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Fibonacci sequence
...s the sum of the two preceding it. The first few terms are <math>1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...</math>. ...ivial example of a [[linear recursion]] with constant coefficients. There is also an explicit formula [[#Binet's formula|below]].

6 KB (957 words) - 23:49, 7 March 2024
Muirhead's Inequality
The inequality is easier to understand given an example. Since the sequence <math>(5,1)</mat ...lympiad solution; one should use an application of AM-GM instead. Thus, it is suggested that Muirhead be used only to verify that an inequality ''can'' b

8 KB (1,346 words) - 12:53, 8 October 2023
Cardinality
..., 3\}, \{1, 2, 3\}\}</math> is 3, and the cardinality of the [[empty set]] is 0. ...In the above example, the cardinality of <math>\{3, 4\}</math> is <math>|\{3, 4\}| = 2</math>. Sometimes, the notations <math>n(A)</math> and <math>\# (

2 KB (263 words) - 00:54, 17 November 2019
Fundamental Theorem of Calculus
This section is for people who know what [[integral]]s are but don't know the Fundamental T * Evaluate: <math>\int_2^5 x^3 dx</math> and <math>\int_{.2}^{.4} \cos(x) dx</math>. (The next few questi

11 KB (2,082 words) - 15:23, 2 January 2022
Polygon
A '''polygon''' is a closed [[planar figure]] consisting of straight [[line segment]]s. There A polygon can be [[regular polygon| regular]] or irregular. A polygon is regular if all sides are the same length and all angles are [[congruent]].

2 KB (372 words) - 19:04, 30 May 2015
2011 AMC 10B Problems/Problem 22
...opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? ...27} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad\textbf{(E)}\ \frac{\sqrt{3}}{9}</math>

4 KB (691 words) - 18:38, 19 September 2021
2011 AMC 10B Problems/Problem 4
...the costs equally, LeRoy must give Bernardo half of the difference, which is <math>\boxed{\textbf{(C) } \;\frac{B-A}{2}}</math> .... Quickly, we realize the only way they could pay the same amount of money is if they both pay 45 dollars. This means LeRoy must give Bernardo <math>50 -

1 KB (249 words) - 13:05, 24 January 2024
Vector
...x). Most generally, but also most abstractly, a vector is any object which is an element of a given vector space. ...es, <math>(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]].

7 KB (1,265 words) - 13:22, 14 July 2021
Logarithm
...come down to never having to deal with massive numbers. ex. :<cmath>((((((3^5)^6)^7)^8)^9)^{10})^{11}=\underbrace{1177\ldots 1}_{\text{793549 digits}}< left to right parenthesized exponentiation) is only 7 digits before the decimal point. Comparing the logs of the numbers t

4 KB (680 words) - 12:54, 16 October 2023
Law of Cosines
The '''Law of Cosines''' is a theorem which relates the side-[[length]]s and [[angle]]s of a [[triangle In the case that one of the angles has measure <math>90^\circ</math> (is a [[right angle]]), the corresponding statement reduces to the [[Pythagorea

6 KB (1,003 words) - 00:02, 20 May 2024
Schur's Inequality
...Inequality''' is an [[inequality]] that holds for [[positive number]]s. It is named for Issai Schur. ...ath>a=b=c</math> or when two of <math>a,b,c</math> are equal and the third is <math>{0}</math>.

2 KB (398 words) - 16:57, 29 December 2021
Trigonometric identities
...<math>(\cos (x), \sin (x))</math> is defined to be on the unit circle, it is a distance one away from the origin. Then by the distance formula, <math>\s * <math>\sin 3x = 3\sin x-4\sin^3 x</math>

8 KB (1,397 words) - 21:55, 20 January 2024
Irrational number
An '''irrational number''' is a [[real number]] that cannot be expressed as the [[ratio]] of two [[intege ...entury <math>B.C</math>. The Pythagoreans lived by the doctrine that ''all is number'', or that all things could be explained by relationships between nu

3 KB (368 words) - 19:26, 6 June 2015
Complex number
...ive]], so this equation has no solutions in the real numbers. However, it is possible to define a number, <math> i </math>, such that <math> i = \sqrt{- ...= \sqrt{-1} </math> is the [[imaginary unit]]. The set of complex numbers is denoted by <math>\mathbb{C}</math>. The set of complex numbers contains th

5 KB (860 words) - 15:36, 10 December 2023
Angle bisector
...math> such that the angle between this line and <math>\overline{AB}</math> is congruent to the angle between this line and <math>\overline{AC}</math>: D=(3,4);

3 KB (575 words) - 15:27, 19 March 2023
Without loss of generality
...ten abbreviated to WLOG, is a frequently used expression in math. The term is used to indicate that the following proof emphasizes on a particular case, If you use WLOG in a proof and the statement is not necessarily true, points will get marked off. For example, you can't sa

2 KB (280 words) - 15:30, 22 February 2024
Law of Sines
The '''Law of Sines''' is a useful identity in a [[triangle]], which, along with the [[law of cosines ...math>, <math>c</math> opposite to <math>C</math>, and where <math>R</math> is the circumradius:

4 KB (658 words) - 16:19, 28 April 2024
Geometric sequence
...hat the ratio between any two consecutive terms is constant. This constant is called the '''common ratio''' of the sequence. ...mon ratio <math>-1/2</math>; however, <math>1, 3, 9, -27</math> and <math>-3, 1, 5, 9, \ldots</math> are not geometric sequences, as the ratio between c

4 KB (644 words) - 12:55, 7 March 2022
Arithmetic sequence
...he difference between any two consecutive terms is constant. This constant is called the '''common difference''' of the sequence. ...ence with common difference <math>1</math> and <math>99, 91, 83, 75</math> is an arithmetic sequence with common difference <math>-8</math>; however, <ma

4 KB (736 words) - 02:00, 7 March 2024
Fermat's Last Theorem
...ting that for positive [[integers]] <math>a,b,c,n</math> with <math>n \geq 3</math>, there are no solutions to the equation <math>a^n + b^n = c^n</math> ...vered a truly 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
...piece of length <math>k_i</math> from the end of leg <math>L_i \; (i = 1,2,3,4)</math> and still have a stable table? ...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
2000 AMC 12 Problems/Problem 1
...ger]]s such that the product <math>I \cdot M \cdot O = 2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? ...process on <math>2001</math> to get <math>667 * 3 * 1</math> as our <math>3</math> factors.

2 KB (276 words) - 05:25, 9 December 2023
Diophantine equation
A '''Diophantine equation''' is an [[equation]] relating [[integer]] (or sometimes [[natural number]] or [[ ...a Diophantine 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
Fraction
A '''fraction''' is the [[ratio]] of two [[number]]s. Most commonly, we consider [[rational nu ...numerator is the same as the denominator such as <math>\frac{3}{3}</math> is always equal to <math>1</math>.

3 KB (432 words) - 19:34, 11 June 2020
Functional equation
A '''functional equation''', roughly speaking, is an equation in which some of the unknowns to be solved for are [[function]] ...he '''inverse function'''.) Often the inverse of a function <math>f</math> is denoted by <math>f^{-1}</math>.

2 KB (361 words) - 14:40, 24 August 2021
Exponential form
...(yes, again!) rewrite <math>z</math> as <math>z=re^{i\theta}</math>, which is the general exponential form of a complex number. D=(1/2,sqrt(3)/2);

1 KB (238 words) - 22:51, 20 February 2022
Floor function
...ion is the same as "dropping everything after the decimal point," but this is ''not'' true for negative values. *<math>\lfloor 3.14 \rfloor = 3</math>

3 KB (508 words) - 21:05, 26 February 2024
Pascal's triangle
'''Pascal's triangle''' is a triangle which contains the values from the [[binomial expansion]]; its v ...n</math>, the sum of the values on row <math>n</math> of Pascal's Triangle is <math>2^n</math>.

5 KB (838 words) - 17:20, 3 January 2023
Exponentiation
.../math>, where <math>b</math> is the exponent (or power) and <math>a</math> is the [[base]]. ...ed if a equation has [[parentheses]] or the first one performed when there is no parentheses.

5 KB (803 words) - 16:25, 10 August 2020
Trigonometry
...gths and angles of triangles through the '''trigonometric functions'''. It is a fundamental branch of mathematics, and its discovery paved the way toward In contest math, trigonometry is an integral subfield of both [[geometry]] and [[algebra]]. Many essential r

8 KB (1,217 words) - 20:15, 7 September 2023
Mathematical Olympiad Summer Program
...especially the [[International Mathematical Olympiad]]. While the program is free to participants, invitations are limited to the top finishers on the [ ...d train the US team for the [[International Mathematical Olympiad]]. This is done at the start of MOP via a [[team selection test]] (TST). The results

6 KB (936 words) - 10:37, 27 November 2023
Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
...-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality''' (EM-AM-GM-HM), is an [[inequality]] of the [[root-mean power]], [[arithmetic mean]], [[geomet ...where <math>n_1>1,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math>n</math> is the root mean power.

