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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

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