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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) - 09:11, 7 June 2023
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);

4 KB (709 words) - 01:50, 10 January 2022
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>?

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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>

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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

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