5 KB (912 words) - 20:06, 14 March 2023
Harmonic series
Generally, a '''harmonic series''' is a [[series]] whose terms involve the [[reciprocal]]s of the [[positive inte The the most basic harmonic series is the infinite sum

2 KB (334 words) - 20:52, 13 March 2022
Goldbach Conjecture
...proven [[conjecture]] stating that every [[even integer]] greater than two is the sum of two [[prime number]]s. The conjecture has been tested up to 400 Goldbach's conjecture is one of the oldest unsolved problems in [[number theory]] and in all of math

7 KB (1,201 words) - 16:59, 19 February 2024
Twin Prime Conjecture
The '''Twin Prime Conjecture''' is a [[conjecture]] (i.e., not a [[theorem]]) that states that there are [[inf One possible strategy to prove the infinitude of twin primes is an idea adopted from the proof of [[Dirichlet's Theorem]]. If one can show

2 KB (308 words) - 02:27, 1 May 2024
Quadratic residues
...</math> if there is some integer <math>n</math> so that <math>n^2-a</math> is [[divisibility | divisible]] by <math>m</math>. ...modulo\ }\ p, \\ -1 & \mathrm{if }\ p\nmid a\ \mathrm{ and }\ a\ \mathrm{\ is\ a\ quadratic\ nonresidue\ modulo\ }\ p. \end{cases}</math>

5 KB (778 words) - 13:10, 29 November 2017
Power of a Point Theorem
The '''Power of a Point Theorem''' is a relationship that holds between the lengths of the [[line segment]]s form # One of the lines is [[tangent line|tangent]] to the circle while the other is a [[secant line|secant]] (middle figure). In this case, we have <math> AB^2

5 KB (827 words) - 17:30, 21 February 2024
Collatz Problem
...th>\{n,f(n),f(f(n)),f(f(f(n))),\ldots\}</math> contains 1. This conjecture is still open. Some people have described it as the easiest unsolved problem i ...6m+4\over 2}=3m+2</cmath> we can then observe that; only if <math>m</math> is even will another division by 2 be possible.

1 KB (231 words) - 19:45, 24 February 2020
Algebraic number
...[27]{19}}{\sqrt[3]{4}+\sqrt[7]{97}}</math>. A number that is not algebraic is called a [[transcendental number]], such as <math>e</math> or <math>\pi</ma ...mbers is large, there are only [[countable|countably]] many of them. That is, the algebraic numbers have the same [[cardinality]] as the [[natural numbe

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International Mathematical Olympiad
The '''International Mathematical Olympiad''' is the pinnacle of all high school [[mathematics competition]]s and the oldest ...eakdown=Problem 1/4: 6.5 Problem 2/5: 7.5-8 Problem 3/6: 9.5 Problem SL1-2: 5.5-7 Problem SL3-4: 7-8 <

3 KB (490 words) - 03:32, 23 July 2023
Prime Number Theorem
The '''Prime Number Theorem''' (PNT) is one of the most celebrated results in [[analytic number theory]]. Indeed, it is

10 KB (1,729 words) - 19:52, 21 October 2023
Infinite
...n]] <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 that y

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Isosceles trapezoid
...isosceles trapezoid''' is a geometric figure that lies in a [[plane]]. It is a specific type of [[trapezoid]] in which the legs have the same length. I * the segment joining the midpoints of the bases is perpendicular to the bases

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Mock AMC
A '''Mock AMC''' is a contest intended to mimic an actual [[AMC]] (American Mathematics Competi ...popular in the months leading up to the actual [[AMC]] competition. There is no guarantee that community members will make Mock AMCs in any given year,

51 KB (6,175 words) - 20:58, 6 December 2023
Mock AIME
A '''Mock AIME''' is a contest that is intended to mimic the [[AIME]] competition. (In more recent years, recurrin ...Y2QwOTc3NWZiYjY0LnBkZg==&rn=TWlsZG9yZiBNb2NrIEFJTUUucGRm Mildorf Mock AIME 3]

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Permutation
A '''permutation''' of a [[set]] of <math>r</math> objects is any rearrangement (linear ordering) of the <math>r</math> objects. There a ...of [[infinite]] sets. In this case, a permutation of a set <math>S</math> is simply a [[bijection]] between <math>S</math> and itself.

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Riemann zeta function
The '''Riemann zeta function''' is a function very important in [[number theory]]. In particular, the [[Riemann Hypothesis]] is a conjecture

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Mock USAMO
...hen a mock USAMO is run on [[AoPS]]/[[MathLinks]], a very wide time window is often allowed to take the mock USAMO. ** [http://www.artofproblemsolving.com/blog/2712 Mock USAMO 3 2006]

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Modular arithmetic/Introduction
...of arithmetic that involves only [[integers]]. This goal of this article is to explain the basics of modular arithmetic while presenting a progression <math>1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, \ldots </math>

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Perfect square
An [[integer]] <math>n</math> is said to be a '''perfect square''' if there is an integer <math>m</math> so that <math>m^2=n</math>. The first few perfect ...of the first <math>n</math> square numbers (starting with <math>1</math>) is <math>\frac{n(n+1)(2n+1)}{6}</math>

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Uncountable
...or [[countably infinite]]. The most common example of an uncountable set is the set of [[real number]]s <math>\mathbb{R}</math>. == Proof that <math>\mathbb{R}</math> is uncountable ==

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Modular arithmetic/Intermediate
...quiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. ...<math>n</math>, the relation <math>a \equiv b</math> (mod <math>n</math>) is an [[equivalence relation]] on the set of integers. This relation gives ri

14 KB (2,317 words) - 19:01, 29 October 2021
Right triangle
A '''right triangle''' is any [[triangle]] with an angle of 90 degrees (that is, a [[right angle]]). A = (0, 3);

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2006 AIME I Problems
..., \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <math> \overline{CD}, AB=18, BC=21, </math> and <math> CD Let set <math> \mathcal{A} </math> be a 90-element subset of <math> \{1,2,3,\ldots,100\}, </math> and let <math> S </math> be the sum of the elements o

7 KB (1,173 words) - 03:31, 4 January 2023
2006 AIME I Problems/Problem 15
...n that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math> for all integers <math> k\ge 1, </math> find the minimum possible Suppose <math>b_{i} = \frac {x_{i}}3</math>.

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2006 AIME I Problems/Problem 14
...notation <math> \lfloor x\rfloor </math> denotes the greatest integer that is less than or equal to <math> x. </math>) currentprojection = perspective(1,-10,3.3);

6 KB (980 words) - 21:45, 31 March 2020
2006 AIME I Problems/Problem 13
...atest integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect square]]. ...h>k</math> is odd, then <math>n+1</math> is even, hence <math>k+n-1</math> is odd, and <math>S_n</math> cannot be a perfect square. Hence <math>k</math>

10 KB (1,702 words) - 00:45, 16 November 2023
2006 AIME I Problems/Problem 11
...h> k </math> for each [[integer]] <math> k, 1 \le k \le 8. </math> A tower is to be built using all 8 cubes according to the rules: ...s than can be constructed. What is the [[remainder]] when <math> T </math> is divided by 1000?

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2006 AIME I Problems/Problem 10
...d <math> c </math> are positive integers whose [[greatest common divisor]] is 1. Find <math> a^2+b^2+c^2. </math> int[] array={3,3,2};

4 KB (731 words) - 17:59, 4 January 2022
2006 AIME I Problems/Problem 9
The [[sequence]] <math> a_1, a_2, \ldots </math> is [[geometric sequence|geometric]] with <math> a_1=a </math> and common [[rat ...<math>a, r</math> [[positive integer]]s. <math>a^{12}r^{66}=8^{2006} = (2^3)^{2006} = (2^6)^{1003}</math> so <math>a^{2}r^{11}=2^{1003}</math>.

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2006 AIME I Problems/Problem 8
...he area of rhombus <math> \mathcal{T}</math>. Given that <math> K </math> is a [[positive integer]], find the number of possible values for <math> K</ma ...(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6);

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2006 AIME I Problems/Problem 7
...[[region]] <math> C </math> to the area of shaded region <math> B </math> is 11/5. Find the ratio of shaded region <math> D </math> to the area of shade pair A=(1/3,4), B=A+7.5*dir(-17), C=A+7*dir(10);

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2006 AIME I Problems/Problem 5
...rt{10}+144\sqrt{15}+2006}</math> can be written as <math> a\sqrt{2}+b\sqrt{3}+c\sqrt{5}, </math> where <math> a, b, </math> and <math> c </math> are [[p <cmath> a\sqrt{2}+b\sqrt{3}+c\sqrt{5} = \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}</cmath>

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2006 AIME I Problems/Problem 4
...h> 1!2!3!4!\cdots99!100!. </math> Find the remainder when <math> N </math> is divided by <math>1000</math>. ...ng into our given expression. Since there are clearly more 2s than 5s, it is sufficient to count the number of 5s.

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2006 AIME I Problems/Problem 3
...]] such that when its leftmost [[digit]] is deleted, the resulting integer is <math>\frac{1}{29}</math> of the original integer. ...7.</math> But <math>a_n</math> is a nonzero digit, so the only possibility is <math>a_n = 7.</math> This gives <cmath>7 \cdot 10^n = 28N_0</cmath> or <cm

4 KB (622 words) - 03:53, 10 December 2022
2006 AIME I Problems/Problem 2
...<math> \mathcal{A} </math> be a 90-[[element]] [[subset]] of <math> \{1,2,3,\ldots,100\}, </math> and let <math> S </math> be the sum of the elements o ...995</math> are possible values of S, so the number of possible values of S is <math>4995-4095+1=901</math>.

