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• 2016 AMC 10A Problems/Problem 12
  ...Which of the following is a correct statement about the probability <math>p</math> that the product of the three integers is odd? ...{1}{8}<p<\dfrac{1}{3}\qquad\textbf{(D)}\ p=\dfrac{1}{3}\qquad\textbf{(E)}\ p>\dfrac{1}{3}</math>
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• Math books
  ...https://artofproblemsolving.com/community/user/243060 cargeek9], currently a junior in high school. It covers the basics of algebra, geometry, combinato ...3265/ref=sr_1_1?ie=UTF8&s=books&qid=1204029652&sr=1-1 Complex Numbers from A to... Z] by [[Titu Andreescu]]
  24 KB (3,177 words) - 12:53, 20 February 2024
• Physics books
  .../www.amazon.com/exec/obidos/ASIN/0756613647/artofproblems-20 Chaos: Making a New Science] by James Gleick ...le.amazon.com/Dreams-Final-Theory-Steven-Weinberg/dp/0679419233/ Dreams of a Final Theory] by Weinberg
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• Physics
  <strong>Physics</strong> is a branch of [[science]] that studies the properties of matter, energy, and ma Modern Physics is also a group of different subjects in physics:
  9 KB (1,355 words) - 07:29, 29 September 2021
• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
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• Joining an ARML team
  The SD team is entering just its third year of participation, so a permanent process of selecting team members has yet to be decided upon. The A student who wishes to attend practice should try to take the SDMO. If a student wishes to speak with one of the coaches for the team, they may do s
  21 KB (3,435 words) - 00:56, 23 May 2024
• Wilson's Theorem
  ...math>(p-1)! + 1</math> is divisible by <math>p</math> if and only if <math>p</math> is prime. It was stated by John Wilson. The French mathematician Lag ...<math>(p-1)! + 1</math>. Therefore <math>p</math> does not divide <math>(p-1)! + 1</math>.
  4 KB (639 words) - 01:53, 2 February 2023
• Trivial Inequality
  <cmath>ax^2+bx+c = a(x^2+\frac{b}{a}x+\frac{b^2}{4a^2})+c-\frac{b^2}{4a} = a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a}.</cmath> .../math> is positive and <math>ax^2+bx+c\le c-\frac{b^2}{4a}</math> if <math>a</math> is negative.
  3 KB (560 words) - 22:51, 13 January 2024
• Polynomial
  ...algebra]] often an arbitrary [[field]]). Note that a [[constant]] is also a polynomial. ===A More Precise Definition===
  6 KB (1,100 words) - 01:44, 17 January 2024
• Pigeonhole Principle
  ...ts, the principle may be referred to as the '''Dirichlet box principle'''. A common phrasing of the principle uses balls and boxes and is that if <math> ...nhole principle is as follows: suppose for contradiction that there exists a way to place <math>n</math> balls into <math>k</math> boxes where <math>n>k
  11 KB (1,985 words) - 21:03, 5 August 2023
• Relatively prime
  ...their difference <math>(n+1)-n = 1</math>, which is impossible since <math>p > 1</math>. ...exist some <math>x,y\in \mathbb{Z}</math> such that <math>ax+by=1</math> (a special case of [[Bezout's Lemma]]). The [[Euclidean Algorithm]] can be use
  2 KB (245 words) - 15:51, 25 February 2020
• 1965 AHSME Problems/Problem 40
  ...n</math> be the number of integer values of <math>x</math> such that <math>P = x^4 + 6x^3 + 11x^2 + 3x + 31</math> is the square of an integer. Then <ma <math>\textbf{(A)}\ 4 \qquad
  3 KB (571 words) - 00:42, 22 October 2021
• Combination
  ...'binomial coefficient''', is a way of choosing <math>r</math> objects from a set of <math>n</math> where the order in which the objects are chosen is ir ...ons are, their various types, and how to calculate each type! It serves as a great introductory video to combinations, permutations, and counting proble
  4 KB (615 words) - 11:43, 21 May 2021
• Cauchy-Schwarz Inequality
  ..._2 + \cdots + a_nb_n)^2,</cmath> with equality if and only if there exists a constant <math>t</math> such that <math>a_i = t b_i</math> for all <math>1 ...cdot \overrightarrow{w}|</cmath> with equality if and only if there exists a scalar <math>t</math> such that <math>\overrightarrow{v} = t \overrightarro
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• Factorial
  ...!=n \cdot (n-1) \cdots 2 \cdot 1 = \prod_{i=1}^n i</math>. Alternatively, a [[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n ...</math> (remember! this is 1, not 0! (the '!' was an exclamation mark, not a factorial sign))
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• Discriminant
  ...has two [[nonreal]] roots; and if the discriminant is 0, the equation has a real [[double root]]. ...nant can tell us something about the roots of a given [[polynomial]] <math>p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0</math> of degree <math>n</math> with all t
  4 KB (734 words) - 19:19, 10 October 2023
• Menelaus' theorem
  ...earity]] of points on each of the three sides (extended when necessary) of a [[triangle]]. ...h> intersecting <math>AB</math> on <math>\triangle ABC</math>, where <math>P</math> is on <math>BC</math>, <math>Q</math> is on the extension of <math>A
  5 KB (804 words) - 03:01, 12 June 2023
• Fermat's Little Theorem
  ...] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to ...ibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>.
