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• Math books
  * Intermediate is recommended for students who can expect to pass the AMC 10/12. Before adding any books to this page, please review the [[AoPSWiki:Linking books]] page.
  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\\
  1 KB (193 words) - 21:13, 18 May 2021
• Mathematics
  ...the development of trustworthy theories and equations which are important to people in most modern technical discplines such as engineering and economic ...aticians then use various methods of formal [[proof]] to extend the axioms to come up with surprising and elegant results. Such methods include [[inducti
  6 KB (902 words) - 12:53, 3 September 2019
• Trivial Inequality
  ...ath>. Since both sides of the inequality are nonnegative, it is equivalent to <math>x+y \ge 2\sqrt{xy}</math>, and thus we have <cmath> \frac{x+y}{2} \ge Another application will be to minimize/maximize quadratics. For example,
  3 KB (560 words) - 22:51, 13 January 2024
• Factoring
  ...or [[polynomial]]) as a product of different terms. This often allows one to find information about an expression that was not otherwise obvious. Using the formula for the sum of a [[geometric sequence]], it's easy to derive the general formula for difference of powers:
  3 KB (532 words) - 22:00, 13 January 2024
• Relatively prime
  Two [[positive]] [[integer]]s <math>m</math> and <math>n</math> are said to be '''relatively prime''' or '''coprime''' if they share no [[common diviso ...ive integers less than any given positive integer that is relatively prime to that number.
  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]]. ...being real. But for polynomials of degree 4 or higher it can be difficult to use it.
  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
  ...a hard time following the rest of this article). This theorem is credited to [[Pierre de Fermat]]. ...}</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>.
  3 KB (542 words) - 17:45, 21 March 2023
• 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 Algebra can be used to solve different types of equations, but algebra is also many other things
  3 KB (369 words) - 21:18, 18 June 2021
• Number theory/Advanced
  ...ry]] and [[Galois theory]]. Algebraic methods are particularly well-suited to studying properties of individual prime numbers. From an algebraic perspect ...e techniques of [[analysis]] and [[calculus]] are particularly well-suited to studying large-scale properties of prime numbers. The most famous problem i
  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
  ...r]]) of a given [[circle]] to its [[diameter]]. It is approximately equal to 3.141592653. The number pi is one of the most important [[constant]]s in a ...proximates <math>\frac{\pi}{4}</math>. This can simply be multiplied by 4 to approximate <math>\pi</math>.
  8 KB (1,469 words) - 21:11, 16 September 2022
• 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
  4 KB (691 words) - 18:38, 19 September 2021
• 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
• Complex plane
  ...point two units to the right and one unit down from the origin corresponds to the complex number <math>2 - i</math>.
  892 bytes (134 words) - 16:52, 3 September 2017
• Complex number
  The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. ...solutions of <math> x^2 = -1 </math> but we can now find ''all'' solutions to ''every'' polynomial. (See the [[Fundamental Theorem of Algebra]] for more
  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 Once one gets used to the notation, it is almost always preferred to write <math>re^{i\theta}</math> rather than <math>r\mathrm{cis }(\theta)</m
  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 examine the [[complex numb ...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
  ...at the angle between this line and <math>\overline{AB}</math> is congruent to the angle between this line and <math>\overline{AC}</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>. ...marvelous demonstration of this proposition that this margin is too narrow to contain.''"
  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 ...antine equation has infinitely many solutions, [[parametric form]] is used to express the relation between the variables of the equation.
  9 KB (1,434 words) - 13:10, 20 February 2024
• Area
  ...]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
  ...and larger <math>p</math> and <math>q</math>. So, the reasonable question to ask here is how well can we approximate <math>x</math> by rationals with no ...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
  2 KB (271 words) - 22:06, 12 April 2022
• 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 ...is [[monic]]. Then <math>1/P(z)</math> is an [[entire]] function; we wish to show that it is bounded. It is clearly bounded when <math>n=0</math>; we n
  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
  of primes less than or equal to <math>x</math>. In other words, it states <cmath> \lim_{x\to \infty} \frac{\pi(x) \log x}{x} = 1 . </cmath>
  10 KB (1,729 words) - 19:52, 21 October 2023
• Infinite
  ...h>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 years, recurring competitio ** [https://artofproblemsolving.com/community/c5h2987334 Or(z)IME]
  8 KB (906 words) - 17:30, 26 April 2024
• Commutative property
  ...a [[binary operation]]) is said to have the '''commutative property''' or to ''be commutative'' if the order of its arguments does not affect the value. ...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
• 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>. ...ing a wide range of number-theoretic problems, including finding solutions to [[Diophantine equation|Diophantine equations]], testing whether certain lar
  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}} ...and sodas cost <math>2</math> dollars each. How many dollars will it cost to purchase <math>5</math> sandwiches and <math>8</math> sodas?