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2022 AIME I Problems/Problem 1
...x)</math> and <math>Q(x)</math> cancel, we conclude that <math>R(x)</math> is a linear polynomial. so the slope of <math>R(x)</math> is <math>\frac{106-108}{20-16}=-\frac12.</math>

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2021 JMC 10
What is the value of <cmath>\dfrac{20}{2\cdot1} - \dfrac{2+0}{2/1}?</cmath> <math>\textbf{(A) } 3 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 9 \qquad\te

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2006 AMC 12B Problems
What is <math>( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}</math>? .../math>, define <math>x\spadesuit y = (x + y)(x - y)</math>. What is <math>3\spadesuit(4\spadesuit 5)</math>?

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2006 AMC 12A Problems
Sandwiches at Joe's Fast Food cost <math>3</math> dollars each and sodas cost <math>2</math> dollars each. How many do Define <math>x\otimes y=x^3-y</math>. What is <math>h\otimes (h\otimes h)</math>?

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2005 AMC 12A Problems
...\%</math> of <math>x</math> and <math>20 \%</math> of <math>y</math>. What is <math>x - y</math>? ...+ 7 = 3</math> and <math>bx - 10 = - 2</math> have the same solution. What is the value of <math>b</math>?

13 KB (1,971 words) - 13:03, 19 February 2020
2004 AMC 12A Problems
Alicia earns <math> 20</math> dollars per hour, of which <math>1.45\%</math> is deducted to pay local taxes. How many cents per hour of Alicia's wages are ...ct answer is worth <math>0</math> points, and each problem left unanswered is worth <math>2.5</math> points. If Charlyn leaves <math>8</math> of the <mat

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2003 AMC 12A Problems
What is the difference between the sum of the first <math>2003</math> even counting ...es and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League?

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2002 AMC 12A Problems
<math>(2x+3)(x-4)+(2x+3)(x-6)=0 </math> ...the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the

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2000 AMC 12 Problems
...ntegers such that the product <math>I \cdot M \cdot O = 2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? == Problem 3 ==

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2001 AMC 12 Problems
The sum of two numbers is <math>S</math>. Suppose <math>3</math> is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two

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2002 AMC 12B Problems
...ne 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 disti What is the value of

10 KB (1,547 words) - 04:20, 9 October 2022
2003 AMC 12B Problems
Which of the following is the same as <cmath>\frac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}?</cmath>

13 KB (1,987 words) - 18:53, 10 December 2022
2004 AMC 12B Problems
<math>(\mathrm {A}) 3\qquad (\mathrm {B}) 6 \qquad (\mathrm {C}) 9 \qquad (\mathrm {D}) 12 \qquad ...>d</math> are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result?

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2005 AMC 12B Problems
...property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>? == Problem 3 ==

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2006 AMC 12B Problems/Problem 2
.../math>, define <math>x\spadesuit y = (x + y)(x - y)</math>. What is <math>3\spadesuit(4\spadesuit 5)</math>? <math>3\spadesuit -9=-72 \Rightarrow \text{(A)}</math>

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2006 AMC 12B Problems/Problem 4
...</math>. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the <math>20.00</math> that she will receive i The total price of the items is <math>(8-.01)+(5-.01)+(3-.01)+(2-.01)+(1-.01)=19-.05=18.95</math>

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2006 AMC 12B Problems/Problem 5
...hile Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up ...Bob is catching up to John is <math>5-3=2</math> miles per hour. Since Bob is one mile behind John, it will take <math>\frac{1}{2} \Rightarrow \text{(A)}

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2006 AMC 12B Problems/Problem 7
The first child can be seated in <math>3</math> spaces. <math>3 \times 2 \times 2 = 12 \Rightarrow \text{(B)}</math>

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2006 AMC 12B Problems/Problem 8
...>y = \frac 14x + b</math> intersect at the point <math>(1,2)</math>. What is <math>a + b</math>?<!-- don't remove the following tag, for PoTW on the Wik <math>\frac{3}{4}(x+y)=a+b</math>

1 KB (235 words) - 00:46, 6 January 2022
2006 AMC 12B Problems/Problem 9
...ces for <math>a</math> and <math>b</math>. Thus there are altogether <math>3+10+21=\boxed{34}</math> such integers. If it was 2, there is 1 possibility for the hundreds digit, 3 for the ones digit.

3 KB (409 words) - 17:10, 30 April 2024
2006 AMC 12B Problems/Problem 13
...s <math> ABCD</math> is 24, and <math> \angle BAD = 60^\circ</math>. What is the area of rhombus <math> BFDE</math>? ...n, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3));

3 KB (447 words) - 03:49, 16 January 2021
2006 AMC 12B Problems/Problem 14
...> and <math>N</math> are all positive integers with <math>N>1</math>. What is the cost of the jam Elmo uses to make the sandwiches? ...ply that if <math>B=2</math> and <math>J=3</math>, then <math>4B+5J=4(2)+5(3)=23</math>. The problem asks for the total cost of jam, or <math>N(5J)=11(1

1 KB (227 words) - 17:21, 8 December 2013
2006 AMC 12B Problems/Problem 15
...h> \overline{BC}</math> are common external tangents to the circles. What is the area of hexagon <math> AOBCPD</math>? ...bf{(B) } 24\sqrt {2} \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 24\sqrt {3} \qquad \textbf{(E) } 32\sqrt {2}</math>

3 KB (458 words) - 16:40, 6 October 2019
2006 AMC 12B Problems/Problem 16
...>C</math> at <math>(0,0)</math> and <math>(7,1)</math>, respectively. What is its area? \mathrm{(A)}\ 20\sqrt {3}

1 KB (203 words) - 16:36, 18 September 2023
2006 AMC 12B Problems/Problem 17
...<math>6</math> on each die are in the ratio <math>1:2:3:4:5:6</math>. What is the probability of rolling a total of <math>7</math> on the two dice? The probability of getting an <math>x</math> on one of these dice is <math>\frac{x}{21}</math>.

1 KB (188 words) - 22:10, 9 June 2016
2006 AMC 12B Problems/Problem 18
...can easily be shown that each location that satisfies these two conditions is indeed reachable. If the object only makes <math>1</math> move, it is obvious that there are only 4 possible points that the object can move to.

2 KB (354 words) - 16:57, 28 December 2020
2006 AMC 12B Problems/Problem 19
..."and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children? First, The number of the plate is divisible by <math>9</math> and in the form of

4 KB (696 words) - 09:47, 10 August 2015
2006 AMC 12B Problems/Problem 20
...th>x</math> be chosen at random from the interval <math>(0,1)</math>. What is the probability that Here <math>\lfloor x\rfloor</math> denotes the greatest integer that is less than or equal to <math>x</math>.

3 KB (485 words) - 14:09, 21 May 2021
2006 AMC 12B Problems/Problem 22
...are integers and <math>m</math> is not divisible by <math>10</math>. What is the smallest possible value of <math>n</math>? The power of <math>10</math> for any factorial is given by the well-known algorithm

5 KB (881 words) - 15:52, 23 June 2021
2006 AMC 12B Problems/Problem 23
...ath>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>? MP("90^\circ-\alpha",C,3*dir(30),f);

7 KB (1,169 words) - 14:04, 10 June 2022
2006 AMC 12B Problems/Problem 24
...\le \frac{\pi}{2}</math> and <math>0 \le y \le \frac{\pi}{2}</math>. What is the area of the subset of <math>S</math> for which <cmath> \mathrm{(D)}\ \dfrac{3\pi^2}{16}

3 KB (563 words) - 22:45, 24 October 2021
2006 AMC 12B Problems/Problem 25
A sequence <math>a_1,a_2,\dots</math> of non-negative integers is defined by the rule <math>a_{n+2}=|a_{n+1}-a_n|</math> for <math>n\geq 1</m ...sequence <math>(a_n)</math> completes at <math>i</math> if <math>i</math> is the minimal positive integer such that <math>a_i = a_{i + 1} = 1</math>. Ot

5 KB (924 words) - 12:02, 15 June 2022
2006 AMC 12A Problems/Problem 10
For how many real values of <math>x</math> is <math>\sqrt{120-\sqrt{x}}</math> an integer? <math> \textbf{(A) } 3\qquad \textbf{(B) } 6\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \t

1 KB (167 words) - 23:23, 16 December 2021
2006 AMC 12A Problems/Problem 13
...e centers of three mutually externally tangent [[circle]]s, as shown. What is the sum of the areas of the three circles? <cmath>r_A + r_B = 3</cmath>

1 KB (184 words) - 13:57, 19 January 2021
2006 AMC 12A Problems/Problem 14
...debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way? ...mon divisor]]) of <math>a</math> and <math>b</math>. Therefore, the answer is <math>gcd(300,210)=\boxed{\textbf{(C) }30}.</math>

3 KB (442 words) - 03:13, 8 August 2022
2006 AMC 12A Problems/Problem 15
Suppose <math>\cos x=0</math> and <math>\cos (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) \ } \frac{\pi}{2}\qquad \mathrm{(D) \ } \frac{5\pi}{6} \

919 bytes (138 words) - 12:45, 4 August 2017
2006 AMC 12A Problems/Problem 16
...d <math>CD</math> intersect at <math>E</math>, and <math>AE=5</math>. What is <math>CD</math>? dotfactor=3;

2 KB (286 words) - 10:16, 19 December 2021
2006 AMC 12A Problems/Problem 17
...th> is tangent to the circle, and <math>AF=\sqrt{9+5\sqrt{2}}</math>. What is <math>r/s</math>? ...rac{5}{9}\qquad \mathrm{(C) \ } \frac{3}{5}\qquad \mathrm{(D) \ } \frac{5}{3}\qquad \mathrm{(E) \ } \frac{9}{5}</math>

