @ (x + z)}</cmath> <cmath>\text{III. x @ (y @ z) = (x @ y) @ (x @ z)}</cmath> <math> \textbf{(A)}\ \text{I only} \qquad\textbf{(B)}\ \text{II only} \qquad\textbf{(C)}\ \te
  2 KB (287 words) - 01:48, 26 June 2022
• 2001 AMC 8 Problems/Problem 23
  ...equilateral triangle, and points <math>X</math>, <math>Y</math> and <math>Z</math> are midpoints of its sides. How many noncongruent triangles can be pair SS,R,T,X,Y,Z;
  3 KB (549 words) - 16:35, 23 July 2023
• 2007 AMC 10B Problems
  <math>\textbf{(A) } 678 \qquad\textbf{(B) } 768 \qquad\textbf{(C) } 786 \qquad\textbf{(D) } Define the operation <math>\star</math> by <math>a \star b = (a+b)b.</math> What is <math>(3 \star 5) - (5 \star 3)?</math>
  15 KB (2,297 words) - 12:57, 19 February 2020
• 2007 AMC 10B Problems/Problem 9
  ...at the letter <math>A</math> is one place to the right of the letter <math>Z</math>). The second time this same letter appears in the given message, it <cmath>\text{"Lee's sis is a Mississippi miss, Chriss!"?}</cmath>
  2 KB (280 words) - 13:06, 4 June 2021
• Asymptote: 3D graphics
  ...work when three is active. For example: To fill a surface, one must define a surface and draw that, instead of using <tt>[[asymptote: Filling|filldraw]] * and surface (a surface bounded by a path(3))
  4 KB (568 words) - 10:54, 8 October 2021
• 1958 AHSME Problems/Problem 2
  If <math> \frac {1}{x} - \frac {1}{y} = \frac {1}{z}</math>, then <math> z</math> equals: <math> \textbf{(A)}\ y - x\qquad \textbf{(B)}\ x - y\qquad \textbf{(C)}\ \frac {y - x}{xy}\qq
  661 bytes (115 words) - 06:10, 3 October 2014
• 2011 AMC 10B Problems/Problem 21
  Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of t <math> \textbf{(A)}\ 16 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 62
  8 KB (1,303 words) - 20:29, 5 September 2022
• Olympiad books
  Here is a list of '''Olympiad Books''' that have Olympiad-level problems used to trai *''Inequalities: A Mathematical Olympiad Approach'' - '''Radmila Bulajich Manfrino, Jose Anton
  17 KB (2,261 words) - 00:30, 22 April 2024
• 2007 AMC 10B Problems/Problem 21
  ...on <math>\overline{AC}, W</math> on <math>\overline{AB},</math> and <math>Z</math> on <math>\overline{BC}.</math> What is the side length of the square <math>\textbf{(A) } \frac{3}{2} \qquad\textbf{(B) } \frac{60}{37} \qquad\textbf{(C) } \frac{
  2 KB (335 words) - 16:17, 18 February 2024
• 2011 AMC 12B Problems/Problem 13
  Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of t <math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\
  3 KB (459 words) - 09:55, 8 August 2023
• 2011 AMC 10B Problems/Problem 24
  ...th>\frac{1}{2} < m < a</math>. What is the maximum possible value of <math>a</math>? <math> \textbf{(A)}\ \frac{51}{101} \qquad\textbf{(B)}\ \frac{50}{99} \qquad\textbf{(C)}\ \fr
  4 KB (692 words) - 20:45, 19 March 2023
• 1998 AHSME Problems/Problem 1
  ...7, x = 5, y = 4, z = 8</math>. In rectangle E, <math>w = 9, x = 2, y = 7, z = 0</math>. These five rectangles are placed, without rotating or reflectin ...,(0,6),(-1,0));label("$x$",(1.5,7),(0,1));label("$y$",(3,6),(1,0));label("$z$",(1.5,5),(0,-1));
  2 KB (346 words) - 14:28, 5 July 2013
• 1998 AJHSME Problems/Problem 10
  ...that order. If <math>\dfrac{\text{W}}{\text{X}} - \dfrac{\text{Y}}{\text{Z}}=1</math>, then the sum of <math>\text{W}</math> and <math>\text{Y}</math> <math>\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad
  1 KB (212 words) - 00:29, 5 July 2013
• 2012 AMC 12B Problems/Problem 23
  ...integers, <math>0\le d\le c\le b\le a\le 4</math>, and the polynomial has a zero <math>z_0</math> with <math>|z_0|=1.</math> What is the sum of all val <math>\textbf{(A)}\ 84\qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108\q
  11 KB (1,979 words) - 17:25, 6 September 2021
• 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
• 1997 USAMO Problems/Problem 5
  Prove that, for all positive real numbers <math>a, b, c,</math> <math>\frac{1}{a^3+b^3+abc}+\frac{1}{b^3+c^3+abc}+\frac{1}{a^3+c^3+abc} \le \frac{1}{abc}</math>.