  16 KB (2,658 words) - 16:02, 8 May 2024
• Parabola
  ...[[locus]] of points that are equidistant from a point (the [[focus]]) and a line (the [[directrix]]). ...la. The first is [[polynomial]] form: <math>y = a{x}^2+b{x}+c</math> where a, b, and c are [[constant]]s. This is useful for manipulating the polynomial
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• Geometric inequality
  A '''geometric inequality''' is an [[inequality]] involving various measures ...level geometry problems. It also provides the basis for the definition of a [[metric space]] in [[analysis]].
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• Brahmagupta's Formula
  ...ath>c</math>, <math>d</math> are the four side lengths and <math>s = \frac{a+b+c+d}{2}</math>. ...^2+b^2-2ab\cos B=c^2+d^2-2cd\cos D</math>. <math>\cos B=-\cos D</math>, so a little rearranging gives
  3 KB (465 words) - 18:31, 3 July 2023
• Ptolemy's theorem
  ...orem''' gives a relationship between the side lengths and the diagonals of a [[cyclic quadrilateral]]; it is the [[equality condition | equality case]] Given a [[cyclic quadrilateral]] <math>ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
  7 KB (1,198 words) - 20:39, 9 March 2024
• Number theory/Advanced
  ...in some subfield (like the reals or the rationals). One also needs to add a limit point, called the point at infinity. As <math>x\to \infty</math>, the ...points. This means that given 2 points on the curve, they can be added in a way that satisfies the normal laws of addition, like associativity, commuta
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• Euler's totient function
  '''Euler's totient function''' <math>\phi(n)</math> applied to a [[positive integer]] <math>n</math> is defined to be the number of positive ...eral [[prime factorization]] of <math>{n} = {p}_1^{e_1}{p}_2^{e_2} \cdots {p}_m^{e_m}</math>, one can compute <math>\phi(n)</math> using the formula <cm
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• Probability
  Before reading about the following topics, a student learning about probability should learn about introductory [[counti ...dergraduate level. Once measure theory is covered, probability does become a lot easier to use and understand.
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• Real number
  A '''real number''' is a number that falls on the real number line. It can have any value. Some exam The set of real numbers, denoted by <math>\mathbb{R}</math>, is a subset of [[complex number]]s(<math>\mathbb{C}</math>). Commonly used subse
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• Divisibility rules
  ...its of the number are divisible by <math>2^n</math>. Thus, in particular, a number is divisible by 2 if and only if its units digit is divisible by 2, A number is divisible by 3 or 9 if and only if the sum of its digits is divis
  8 KB (1,298 words) - 15:07, 23 May 2024
• Orthic triangle
  ...h>E</math>, and <math>F</math> denote the feet of the altitudes from <math>A</math>, <math>B</math>, and <math>C</math>, respectively. Then, <math>\tria ...e EFH = \theta = 90^\circ - \angle C</math>. The same argument (with <math>A</math> instead of <math>B</math>) shows that <math>\angle DFH = 90^\circ -
  8 KB (1,408 words) - 11:54, 8 December 2021
• Conic section
  ..., and every conic section can be described in [[Cartesian coordinates]] by a [[polynomial]] [[equation]] of degree two or less. ...cone or equivalently [[perpendicular]] to the axis. (This is really just a special case of the ellipse -- see the next bullet point.)