6 KB (958 words) - 23:29, 28 September 2023
2006 AMC 12A Problems/Problem 20
...s equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vert Therefore, starting at <math>A</math>, the bug has a <math>\frac{3}{3}</math> chance of finding a good path to the next vertex, and call it <math

5 KB (908 words) - 19:23, 22 September 2022
2006 AMC 12A Problems/Problem 22
...sible from a randomly chosen point on the circle is <math>1/2</math>. What is <math>r</math>? ...quad \rm{(D) \ } 3\sqrt{2}+\sqrt{6}\qquad \mathrm{(E) \ } 6\sqrt{2}-\sqrt{3}</math>

2 KB (343 words) - 15:39, 14 June 2023
2006 AMC 12A Problems/Problem 23
...,\ldots ,x^{100})</math>. If <math>A^{100}(S)=(1/2^{50})</math>, then what is <math>x</math>? <cmath>A^2(S)=\left(\frac{1+2x+x^2}{2^2},\frac{x+2x^2+x^3}{2^2},...,\frac{x^{98}+2x^{99}+x^{100}}{2^2}\right)</cmath>

3 KB (466 words) - 22:40, 29 September 2023
2006 AMC 12A Problems/Problem 24
is simplified by expanding it and combining like terms. How many terms are in if the exponent of <math>y</math> is <math>1</math>, the exponent of <math>z</math> can be all even integers up

8 KB (1,332 words) - 17:37, 17 September 2023
2006 AMC 12A Problems/Problem 25
How many non-[[empty set | empty]] [[subset]]s <math>S</math> of <math>\{1,2,3,\ldots ,15\}</math> have the following two properties? ...k+1</math>, with no restriction on consecutive numbers. Since this process is easily reversible, we have a [[bijection]].

8 KB (1,405 words) - 11:52, 27 September 2022
2005 AMC 12A Problems/Problem 20
...\geq 2</math>. For how many values of <math>x</math> in <math>[0,1]</math> is <math>f^{[2005]}(x) = \frac {1}{2}</math>? ...<math>f(x)=2-2x,\frac{1}{2}\le x\le 1</math>,as long as <math>f(x)</math> is between <math>0</math> and <math>1</math>, <math>x</math> will be in the ri

3 KB (437 words) - 23:49, 28 September 2022
2005 AMC 12A Problems/Problem 25
...ly possible side length (red triangle in diagram). Each of these triangles is determined by one vertex of the cube, so in one cube we have 8 equilateral currentprojection=perspective(1/3,-1,1/2);

4 KB (498 words) - 00:46, 4 August 2023
2005 AMC 12B Problems/Problem 2
...property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>? ...h> means <math>0.01x</math>, the statement "<math>x\% \text{ of } x \text{ is 4}</math>" can be rewritten as "<math>0.01x \cdot x = 4</math>":

1 KB (145 words) - 13:56, 14 December 2021
2005 AMC 12B Problems/Problem 4
...>A</math> on <math>22</math> of the first <math>30</math> quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she e \textbf{(C) }\ 3 \qquad

1 KB (197 words) - 14:16, 14 December 2021
2005 AMC 12B Problems/Problem 6
...lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>? \textbf{(A) }\ 3 \qquad

2 KB (299 words) - 15:29, 5 July 2022
2005 AMC 12B Problems/Problem 7
What is the area enclosed by the graph of <math>|3x|+|4y|=12</math>? ...equations (using the logic that if <math>|a|=b</math>, then <math>a</math> is either <math>b</math> or <math>-b</math>):

2 KB (357 words) - 20:15, 27 December 2020
2005 AMC 12B Problems/Problem 9
...got <math>90</math> points, and the rest got <math>95</math> points. What is the difference between the [[mean]] and the [[median]] score on this exam? ...720}{20}=86</math>. The difference between the mean and median, therefore, is <math>\boxed{\textbf{(B)}\ 1}</math>.

2 KB (280 words) - 15:35, 16 December 2021
2005 AMC 12B Problems/Problem 10
...ding term is the sum of the cubes of the digits of the previous term. What is the <math>{2005}^{\text{th}}</math> term of the sequence? ...<math>250</math>. It just so happens that <math>2005\equiv 1\ (\text{mod}\ 3)</math>, which leads us to the answer of <math>\boxed{\textbf{(E) } 250}</m

1 KB (204 words) - 14:37, 15 December 2021
2005 AMC 12B Problems/Problem 11
...awn at random without replacement. What is the probability that their sum is $<math>20</math> or more? ...\qquad \textbf{(D) }\ {{{\frac{1}{2}}}} \qquad \textbf{(E) }\ {{{\frac{2}{3}}}}</math>

4 KB (607 words) - 21:01, 20 May 2023
2005 AMC 12B Problems/Problem 13
...math>, <math>6^{x_3}=7</math>, ... , <math>127^{x_{124}}=128</math>. What is <math>x_1x_2...x_{124}</math>? ...)}\ {{{2}}} \qquad \mathrm{(B)}\ {{{\frac{5}{2}}}} \qquad \mathrm{(C)}\ {{{3}}} \qquad \mathrm{(D)}\ {{{\frac{7}{2}}}} \qquad \mathrm{(E)}\ {{{4}}}</mat

1 KB (203 words) - 19:57, 24 December 2020
2005 AMC 12B Problems/Problem 14
...o the lines <math>y=x</math>, <math>y=-x</math> and <math>y=6</math>. What is the radius of this circle? ...</math> and the diagonal is <math>k = R+6</math>. The diagonal of a square is <math>\sqrt{2}</math> times the side length. Therefore, <math>R+6 = R\sqrt{

2 KB (278 words) - 21:12, 24 December 2020
2005 AMC 12B Problems/Problem 15
...is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits? \mathrm{(C)}\ 3 \qquad

2 KB (411 words) - 21:02, 21 December 2020
2005 AMC 12B Problems/Problem 16
...radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains th \mathrm {(B)}\ \sqrt{3} \qquad

2 KB (364 words) - 04:54, 16 January 2023
2005 AMC 12B Problems/Problem 17
<cmath>a\cdot\log_{10}2+b\cdot\log_{10}3+c\cdot\log_{10}5+d\cdot\log_{10}7=2005?</cmath> <cmath>\log_{10}2^{a}+\log_{10}3^{b}+\log_{10}5^{c}+\log_{10}7^{d}=2005</cmath>

1 KB (159 words) - 21:18, 21 December 2020
2005 AMC 12B Problems/Problem 20
...g</math> and <math>h</math> be distinct elements in the set <math>\{-7,-5,-3,-2,2,4,6,13\}.</math> What is the minimum possible value of <math>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</math>

3 KB (463 words) - 19:28, 6 November 2022
2005 AMC 12B Problems/Problem 21
...60</math> divisors and <math>7n</math> has <math>80</math> divisors. What is the greatest integer <math>k</math> such that <math>7^k</math> divides <mat ...\mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad \mathrm{(D)}\ {{{3}}} \qquad \mathrm{(E)}\ {{{4}}}</math>

888 bytes (140 words) - 20:04, 24 December 2020
2005 AMC 12B Problems/Problem 22
A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is defined by the rule where <math>\overline {z_{n}}</math> is the [[complex conjugate]] of <math>z_{n}</math> and <math>i^{2}=-1</math>.

4 KB (660 words) - 17:40, 24 January 2021
2005 AMC 12B Problems/Problem 23
...> we have <math>x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.</math> What is the value of <math>a+b?</math> Therefore, <math>x^3 + y^3 = s\cdot\dfrac{3t-s^2}{2} = s(15s-\dfrac{s^2}{2})</math>.

5 KB (786 words) - 11:36, 19 May 2024
2005 AMC 12B Problems/Problem 24
...h>m</math> and <math>n</math> are relatively prime positive integers. What is the value of <math>m + n</math>? ...that the slope between the first two is <math>2</math>, and <math>A</math> is the point with the least <math>y</math>-coordinate.

4 KB (761 words) - 09:10, 1 August 2023
2005 AMC 12B Problems/Problem 25
...o one of the four adjacent vertices, each with equal [[probability]]. What is the probability that no two ants arrive at the same vertex? \qquad\mathrm{(E)}\ \frac {3}{128}</math>

10 KB (1,840 words) - 21:35, 7 September 2023
2006 AMC 10A Problems
Sandwiches at Joe's Fast Food cost <math> \textdollar 3 </math> each and sodas cost <math> \textdollar 2 </math> each. How many dol Define <math>x\otimes y=x^3-y</math>. What is <math>h\otimes (h\otimes h)</math>?

13 KB (2,028 words) - 16:32, 22 March 2022
1995 AHSME Problems/Problem 6
...to the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked x? path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3);

1 KB (168 words) - 00:49, 14 October 2013
2006 AMC 10A Problems/Problem 8
...es through the points <math> (2,3) </math> and <math> (4,3) </math>. What is <math>c</math>? Substitute the points <math> (2,3) </math> and <math> (4,3) </math> into the given equation for <math> (x,y) </math>.