  3 KB (581 words) - 13:55, 25 March 2024
• 2011 IMO Problems/Problem 3
  Let <math>f: \mathbb R \to \mathbb R</math> be a real-valued function defined on the set of real numbers that satisfies <cma <math>P(x,z-x)\implies f(z)\leq (z-x)f(x)+f(f(x)).</math>
  1 KB (234 words) - 01:20, 19 November 2023
• 2011 IMO Problems/Problem 5
  Let <math>f</math> be a function from the set of integers to the set of positive integers. Suppose Let <math>f</math> be a function from the set of integers to the set of positive integers. Suppose
  4 KB (825 words) - 01:21, 19 November 2023
• 2011 IMO Problems/Problem 4
  ...h of the <math>n</math> weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we ...k}}</math>, where <math>a_k \in \mathbb{Z}</math>, and vice versa, forming a <math>1:1</math> relation. With this in mind, we use recursion upon the las
  4 KB (632 words) - 01:21, 19 November 2023
• 1966 IMO Problems/Problem 6
  ...hs of sides <math>BC</math>, <math>CA</math>, and <math>AB</math> be <math>a</math>, <math>b</math>, and <math>c</math>, respectively. Let <math>BK=d</m ...rac{bc}{4}</math>. Similarly, <math>d(c-f)>\frac{ac}{4}</math> and <math>e(a-d)>\frac{ab}{4}</math>. Multiplying these three inequalities together yield
  2 KB (397 words) - 00:37, 17 May 2015
• 2002 AMC 8 Problems/Problem 16
  ...riangles are constructed on the sides of a 3-4-5 right triangle, as shown. A capital letter represents the area of each triangle. Which one of the follo label(scale(0.8)*"$Z$", (0, 3), S);
  2 KB (336 words) - 20:00, 15 April 2023
• 2002 AMC 8 Problems/Problem 20
  The area of triangle <math>XYZ</math> is 8 square inches. Points <math>A</math> and <math>B</math> are midpoints of congruent segments <math>\overli label(scale(0.8)*"$Z$", (10,0), E);
  9 KB (1,488 words) - 23:09, 28 December 2023
• 1997 AJHSME Problems/Problem 23
  <math>\text{(A)}\ 7 \qquad \text{(B)}\ 25 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 48 \qqu ...f their squares if the five digits are distinct and non-zero. If there is a zero, it will be forced to the left by rule #2.
  2 KB (297 words) - 15:15, 6 May 2023
• 1997 AHSME Problems
  If <math>\texttt{a}</math> and <math>\texttt{b}</math> are digits for which <math> \begin{array}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{array} </m
  17 KB (2,590 words) - 13:38, 19 February 2020
• 1997 AHSME Problems/Problem 3
  If <math>x</math>, <math>y</math>, and <math>z</math> are real numbers such that <center><math>(x-3)^2 + (y-4)^2 + (z-5)^2 = 0</math>,</center>
  796 bytes (123 words) - 14:12, 5 July 2013
• 1997 AHSME Problems/Problem 25
  ...22</math> and <math>M</math> and <math>N</math> are the midpoints of <math>A^{\prime} C^{\prime}</math> and <math>B^{\prime}D^{\prime}</math>, respectiv <math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\t
  2 KB (307 words) - 22:13, 7 December 2023
• 1994 AJHSME Problems/Problem 20
  Let <math>W,X,Y</math> and <math>Z</math> be four different digits selected from the set ...</math> is to be as small as possible, then <math>\dfrac{W}{X} + \dfrac{Y}{Z}</math> must equal
  1 KB (206 words) - 11:56, 27 June 2023
• 1994 AJHSME Problems/Problem 23
  If <math>X</math>, <math>Y</math> and <math>Z</math> are different digits, then the largest possible <math>3-</math>digit <math>\text{(A)}\ XXY \qquad \text{(B)}\ XYZ \qquad \text{(C)}\ YYX \qquad \text{(D)}\ YYZ
  910 bytes (151 words) - 00:14, 5 July 2013
• 1979 USAMO Problems/Problem 2
  ...cts the angle <math>ACB</math> in the spherical triangle <math>ABC</math> (a spherical triangle has great circle arcs as sides). Since <math>N</math> is the north pole, we define the Earth with a sphere of radius one in space with <math>N=(0,0,1)</math> and sphere center