  5 KB (891 words) - 01:14, 9 January 2023
• Prime number
  A '''prime number''' (or simply '''prime''') is a [[positive integer]] <math>p>1</math> whose only positive [[divisor | divisors]] are 1 and itself. ...</math> must [[#Importance of Primes|have]] a prime factor, which leads to a direct contradiction.
  6 KB (985 words) - 12:38, 25 February 2024
• Function
  A '''function''' is a rule that maps one set of values to another set of values, assigning to eac ...is a ''function from <math>A</math> to <math>B</math>'' (written <math>f: A \to B</math>) if and only if
  10 KB (1,761 words) - 03:16, 12 May 2023
• Complementary counting
  ...ting may lead to a quick solution is the phrase "not" or "at least" within a problem statement. ...is the [[complement]] of <math>B</math>. In most instances, though, <math>A</math> is obvious from context and is committed from mention.
  8 KB (1,192 words) - 17:20, 16 June 2023
• Combinatorial identity
  ...binatorially by noting that any combination of <math>r</math> objects from a group of <math>m+n</math> objects must have some <math>0\le k\le r</math> o if(n==8){label(string(chew(n,i)),(11+n/2-i,-n),p=red+2.5);}
  12 KB (1,996 words) - 12:01, 18 May 2024
• Circle
  A '''circle''' is a geometric figure commonly used in Euclidean [[geometry]]. {{asy image|<asy>unitsize(2cm);draw(unitcircle,blue);</asy>|right|A basic circle.}}
  9 KB (1,581 words) - 18:59, 9 May 2024
• Ellipse
  An '''ellipse''' is a type of [[conic section]]. An ellipse is formed by cutting through a [[cone]] at an [[angle]].
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• Base numbers
  ...each number in 2746 is actually just a placeholder which shows how many of a certain power of 10 there are. The first digit to the left of the decimal ...digits 0-9. Usually, the base, or '''radix''', of a number is denoted as a subscript written at the right end of the number (e.g. in our example above
  4 KB (547 words) - 17:23, 30 December 2020
• Partition of an interval
  A '''partition of an interval''' is a division of an [[interval]] into several disjoint sub-intervals. Partition Let <math>[a,b]</math> be an interval of [[real number]]s.
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• Henstock-Kurzweil integral
  ...ble generalizations of the [[Integral|Riemann integral]], but it also uses a strikingly simple and elegant idea. It was developed independently by [[Ral Let <math>f:[a,b]\rightarrow\mathbb{R}</math>
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• 2001 AMC 12 Problems/Problem 2
  Let <math>P(n)</math> and <math>S(n)</math> denote the product and the sum, respectivel ...ath>P(23) = 6</math> and <math>S(23) = 5</math>. Suppose <math>N</math> is a
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• 2001 AMC 12 Problems/Problem 3
  The state income tax where Kristin lives is levied at the rate of <math>p\%</math> of the first <math>\textdollar 28000</math> of annual income plus <math>(p + 2)\%</math> of any amount above <math>\textdollar 28000</math>. Kristin
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• Square (geometry)
  A '''square''' is a [[quadrilateral]] in which all [[edge|sides]] have equal length and all [[a pair A, B, C, D;
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• Ptolemy's Inequality
  '''Ptolemy's Inequality''' is a famous inequality attributed to the Greek mathematician Ptolemy. The inequality states that in for four points <math>A, B, C, D </math> in the plane,
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• Pi
  Pi is the [[ratio]] of the [[circumference]] ([[perimeter]]) of a given [[circle]] to its [[diameter]]. It is approximately equal to 3.14159 ...i</math> is to inscribe a unit circle in a square of side length 2. Using a computer, random points are placed inside the square. Because the area of
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• Chain Rule
  ...<math>h'(x_0)</math>,<math>f'(g(x_0))</math>, and <math>g'(x_0)</math> is a matrix.) ...as <math>\Delta x</math> approaches <math>0</math>. This can be made into a rigorous proof. (But we do have to worry about the possibility that <math>
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• Muirhead's Inequality
  ...>p</math> [[Majorization|majorizes]] a sequence <math>q</math>, then given a set of positive reals <math>x_1,x_2,\cdots,x_n</math>: A common [[Brute forcing|bruteforce]] technique with inequalities is to clear
  8 KB (1,346 words) - 12:53, 8 October 2023
• Fundamental Theorem of Calculus
  The '''Fundamental Theorem of Calculus''' establishes a link between the two central operations of [[calculus]]: [[derivative|diffe ...imes <math>t=1</math> and <math>t=2</math> geometrically, as an area under a curve.