2 KB (348 words) - 23:10, 16 December 2021
2006 AMC 12A Problems/Problem 12
...ove it. The bottom ring has an outside diameter of <math>3</math> cm. What is the distance, in cm, from the top of the top ring to the bottom of the bott D(CR((0,-39),3));

2 KB (292 words) - 11:56, 17 December 2021
2006 AMC 10A Problems/Problem 15
.../math> meters in the opposite direction and the circumference of his track is <math>100\pi</math>. ...will meet again in <math>k</math> minutes. So the total amount of meetings is <math>\lfloor\frac{30}{k}\rfloor=\lfloor\frac{150}{\pi}\rfloor=\boxed{\text

3 KB (532 words) - 17:49, 13 August 2023
2006 AMC 10A Problems/Problem 16
...h>\overline{AB}</math> and <math>\overline{AC}</math> are congruent. What is the area of <math>\triangle ABC</math>? MP('2', (2*t,3), W); MP('1',(2*t, 5.5), W);</asy>

5 KB (732 words) - 23:19, 19 September 2023
2006 AMC 10A Problems/Problem 17
...HE}</math>. In addition, <math>AH=AC=2</math>, and <math>AD=3</math>. What is the area of quadrilateral <math>WXYZ</math> shown in the figure? A=(0,2); B=(1,2); C=(2,2); D=(3,2);

6 KB (1,066 words) - 00:21, 2 February 2023
2006 AMC 10A Problems/Problem 18
...quad\textbf{(D) } 10^2\times 26^4\qquad\textbf{(E) } 5\times 10^3\times 26^3\qquad</math> Therefore, the number of distinct license plates is <math> 5\times 10^4\times 26^2 \Longrightarrow \boxed{\mathrm{C}}</math>.

2 KB (254 words) - 14:39, 5 April 2024
2006 AMC 10A Problems/Problem 19
...le value for the smallest angle is <math>1</math> and the highest possible is <math>59</math> (since the numbers are distinct), so there are <math>\boxed ==Solution 3 (Quick Summation)==

2 KB (259 words) - 03:10, 22 June 2023
2006 AMC 10A Problems/Problem 20
...is the probability that some pair of these integers has a difference that is a multiple of <math>5</math>? ...) } \frac{1}{2}\qquad\textbf{(B) } \frac{3}{5}\qquad\textbf{(C) } \frac{2}{3}\qquad\textbf{(D) } \frac{4}{5}\qquad\textbf{(E) } 1\qquad</math>

1 KB (187 words) - 08:21, 17 March 2023
2006 AMC 10A Problems/Problem 21
...itive integers have at least one digit that is a <math>2</math> or a <math>3</math>? ...s and subtracting off those which do not have any <math>2</math>s or <math>3</math>s as digits.

3 KB (525 words) - 20:25, 30 April 2024
2006 AMC 10A Problems/Problem 24
...ent faces of a unit cube are joined to form a regular [[octahedron]]. What is the volume of this octahedron? ...) } \frac{1}{6}\qquad\textbf{(C) } \frac{1}{4}\qquad\textbf{(D) } \frac{1}{3}\qquad\textbf{(E) } \frac{1}{2}\qquad</math>

2 KB (292 words) - 10:19, 19 December 2021
Set
...ames really do not define the meaning of the word ''set''; all they can do is replace it in various sentences. So, instead of defining what sets are, one ...uch as the following: <math>\{1,4,5,3,24,4,4,5,6,2\}</math> Such an entity is actually called a multiset.

11 KB (2,021 words) - 00:00, 17 July 2011
Newman's Tauberian Theorem
'''Newman's Tauberian Theorem''' is a [[tauberian theorem]] (which is well-defined by this formula for <math>\Re s>0</math>) admits an

6 KB (1,034 words) - 07:55, 12 August 2019
2006 USAMO Problems
if and only if <math>s</math> is not a divisor of <math>p-1</math>. ...rms of <math>k</math>, the minimum value of <math>N</math> for which there is a set of <math>2k+1</math> distinct positive integers that has sum greater

3 KB (520 words) - 09:24, 14 May 2021
1991 AJHSME Problems
== Problem 3 == [[1991 AJHSME Problems/Problem 3|Solution]]

17 KB (2,246 words) - 13:37, 19 February 2020
1990 AJHSME Problems/Problem 1
What is the smallest sum of two <math>3</math>-digit numbers that can be obtained by placing each of the six digits draw((1,1)--(3,1)--(3,3)--(1,3)--cycle); draw((1,4)--(3,4)--(3,6)--(1,6)--cycle);

1 KB (191 words) - 17:12, 29 October 2016
2022 USAJMO Problems/Problem 1
<math>\bullet</math> <math>a_n-g_n</math> is divisible by <math>m</math> for all integers <math>n>1</math>; <math>\bullet</math> <math>a_2-a_1</math> is not divisible by <math>m</math>.

4 KB (792 words) - 00:29, 13 April 2024
2006 AIME II Problems/Problem 2
...th>\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>. ...number of positive integer <math>n</math> which satisfies this requirement is <math>\boxed{893}</math>.

1 KB (164 words) - 14:58, 14 April 2020
2005 AIME II Problems/Problem 1
...that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find <math> n. </math> ...ial coefficient]] <math>{n \choose 6} = \frac{n\cdot(n-1)\cdot(n-2)\cdot(n-3)\cdot(n-4)\cdot(n-5)}{6\cdot5\cdot4\cdot3\cdot2\cdot1}</math>.

1 KB (239 words) - 11:54, 31 July 2023
2005 AIME II Problems/Problem 2
...uests. Given that the [[probability]] each guest got one roll of each type is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are [[ *Person 1: <math>\frac{9 \cdot 6 \cdot 3}{9 \cdot 8 \cdot 7} = \frac{9}{28}</math>

4 KB (628 words) - 11:28, 14 April 2024
2005 AIME II Problems/Problem 4
<math>15^7 = 3^7\cdot5^7</math> so <math>15^7</math> has <math>8\cdot8 = 64</math> divisor <math>\gcd(15^7, 18^{11}) = 3^7 </math> which has 8 divisors.

3 KB (377 words) - 18:36, 1 January 2024
2005 AIME II Problems/Problem 13
...th> P(17)=10 </math> and <math> P(24)=17. </math> Given that <math> P(n)=n+3 </math> has two distinct integer solutions <math> n_1 </math> and <math> n_ ...h>(x-17)(x-24)</math> to be a factor of <math>10</math>. Hence the answer is <math>19\cdot 22=\boxed{418}</math>.

4 KB (642 words) - 14:55, 12 August 2019
2005 AIME II Problems/Problem 5
...og b=3\log a </math> or <math>\log b=2\log a </math>, so either <math> b=a^3 </math> or <math> b=a^2 </math>. ...e <math> b=a^3 </math>, note that <math> 12^3=1728 </math> while <math> 13^3=2197 </math>. Therefore, for this case, all values of <math>a</math> from <

3 KB (547 words) - 19:15, 4 April 2024
2005 AIME II Problems/Problem 6
...agical. For example, eight cards form a magical stack because cards number 3 and number 6 retain their original positions. Find the number of cards in t ...s suggests that <math>n = 131 + 65 = 196</math>; the total number of cards is <math>196 \cdot 2 = \boxed{392}</math>.

2 KB (384 words) - 00:31, 26 July 2018
2005 AIME II Problems
...that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find <math> n. </math> ...he guests. Given that the probability each guest got one roll of each type is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are re

7 KB (1,119 words) - 21:12, 28 February 2020
2005 AIME II Problems/Problem 7
It follows that <math>(x + 1)^{48} = (\sqrt[16]5)^{48} = 5^3 = \boxed{125}</math>. ...+1) = (y^{15}+y^{14}+y^{13}+y^{12}+y^{11}+y^{10}+y^9+y^8+y^7+y^6+y^5+y^4+y^3+y^2+y+1)=\frac{y^{16}-1}{y-1}</cmath>

2 KB (279 words) - 12:33, 27 October 2019
2005 AIME II Problems/Problem 8
.../math> and <math> p </math> are [[relatively prime]], and <math> n </math> is not divisible by the square of any [[prime]], find <math> m+n+p. </math> ...= (-10,0), C2 = (4,0), C3 = (0,0), H = (-10-28/3,0), T = 58/7*expi(pi-acos(3/7));

4 KB (693 words) - 13:03, 28 December 2021
2005 AIME II Problems/Problem 9
...positive integers <math> n </math> less than or equal to <math>1000</math> is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true for all rea ...t certainly hold for <math>t = \frac{\pi}2 - u</math>. Thus, the question is equivalent to asking for how many [[positive integer]]s <math>n \leq 1000</

6 KB (1,154 words) - 03:30, 11 January 2024
2005 AIME II Problems/Problem 12
...h> and <math> r </math> are [[positive]] [[integer]]s and <math> r </math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math ...- y</math> again, we know have <math>xy = (400 - y)y = 150^2</math>. This is a quadratic with roots <math>200 \pm 50\sqrt{7}</math>. Since <math>y < x</

13 KB (2,080 words) - 21:20, 11 December 2022
2005 AIME II Problems/Problem 10
...at the ratio of the volume of <math> O </math> to that of <math> C </math> is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are re ...,0,-3)--(0,-3,0)--(3,0,0)--(0,0,-3)--(0,3,0)--(0,0,3)--(3,0,0)--(0,3,0)--(-3,0,0));

3 KB (436 words) - 03:10, 23 September 2020
2005 AIME II Problems/Problem 11
...>a_0 = 37, a_1 = 72, a_m = 0, </math> and <math> a_{k+1} = a_{k-1} - \frac 3{a_k} </math> for <math> k = 1,2,\ldots, m-1. </math> Find <math>m. </math> <math>a_{k}a_{k+1} = a_{k-1}a_{k} - 3 </math>.