  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}} ...ed to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
  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 17
  ...\overline{BE}</math> as <math>A</math>. Segment <math>AF</math> is tangent to the circle, and <math>AF=\sqrt{9+5\sqrt{2}}</math>. What is <math>r/s</math 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}) ...ath>1</math>, <math>x</math> will be in the right domain, so we don't need to worry about the domain of <math>x</math>.
  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
  Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered \mathrm {(A)}\ \sqrt{2} \qquad
  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}} ...odas cost <math> \textdollar 2 </math> each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas?
  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
  ...h> by <math>(\sqrt[16]{5} - 1) </math>, the denominator will [[telescope]] to <math>\sqrt[1]{5} - 1 = 4 </math>, so Like Solution <math>2</math>, let <math>z=\sqrt[16]{5}</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> ...math>9</math> of them may be <math>A: (-1)</math> otherwise we return back to <math>x_{10} = 1</math>, contradicting our assumption that <math>20</math>
  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
  ...ath> are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when <math> n </math> is ...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
  ...d <math> D. </math> A circle contained within the trapezoid is [[tangent]] to all four of these circles. Its radius is <math> \frac{-k+m\sqrt{n}}p, </mat 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 ...math> because we’re choosing 2 1s to go in 40 digit slots. This is equal to 780; we have found <math>q</math>, our denominator.
  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 ...by <math>8</math>). Thus, the minimal value is when each fraction is equal to <math>8</math>, and the solution is <math>8(11 + 13 + 27) = \boxed{408}</ma
  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 ...ngle. Find the square of the distance (in centimeters) from <math>B</math> to the center of the circle.
  7 KB (1,104 words) - 12:53, 6 July 2022
• 1984 AIME Problems
  ...ABC</math> such that when lines are drawn through <math>P</math> parallel to the sides of <math>\triangle ABC</math>, the resulting smaller triangles <m 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
  ...ation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there? ...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 ...r is 1 or 2 and whose denominator is an integer. The closest such fraction to <math>r</math> is <math>\frac 27.</math> What is the number of possible va
  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
  ...three-digit arrangement that reads the same left-to-right as it does right-to-left) is <math>m/n</math>, where <math>m</math> and <math>n</math> are rela ...s shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as <math>\frac{1}{2}\left(\sqrt{p}-q\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</ ...rithm]]ic notation doesn't tell us much, so we'll first convert everything to the equivalent exponential forms.
  4 KB (642 words) - 03:14, 17 August 2022
• 1983 AIME Problems/Problem 5
  One way to solve this problem is by [[substitution]]. We have Hence observe that we can write <math>w=x+y</math> and <math>z=xy</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, ...f(y)</math>, take the [[derivative]] of <math>f(y)</math> and set it equal to zero.
  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
  ...th> such that when [[line]]s are drawn through <math>P</math> [[parallel]] to the sides of <math>\triangle ABC</math>, the resulting smaller [[triangle]] 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 ...r <math>m</math> (the ninth degree [[roots of unity]]). Now we simply need to find the root within the desired range that satisfies our original equation
  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)= ...e roots, and anything congruent to <math>4\pmod{10}</math> are also roots. To see that these may be the only integer roots, observe that the function
  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
• 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>? ...any integer <math>n</math> (same reasoning as above). So now we only need to test every 10 numbers; and our answer will be 100 times the number of integ
  12 KB (1,859 words) - 18:16, 28 March 2022
• 1986 AIME Problems/Problem 9
  ...then drawn, and [[segment]]s are drawn through <math>P</math> [[parallel]] to the sides of the [[triangle]]. If these three segments are of an equal leng pair C=(0,0),A=(510,0),B=IP(circle(C,450),circle(A,425));
  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
  ...1}T_2</math>. Additionally, the area of triangle <math>ABC</math> is equal to both <math>T_1 + T_2 + 441</math> and <math>T_3 + T_4 + T_5 + 440.</math> ...sqrt {440} = \sqrt {AC^2 + BC^2}</math>. We now need <math>(AC)(BC)</math> to find the value of <math>AC + BC</math>, because <math>AC^2 + BC^2 + 2(AC)(B
  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>. ...e must have <math>p = 4</math> and at least one of <math>m, j</math> equal to 3. This gives 7 possible triples <math>(j, m, p)</math>: <math>(0, 3, 4),
  3 KB (547 words) - 22:54, 4 April 2016

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