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• Isoperimetric Inequalities
  ...qualities''' are [[inequalities]] concerning the [[area]] of a figure with a given [[perimeter]]. They were worked on extensively by [[Lagrange]]. ...cle has the largest area. Conversely, of all plane figures with area <math>A</math>, the circle has the least perimeter.
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• Semiperimeter
  ...r <math>\frac{P}{2}</math>, where <math>P</math> is the total perimeter of a figure. It is typically denoted <math>s</math>. ...the simplest is <math>A=rs</math>, where <math>A</math> is the [[area]] of a [[triangle]] and <math>r</math> is the triangle's [[inradius]] (that is, th
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• Bretschneider's formula
  ...gth <math>a,b,c,d</math> (in that order) and [[diagonal]]s of length <math>p, q</math>. '''Bretschneider's formula''' states that the [[area]] <math>[ABCD]=\frac{1}{4} \cdot \sqrt{4p^2q^2-(b^2+d^2-a^2-c^2)^2}</math>.
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• Vector
  ...y, but also most abstractly, a vector is any object which is an element of a given vector space. ...(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]].
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• Law of Cosines
  ...sines''' is a theorem which relates the side-[[length]]s and [[angle]]s of a [[triangle]]. It can be derived in several different ways, the most common ...h>, <math>b</math> and <math>c</math> opposite [[angle]]s of measure <math>A</math>, <math>B</math> and <math>C</math>, respectively, the Law of Cosines
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• Iff
  It is also known as a [[conditional|biconditional]] statement. ...h>p\iff q</math> means <math>p\implies q</math> <b>and</b> <math>q\implies p</math> at the same time.
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• Angle bisector
  ...internal) angle bisector of <math>\angle BAC</math> is the line from <math>A</math> such that the angle between this line and <math>\overline{AB}</math> pair A,B,C,D,E,F;
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• Diophantine equation
  A '''Diophantine equation''' is an [[equation]] relating [[integer]] (or some ...closely tied to [[modular arithmetic]] and [[number theory]]. Often, when a Diophantine equation has infinitely many solutions, [[parametric form]] is
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• Exponential form
  ...> as <math>z=re^{i\theta}</math>, which is the general exponential form of a complex number. pair A,B,C,D,E;
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• Pascal's triangle
  ...ns the values from the [[binomial expansion]]; its various properties play a large role in [[combinatorics]]. ...2 is <math>{4 \choose 2} = 6</math>. Pascal's Triangle thus can serve as a "look-up table" for binomial expansion values. Also, many of the character
  5 KB (838 words) - 17:20, 3 January 2023
• Newton's Sums
  ...ficient way of finding the sums of [[root]]s of a [[polynomial]] raised to a power. They can also be used to derive several [[factoring]] [[identity|id Consider a polynomial <math>P(x)</math> of degree <math>n</math>,
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• 1985 IMO Problems/Problem 5
  A circle with center <math>O</math> passes through the vertices <math>A</math> and <math>C</math> of the triangle <math>ABC</math> and intersects t .../math> is the Miquel Point of quadrilateral <math>ACNK</math>, so there is a spiral similarity centered at <math>M</math> that takes <math>KN</math> to
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• Rational approximation
  ...r and closer to <math>x</math>, we may have to use larger and larger <math>p</math> and <math>q</math>. So, the reasonable question to ask here is how w ...th> can be approximated by a rational number <math>\frac{p}{q}</math> with a given denominator <math>q\ge 1</math> with an error not exceeding <math>\fr
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• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  ...power]], [[arithmetic mean]], [[geometric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> that says ..._1}+\cdots+\frac{1}{x_n}} \ge \sqrt[n_4]{\frac{x_1^{n_4}+\cdots+x_a^{n_4}}{a}}</cmath> where <math>n_1>1,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math
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• Goldbach Conjecture
  The '''Goldbach Conjecture''' is a yet unproven [[conjecture]] stating that every [[even integer]] greater tha In 1742, the Prussian mathematician Christian Goldbach wrote a letter to [[Leonhard Euler]] in which he proposed the following conjecture:
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• Quadratic residues