3 KB (499 words) - 18:52, 21 November 2022
Euler's identity
...er's Formula''' is <math>e^{i\theta}=\cos \theta+ i\sin\theta</math>. It is named after the 18th-century mathematician [[Leonhard Euler]]. ...umbers]] and/or [[trigonometry]]. Euler's formula replaces "[[cis]]", and is a superior notation, as it encapsulates several nice properties:

3 KB (452 words) - 23:17, 4 January 2021
2006 AMC 12A Problems/Problem 4
A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display? 1+2&9&6&3\\

2 KB (257 words) - 11:20, 2 January 2022
2005 AIME I Problems
...ose common difference is <math> k. </math> For example, <math> S_3 </math> is the sequence <math> 1,4,7,10,\ldots. </math> For how many values of <math> == Problem 3 ==

6 KB (983 words) - 05:06, 20 February 2019
2005 AIME I Problems/Problem 2
...ose common difference is <math> k</math>. For example, <math> S_3 </math> is the [[sequence]] <math> 1,4,7,10,\ldots. </math> For how many values of <ma ...h>. Thus the requested number of values is <math>12</math>, and the answer is <math>\boxed{012}</math>.

2 KB (303 words) - 01:31, 5 December 2022
2005 AIME I Problems/Problem 3
...ivisor]]s (positive integral [[divisor]]s excluding itself), each of which is less than 50? ...so <math>n</math> must be in the form <math>n=p\cdot q</math> or <math>n=p^3</math> for distinct [[prime number]]s <math>p</math> and <math>q</math>.

2 KB (249 words) - 09:37, 23 January 2024
2005 AIME I Problems/Problem 4
...5</math>, so this number works and no larger number can. Thus, the answer is <math>\boxed{294}</math>. ...factors of <math>69</math> are <math>(1,69), (3,23)</math>; <math>x</math> is maximized for the first case. Thus, <math>x = \frac{69 + 1}{2} = 35</math>,

8 KB (1,248 words) - 11:43, 16 August 2022
2005 AIME I Problems/Problem 5
...te parts to this problem: one is the color (gold vs silver), and the other is the orientation. ...t occur at all, for <math>9</math> total configurations. Thus, the answer is <math>70 \cdot 9 = \boxed{630}</math>.

5 KB (830 words) - 01:51, 1 March 2023
2005 AIME I Problems/Problem 6
Let <math> P </math> be the product of the nonreal roots of <math> x^4-4x^3+6x^2-4x=2005. </math> Find <math> \lfloor P\rfloor. </math> The left-hand side of that [[equation]] is nearly equal to <math>(x - 1)^4</math>. Thus, we add 1 to each side in ord

4 KB (686 words) - 01:55, 5 December 2022
2005 AIME I Problems/Problem 7
...DE</math> is concurrent with line <math>BC</math>. Then, <math>ABED</math> is an isosceles trapezoid so <math>AD=BE=10</math>, and <math>BC=8</math> and ...</math>. The [[Pythagorean Theorem]] yields that <math>GC^2 = 12^2 - \sqrt{3}^2 = 141</math>, so <math>EF = GC = \sqrt{141}</math>. Therefore, <math>AB

4 KB (567 words) - 20:20, 3 March 2020
2005 AIME I Problems/Problem 8
...2^{222x+1} + 1 </math> has three [[real]] [[root]]s. Given that their sum is <math>m/n</math> where <math> m </math> and <math> n </math> are [[relative ...</math> and <math>x_1 + x_2 + x_3 = \frac{2}{111}</math>. Thus the answer is <math>111 + 2 = \boxed{113}</math>.

1 KB (161 words) - 19:50, 2 January 2022
2005 AIME I Problems/Problem 9
...[probability]] of the entire [[surface area]] of the larger cube is orange is <math> \frac{p^a}{q^br^c}, </math> where <math> p,q, </math> and <math> r < ...orientations, so from these cubes we gain a factor of <math>\left(\frac{2}{3}\right)^6</math>.

4 KB (600 words) - 21:44, 20 November 2023
2005 AIME I Problems/Problem 10
...[[midpoint]] <math>M</math> of [[line segment]] <math>\overline{BC}</math> is <math>\left(\frac{35}{2}, \frac{39}{2}\right)</math>. The equation of the m ...tion for the triangle will give a smaller value of <math>p+q</math>, which is provable by following these steps over again) (alternatively, we could use

5 KB (852 words) - 21:23, 4 October 2023
2005 AIME I Problems/Problem 11
...e]] whose sides have length 8. Given the maximum value of <math> d </math> is <math> m - \sqrt{n},</math> find <math> m+n. </math> ...n it touches both other sides of the square. This can happen only when it is arranged so that the center of the semicircle lies on one diagonal of the s

4 KB (707 words) - 11:11, 16 September 2021
2005 AIME I Problems/Problem 12
...squares less than <math>n</math>. So <math>S(1), S(2)</math> and <math>S(3)</math> are odd, while <math>S(4), S(5), \ldots, S(8)</math> are even, and ...t the numbers between <math>1^2</math> and <math>2^2</math>, between <math>3^2</math> and <math>4^2</math>, and so on, all the way up to the numbers bet

4 KB (647 words) - 02:29, 4 May 2021
2005 AIME I Problems/Problem 13
...th>U</math> represent a move upwards, and <math>D</math> to be a move that is diagonal. [[Casework]] upon the number of diagonal moves: *'''Case ''' <math>d = 1</math>: It is easy to see only <math>2</math> cases.

5 KB (897 words) - 00:21, 29 July 2022
2005 AIME I Problems/Problem 14
...e the area of <math> S. </math> Find the remainder when <math> 10K </math> is divided by <math>1000</math>. Consider a point <math>E</math> such that <math>AE</math> is [[perpendicular]] to <math>BD</math>, <math>AE</math> intersects <math>BD</

3 KB (561 words) - 14:11, 18 February 2018
2005 AIME I Problems/Problem 15
...re <math> m </math> and <math> n </math> are integers and <math> n </math> is not [[divisor | divisible]] by the [[perfect square | square]] of a prime, ...thout loss of generality, let <math>AC < AB</math>, so that <math>E</math> is between <math>D</math> and <math>C</math>. Let the length of the median be

5 KB (906 words) - 23:15, 6 January 2024
2005 AIME II Problems/Problem 15
...or which the line <math> y=ax </math> contains the center of a circle that is externally [[tangent (geometry)|tangent]] to <math> w_2 </math> and interna ...centers is <math>r_1 + r_2</math>, and if they are internally tangent, it is <math>|r_1 - r_2|</math>. So we have

12 KB (2,000 words) - 13:17, 28 December 2020
2005 AIME II Problems/Problem 14
...th> \overline{BC} </math> with <math> CD=6. </math> Point <math> E </math> is on <math> \overline{BC} </math> such that <math> \angle BAE\cong \angle CAD ...{BE} - 1 \Longrightarrow BE = \frac{13^2 \cdot 15}{463}</math>. The answer is <math>q = \boxed{463}</math>.

13 KB (2,129 words) - 18:56, 1 January 2024
2004 AIME I Problems/Problem 15
f(x)=\begin{cases}1 & \text{if }x = 1\\ \frac x{10} & \text{if }x\text{ is divisible by 10}\\ x+1 & \text{otherwise}\end{cases} ...st <math>n</math> such that <math>x_n=1</math>. (For example, <math>d(100)=3</math> and <math>d(87)=7</math>.) Let <math>m</math> be the number of posit

9 KB (1,491 words) - 01:23, 26 December 2022
2004 AIME I Problems/Problem 14
...> and <math> c </math> are [[positive]] [[integer]]s, and <math> c </math> is prime. Find <math> a+b+c. </math> real x = 20 - ((750)^.5)/3, CE = 8*(6^.5) - 4*(5^.5), CD = 8*(6^.5), h = 4*CE/CD;

4 KB (729 words) - 01:00, 27 November 2022
2004 AIME I Problems/Problem 13
...oots of the form <math> z_k = r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34, </math> with <math> 0 < a_1 \le a_2 \le a_3 \le \cdots \le a_{3 ...nomial]] <math>P</math> is very difficult to work with directly, but there is one obvious transformation to make: sum the [[geometric series]]:

2 KB (298 words) - 20:02, 4 July 2013
2004 AIME I Problems/Problem 12
...on <math> [z] </math> denotes the [[floor function|greatest integer]] that is less than or equal to <math> z. </math> <math>\left\lfloor\log_2\left(\frac{1}{x}\right)\right\rfloor</math> is even when

2 KB (303 words) - 22:28, 11 September 2020
2004 AIME I Problems/Problem 11
...s a 3-inch radius. The entire [[surface]] of the cone, including its base, is painted. A [[plane]] [[parallel]] to the base of the cone divides the cone ...face area]] <math>A = \pi r^2 + \pi r \ell</math>, where <math>\ell</math> is the [[slant height]] of the cone. Using the [[Pythagorean Theorem]], we ge

5 KB (839 words) - 22:12, 16 December 2015
2004 AIME I Problems/Problem 10
...[[probability]] that the circle will not touch diagonal <math> AC </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relativ ...nter of the circle must be in the <math>34 \times 13</math> rectangle that is one unit away from the sides of rectangle <math>ABCD</math>. We want to fin

5 KB (836 words) - 07:53, 15 October 2023
2004 AIME I Problems/Problem 9
...<math> U_1 </math> is similar to <math> U_2 </math> and <math> V_1 </math> is similar to <math> V_2. </math> The minimum value of the area of <math> U_1 ...h>ABC</math>. Thus <math>U_1</math>, and hence <math>U_2</math>, are <math>3-4-5\,\triangle</math>s.