  6 KB (1,013 words) - 22:09, 21 November 2023
• 1979 USAMO Problems/Problem 3
  ..._1, a_2, \ldots, a_n</math> is an arbitrary sequence of positive integers. A member of the sequence is picked at .... Then a third value, <math>c</math>. Show that the probability that <math>a + b +c</math> is divisible by <math>3</math> is at least <math>\frac14</mat
  2 KB (307 words) - 14:14, 7 June 2021
• 1996 AHSME Problems/Problem 9
  pair A=(6,0), B=(2,0), C=(2,4), D=(6,4), P=B+1*dir(-65); draw(A--P--B--C--D--cycle);
  3 KB (425 words) - 14:07, 5 July 2013
• 1996 AHSME Problems/Problem 23
  The sum of the lengths of the twelve edges of a rectangular box is <math>140</math>, and <math> \text{(A)}\ 776\qquad\text{(B)}\ 784\qquad\text{(C)}\ 798\qquad\text{(D)}\ 800\qquad
  1 KB (220 words) - 16:24, 28 June 2021
• 1996 AHSME Problems/Problem 25
  <math> \text{(A)}\ 72\qquad\text{(B)}\ 73\qquad\text{(C)}\ 74\qquad\text{(D)}\ 75\qquad\tex The first equation is a [[circle]], so we find its center and [[radius]] by [[completing the square
  9 KB (1,441 words) - 17:51, 22 October 2023
• 1996 AHSME Problems/Problem 27
  <math> \text{(A)}\ 7\qquad\text{(B)}\ 9\qquad\text{(C)}\ 11\qquad\text{(D)}\ 13\qquad\text{ <cmath>x^2 + y^2 + \left(z - \frac{21}{2}\right)^2 \le 36</cmath>
  7 KB (1,046 words) - 11:42, 30 September 2023
• 1996 AHSME Problems/Problem 28
  ...math> 4\times 4\times 3 </math> rectangular parallelepiped, vertices <math>A</math>, <math>B</math>, and <math>C</math> are adjacent to vertex <math>D</ <math>A</math>, <math>B</math>, and <math>C</math> is closest to
  3 KB (545 words) - 10:21, 16 September 2022
• 2010 PUMaC Problems
  == Division A == Assume that <math>f(a+b) = f(a) + f(b) + ab</math>, and that <math>f(75) - f(51) = 1230</math>. Find <mat
  22 KB (3,694 words) - 23:58, 3 June 2022
• 2016 AMC 10A Problems/Problem 20
  ...ains exactly <math>1001</math> terms that include all four variables <math>a, b,c,</math> and <math>d</math>, each to some positive power. What is <math <math>\textbf{(A) }9 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 17
  4 KB (630 words) - 15:13, 8 October 2023
• 1995 AJHSME Problems/Problem 24
  pair A,B,C,D,P,Q,W,X,Y,Z; A = (0,0); B = (12,0); C = (20,6); D = (8,6);
  1 KB (185 words) - 13:33, 21 April 2024
• 1980 USAMO Problems
  ...t are <math>B</math> and <math>b</math>. The third object balances against a weight <math>C</math>, when placed in the left pan. What is its true weight Find the maximum possible number of three term arithmetic progressions in a monotone sequence of <math>n</math> distinct reals.
  2 KB (276 words) - 13:41, 26 December 2015
• 1982 USAMO Problems
  In a party with <math>1982</math> persons, among any group of four there is at l Let <math>S_r=x^r+y^r+z^r</math> with <math>x,y,z</math> real. It is known that if <math>S_1=0</math>,
  2 KB (348 words) - 22:36, 19 March 2020
• 2003 AMC 8 Problems/Problem 25
  ...gle ABC</math> is folded over side <math>\overline{BC}</math>, point <math>A</math> coincides with <math>O</math>, the center of square <math>WXYZ</math pair Z=origin, W=(0,10), X=(10,10), Y=(10,0), O=(5,5), B=(-4,8), C=(-4,2), A=(-13,5);
  2 KB (387 words) - 19:58, 15 April 2023
• 2009 PUMaC Problems
  '''Algebra A''' Find the number of functions <math>f:\mathbb{Z}\mapsto\mathbb{Z}</math> for which <math>f(h+k)+f(hk)=f(h)f(k)+1</math>, for all integers <m
  25 KB (4,154 words) - 16:27, 2 September 2011

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