4 KB (618 words) - 20:01, 4 July 2013
2004 AIME I Problems/Problem 1
...to right. What is the sum of the possible remainders when <math> n </math> is divided by <math>37</math>? ...2) + 10(n + 1) + n = 3210 + 1111n</math>, for <math>n \in \lbrace0, 1, 2, 3, 4, 5, 6\rbrace</math>.

2 KB (374 words) - 14:53, 27 December 2019
2004 AIME I Problems/Problem 2
...atest element of <math>A</math> and the greatest element of <math>B</math> is <math>99</math>. Find <math>m.</math> ...must be <math>2</math>. Therefore, the largest element in <math>A</math> is <math>2 + \frac{m-1}{2}</math>.

8 KB (1,437 words) - 21:53, 19 May 2023
2004 AIME I Problems/Problem 4
...<math> S </math> enclose a region whose [[area]] to the nearest hundredth is <math>k</math>. Find <math> 100k</math>. ...e at each corner of the square. The area enclosed by all of the midpoints is <math>4-4\cdot \left(\frac{\pi}{4}\right)=4-\pi \approx .86</math> to the n

3 KB (532 words) - 09:22, 11 July 2023
2004 AIME I Problems/Problem 5
...</math> and <math> n </math> are relatively prime positive integers. What is <math> m+n </math>? From here, we see the largest possible value of <math>a+b</math> is <math>349</math>.

3 KB (436 words) - 18:31, 9 January 2024
2004 AIME I Problems/Problem 6
...s [[odd integer | odd]] and <math> a_i>a_{i+1} </math> if <math> i </math> is [[even integer | even]]. How many snakelike integers between 1000 and 9999 ...into two cases: one in which zero is one of the digits and one in which it is not. In the latter case, suppose we pick digits

3 KB (562 words) - 18:12, 4 March 2022
2004 AIME I Problems/Problem 7
...ath>Q(x)</math> is some polynomial [[divisibility | divisible]] by <math>x^3</math>. ...x)</math>, where <math>R(x)</math> is some polynomial divisible by <math>x^3</math>.

5 KB (833 words) - 19:43, 1 October 2023
2004 AIME I Problems/Problem 8
There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are sim ...of this <math>n</math>-gon in a counterclockwise direction: <math>0, 1, 2, 3, \ldots, n-1.</math>

4 KB (620 words) - 21:26, 5 June 2021
2004 AIME I Problems
...to right. What is the sum of the possible remainders when <math> n </math> is divided by 37? ...t element of <math> A </math> and the greatest element of <math> B </math> is 99. Find <math> m. </math>

9 KB (1,434 words) - 13:34, 29 December 2021
2004 AIME II Problems/Problem 15
...th>256</math> by <math>1</math> strip of quadruple thickness. This process is repeated <math>8</math> more times. After the last fold, the strip has beco Number the squares <math>0, 1, 2, 3, ... 2^{k} - 1</math>. In this case <math>k = 10</math>, but we will consi

6 KB (899 words) - 20:58, 12 May 2022
2004 AIME II Problems/Problem 14
...ht <math> 7 </math>'s in this way. For how many values of <math> n </math> is it possible to insert <math> + </math> signs so that the resulting expressi ...g by <math>7</math>, <math>a + 11b + 111c = 1000</math>. Then the question is asking for the number of values of <math>n = a + 2b + 3c</math>.

11 KB (1,857 words) - 21:55, 19 June 2023
2004 AIME II Problems/Problem 13
...of triangle <math> ABC </math> and the area of triangle <math> EBD </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relativ ...B \parallel CE, BC \parallel AD, </math> it follows that <math>ABCF</math> is a [[parallelogram]], and so <math>\triangle ABC \cong \triangle CFA</math>.

3 KB (486 words) - 22:15, 7 April 2023
2004 AIME II Problems/Problem 12
..., </math> and <math> p </math> are [[positive integer]]s, <math> n </math> is not [[divisibility | divisible]] by the [[perfect square | square]] of any real r = (-60 + 48 * 3^.5)/23;

3 KB (431 words) - 23:21, 4 July 2013
2004 AIME II Problems/Problem 10
...ath> S, </math> the [[probability]] that it is divisible by <math>9</math> is <math> p/q, </math> where <math> p </math> and <math> q </math> are relativ ...{40}{2}</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 9
...ression. Let <math> a_n </math> be the greatest term in this sequence that is less than <math>1000</math>. Find <math> n+a_n. </math> ...th>. This happens with <math>f(7)f(8) = 29 \cdot 33 = 957</math>, and this is the <math>2(8) = 16</math>th term of the sequence.

3 KB (538 words) - 21:33, 30 December 2023
2004 AIME II Problems/Problem 8
...the prime factorization of <math>2004^{2004}</math> is <math>2^{4008}\cdot 3^{2004}\cdot 167^{2004}</math>. ...ample, the number of divisors of <math>2004=2^2\cdot 3^1\cdot 167^1</math> is <math>(2+1)(1+1)(1+1)=12</math>.

2 KB (353 words) - 18:08, 25 November 2023
2004 AIME II Problems/Problem 7
...th> CF=3 </math> are given. The perimeter of rectangle <math> ABCD </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relativ pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C;

9 KB (1,501 words) - 05:34, 30 October 2023
2004 AIME II Problems/Problem 6
...t the end of the process are in the [[ratio]] <math> 3: 2: 1, </math>what is the least possible total for the number of bananas? ...c{11}{24}b_3</math>, and the third monkey got <math>\frac{1}{8}b_1 + \frac{3}{8}b_2 + \frac{1}{12}b_3</math>.

6 KB (950 words) - 14:18, 15 January 2024
2004 AIME II Problems/Problem 5
...urther behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the <math>800</math> worke ...0}{800}(60)=\frac{150}{8}</math>. The train then has <math>60-15-\frac{50}{3}-\frac{150}{8}=230/24</math> minutes left to travel 250 miles, and doing th

4 KB (592 words) - 19:02, 26 September 2020
2004 AIME II Problems/Problem 4
...ath>n</math>-digit number, for a total of <math>(2^1 - 2) + (2^2 - 2) + (2^3 -2) + (2^4 - 2) = 22</math> such numbers (or we can list them: <math>AB, BA ...s we can form, for a total of <math>(2^0 - 1) + (2^1 - 1) + (2^2 - 1) + (2^3 - 1) = 11</math> such numbers (or we can list them: <math>A0, A00, A0A, AA0

3 KB (508 words) - 01:16, 19 January 2024
2004 AIME II Problems/Problem 3
...gruent]] 1-cm [[cube (geometry) | cube]]s [[face]] to face. When the block is viewed so that three of its faces are visible, exactly <math>231</math> of ...s close together as possible, which occurs when the smaller block is <math>3 \times 7 \times 11</math>. Then the extra layer makes the entire block <ma

2 KB (377 words) - 11:53, 10 March 2014
2004 AIME II Problems/Problem 2
...ability]] that they get the same color combination, irrespective of order, is <math> m/n, </math> where <math> m </math> and <math> n </math> are [[relat ...c{28}{153}</math>. So the probability that they both pick two red candies is <math>\frac{9}{38} \cdot \frac{28}{153} = \frac{14}{323}</math>. The same

2 KB (330 words) - 13:42, 1 January 2015
2004 AIME II Problems/Problem 1
...y [[prime]]. Find the [[remainder]] when the product <math> abcdef </math> is divided by 1000. .../math>; the rest of the area of the circle is then equal to <math>\frac{2}{3}r^2\pi</math>.

2 KB (329 words) - 23:20, 4 July 2013
2004 AIME II Problems
...re of any prime. Find the remainder when the product <math> abcdef </math> is divided by 1000. ...obability that they get the same color combination, irrespective of order, is <math> m/n, </math> where <math> m </math> and <math> n </math> are relativ

9 KB (1,410 words) - 05:05, 20 February 2019
1983 AIME Problems
== Problem 3 == What is the product of the real roots of the equation <math>x^2 + 18x + 30 = 2 \sqr

7 KB (1,104 words) - 12:53, 6 July 2022
2019 AIME I Problems/Problem 2
...9, 20</math> distinct from <math>J</math>. The value of <math>B - J</math> is at least <math>2</math> with a probability that can be expressed in the for ...because <math>B \ne J</math>, so the probability that <math>B-J < 0</math> is <math>\frac{1}{2}</math> by symmetry.

5 KB (830 words) - 22:15, 28 December 2023
1984 AIME Problems
..._{98}</math> if <math>a_1</math>, <math>a_2</math>, <math>a_3\ldots</math> is an arithmetic progression with common difference 1, and <math>a_1+a_2+a_3+\ ...sitive multiple of <math>15</math> such that every digit of <math>n</math> is either <math>8</math> or <math>0</math>. Compute <math>\frac{n}{15}</math>.

6 KB (933 words) - 01:15, 19 June 2022
1986 AIME Problems
What is the sum of the solutions to the equation <math>\sqrt[4]{x} = \frac{12}{7 - == Problem 3 ==

5 KB (847 words) - 15:48, 21 August 2023
1987 AIME Problems
An ordered pair <math>(m,n)</math> of non-negative integers is called "simple" if the addition <math>m+n</math> in base <math>10</math> re What is the largest possible distance between two points, one on the sphere of radi

6 KB (869 words) - 15:34, 22 August 2023
1988 AIME Problems
...rder -- the correct five buttons. The sample shown below has <math>\{1, 2, 3, 6, 9\}</math> as its combination. Suppose that these locks are redesigned == Problem 3 ==

6 KB (902 words) - 08:57, 19 June 2021
1989 AIME Problems
== Problem 3 == Suppose <math>n_{}^{}</math> is a positive integer and <math>d_{}^{}</math> is a single digit in base 10. Find <math>n_{}^{}</math> if

7 KB (1,045 words) - 20:47, 14 December 2023
1990 AIME Problems
The [[increasing sequence]] <math>2,3,5,6,7,10,11,\ldots</math> consists of all [[positive integer]]s that are ne Find the value of <math>(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}</math>.

6 KB (870 words) - 10:14, 19 June 2021
1991 AIME Problems
...overline {AB}</math> of length 4 and <math>\overline {CB}</math> of length 3. Divide <math>\overline {AB}</math> into 168 congruent segments with points == Problem 3 ==

7 KB (1,106 words) - 22:05, 7 June 2021
1992 AIME Problems
...n its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are == Problem 3 ==

8 KB (1,117 words) - 05:32, 11 November 2023
1993 AIME Problems
== Problem 3 == <center><math>\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\

8 KB (1,275 words) - 06:55, 2 September 2021
1994 AIME Problems
...erfect square. What is the remainder when the 1994th term of the sequence is divided by 1000? ...<math>P^{}_{}</math> to a circle of radius 20. Square <math>ABCD\,</math> is constructed with <math>A\,</math> and <math>B\,</math> on the larger circle

7 KB (1,141 words) - 07:37, 7 September 2018
1995 AIME Problems
...</math> The total area enclosed by at least one of <math>S_{1}, S_{2}, S_{3}, S_{4}, S_{5}</math> can be written in the form <math>m/n,</math> where <m == Problem 3 ==

6 KB (1,000 words) - 00:25, 27 March 2024
1996 AIME Problems
...magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Fin ...that <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
== Problem 3 == ...number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?

7 KB (1,098 words) - 17:08, 25 June 2020
1998 AIME Problems
For how many values of <math>k</math> is <math>12^{12}</math> the [[least common multiple]] of the positive integers == Problem 3 ==

7 KB (1,084 words) - 02:01, 28 November 2023
1999 AIME Problems
Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding ter ...rigin cuts this figure into two congruent polygons. The slope of the line is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relativ

7 KB (1,094 words) - 13:39, 16 August 2020
2000 AIME I Problems
...ositive integer <math>n</math> such that no matter how <math>10^{n}</math> is expressed as the product of any two positive integers, at least one of thes ...<math>D</math> across the y-axis. The area of pentagon <math>ABCDE</math> is <math>451</math>. Find <math>u + v</math>.

7 KB (1,204 words) - 03:40, 4 January 2023
2001 AIME I Problems
...h>\mathcal{S}</math>, and the mean of <math>\mathcal{S}\cup\{2001\}</math> is <math>27</math> more than the mean of <math>\mathcal{S}</math>. Find the me == Problem 3 ==

7 KB (1,212 words) - 22:16, 17 December 2023
2002 AIME I Problems
...it 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 relatively pr size(250);real x=sqrt(3);

8 KB (1,374 words) - 21:09, 27 July 2023
2003 AIME I Problems
<center><math> \frac{((3!)!)!}{3!} = k \cdot n!, </math></center> ...k </math> and <math> n </math> are positive integers and <math> n </math> is as large as possible, find <math> k + n. </math>

6 KB (965 words) - 16:36, 8 September 2019
2000 AIME II Problems
<center><math>\frac 2{\log_4{2000^6}} + \frac 3{\log_5{2000^6}}</math></center> A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola <math

6 KB (947 words) - 21:11, 19 February 2019
2001 AIME II Problems
...t and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let <math>m</math> be the smallest numbe == Problem 3 ==

8 KB (1,282 words) - 21:12, 19 February 2019
2002 AIME II Problems
...s between <math>100</math> and <math>999</math>, inclusive; <math>y</math> is the number formed by reversing the digits of <math>x</math>; and <math>z=|x ...7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=(11,3,9)</math>. What is the [[surface area]] of the cube?

7 KB (1,177 words) - 15:42, 11 August 2023
2003 AIME II Problems
...> of three positive integers is 6 times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of <math>N</m ...whose digits are all different. What is the remainder when <math>N</math> is divided by 1000?

7 KB (1,127 words) - 09:02, 11 July 2023
1983 AIME Problems/Problem 1
<math>x^{120}=w^5</math>, <math>y^{120}=w^3</math>, and <math>(xyz)^{120}=w^{10}</math>. ...=w</math>. It now becomes clear that one way to find <math>\log_z w</math> is to find what <math>x^{12}</math> and <math>y^{12}</math> are in terms of <m

4 KB (642 words) - 03:14, 17 August 2022
1983 AIME Problems/Problem 2
It is best to get rid of the [[absolute value]]s first. Adding these together, we find that the sum is equal to <math>30-x</math>, which attains its minimum value (on the given i

1 KB (184 words) - 20:16, 14 January 2023
1983 AIME Problems/Problem 3
What is the product of the [[real]] [[root]]s of the [[equation]] <math>x^2 + 18x + ...d moreover, plugging in <math>y=-6</math>, we get <math>-6=6</math>, which is obviously false). Hence we have <math>y=10</math> as the only solution for

3 KB (532 words) - 05:18, 21 July 2022
1983 AIME Problems/Problem 4
...d that 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> ...tem to get <math>x = 1</math> and <math>y = 5</math>, such that the answer is <math>1^2 + 5^2 = \boxed{026}</math>.

11 KB (1,741 words) - 22:40, 23 November 2023
1983 AIME Problems/Problem 5
.../math> is <math>7</math> and the sum of the cubes is <math>10</math>. What is the largest real value that <math>x + y</math> can have? One way to solve this problem is by [[substitution]]. We have

4 KB (672 words) - 10:17, 17 March 2023
1983 AIME Problems/Problem 6
After some quick division, our answer is <math>\boxed{035}</math>. === Solution 3 (cheap and quick) ===

3 KB (361 words) - 20:20, 14 January 2023
1983 AIME Problems/Problem 7
...ch other. If <math>P</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? ...24}=1-\frac{420}{552}=1-\frac{35}{46}=\frac{11}{46}</math>, and the answer is <math>11+46=\boxed{057}</math>.

9 KB (1,392 words) - 20:37, 19 January 2024
1983 AIME Problems/Problem 8
What is the largest <math>2</math>-digit [[prime]] factor of the integer <math>n = ...h>3p<200</math>. The largest such prime is <math>\boxed{061}</math>, which is our answer.

2 KB (249 words) - 23:25, 11 May 2024
1983 AIME Problems/Problem 9
...e x\sin x \le \frac{\pi}{2}</math>, this value of <math>\frac{2}{3}</math> is attainable by the [[Intermediate Value Theorem]]). ...We show this possible with the same methods in Solution 1; thus the answer is <math>\boxed{012}</math>.

4 KB (722 words) - 20:25, 14 January 2023
1983 AIME Problems/Problem 10
...h>, <math>1005</math> and <math>1231</math> have something in common: each is a <math>4</math>-digit number beginning with <math>1</math> that has exactl ...ath>, <math>x\neq1</math>, and <math>y\neq1</math>. Hence, there are <math>3\cdot9\cdot8=216</math> numbers of this form.

5 KB (855 words) - 20:26, 14 January 2023
1983 AIME Problems/Problem 11
...dges have length <math>s</math>. Given that <math>s=6\sqrt{2}</math>, what is the volume of the solid? triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);

5 KB (865 words) - 21:11, 6 February 2023
1983 AIME Problems/Problem 12
...from their intersection point <math>H</math> to the center <math>O</math> is a positive rational number. Determine the length of <math>AB</math>. draw((-2,-2*sqrt(3))--(-2,2*sqrt(3)));

2 KB (412 words) - 18:23, 1 January 2024
1983 AIME Problems/Problem 13
...3, 6,9\}</math> is <math>9-6+3-2+1=5</math> and for <math>\{5\}</math> it is simply <math>5</math>. Find the sum of all such alternating sums for <math> Let <math>S</math> be a non-[[empty set | empty]] [[subset]] of <math>\{1,2,3,4,5,6\}</math>.

5 KB (894 words) - 22:02, 5 April 2024
1983 AIME Problems/Problem 14
...units apart. At <math>P</math>, one of the points of intersection, a line is drawn in such a way that the chords <math>QP</math> and <math>PR</math> hav <asy>size(160); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=3; pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=in

13 KB (2,149 words) - 18:44, 5 February 2024
1983 AIME Problems/Problem 15
...is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <math>mn</math>? add(pathticks(A--F,1,0.5,0,3.5));

19 KB (3,221 words) - 01:05, 7 February 2023

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