  ...] <math>m</math> if there is some integer <math>n</math> so that <math>n^2-a</math> is [[divisibility | divisible]] by <math>m</math>. ...d a\ \mathrm{ and }\ a\ \mathrm{\ is\ a\ quadratic\ nonresidue\ modulo\ }\ p. \end{cases}</math>
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• Power of a Point Theorem
  ...e lengths of the [[line segment]]s formed when two [[line]]s [[intersect]] a [[circle]] and each other. ...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
• AMC 12 Problems and Solutions
  ! Year || Test A || Test B | colspan="2"| [[2002 AMC 12P | AMC 12 P]]
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• Fundamental Theorem of Algebra
  ...tant]] [[polynomial]] with [[complex number|complex]] [[coefficient]]s has a complex [[root]]. In fact, every known proof of this theorem involves some ...oss of generality, suppose that <math>P</math> is [[monic]]. Then <math>1/P(z)</math> is an [[entire]] function; we wish to show that it is bounded. I
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• Triangle Inequality
  The '''Triangle Inequality''' says that in a [[nondegenerate]] [[triangle]] <math>ABC</math>: ...re <math>P</math> is the sum of the <math>a_j</math>, or <math>a_i < \frac{P}{2}</math>. Stated in another way, it says that in every polygon, each sid
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• Algebraic geometry
  ...[prime ideal]] of <math>R</math>. Then <math>V(I)=\{p\in\mathbb{A}^n\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine algebraic va ...s are algebraic varieties. A projective space <math>\mathbb{P}^n</math> is a quotient set with an equivalence class satisfying
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• Prime Number Theorem
  constants <math>A</math> and <math>B</math>, <cmath> \pi(x) \sim \frac{x}{A \log x - B} . </cmath>
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• Note that with two sequences <math>\mathbf{a}</math> and <math>\mathbf{b}</math>, and <math>\lambda_a = \lambda_b = 1/2< ...nces of nonnegative reals, and let <math>\{ \lambda_i \}_{i=1}^n</math> be a sequence of nonnegative reals such that <math>\sum \lambda = 1</math>. The
  4 KB (774 words) - 12:12, 29 October 2016
• Euler's Polyhedral Formula
  Let <math>P</math> be any [[convex]] [[polyhedron]], and let <math>V</math>, <math>E</m ...yhedron lie in the interior of the polyhedron rather than along an edge or a face? (1988 AIME #10)
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• Mock AMC
  ...an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Prob == Tips for Writing a Mock AMC ==
  51 KB (6,175 words) - 20:58, 6 December 2023
• Permutation
  ...are <math>r!</math> (the [[factorial]] of <math>r</math>) permutations of a set with <math>r</math> distinct objects. ...ite]] 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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• Algebraic topology
  ...of [[topology]] using methods from [[abstract algebra]]. In general, given a [[topological space]], we can associate various algebraic objects, such as ...a,1)=q(a)</math>, and <math>g(0,b)=g(1,b)=x</math>. We call <math>g</math> a [[homotopy]]. Now define <math>\pi_1(X)=L/\sim</math>. That is, we equate a
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• Bertrand's Postulate
  ...t is, in fact, a theorem. A more widely known version states that there is a prime between <math>n</math> and <math>2n</math>. ...in the [[Prime Number Theorem|prime number theorem]] article but requires a closer look at the [[combinations|binomial coefficient]] <math>\binom{2n}{n
  2 KB (309 words) - 21:43, 11 January 2010
• 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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• Uncountable
  A [[set]] <math>S</math> is said to be '''uncountable''' if there is no [[inj ...the elements of <math>A</math> (in other words, take an injection <math>f: A \to \mathbb{N}</math>, and denote <math>\omega_i = f(i)</math>).
  2 KB (403 words) - 20:53, 13 October 2019
• Subset
  ...r than <math>B</math> itself. In the latter case, <math>A</math> is called a ''proper subset''. The following is a true statement:
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• 2006 AIME I Problems
  In quadrilateral <math> ABCD , \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <mat ...th> and let <math> S </math> be the sum of the elements of <math> \mathcal{A}. </math> Find the number of possible values of <math> S. </math>
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• 2006 AIME I Problems/Problem 14
  A tripod has three legs each of length <math>5</math> feet. When the tripod i triple O=(0,0,0),T=(0,0,5),C=(0,3,0),A=(-3*3^.5/2,-3/2,0),B=(3*3^.5/2,-3/2,0);
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• LaTeX:Symbols
  This article provides a short list of commonly used LaTeX symbols. .../math> on the web, (technically an AJAX library simulating it.)) maintains a [http://docs.mathjax.org/en/latest/tex.html#supported-latex-commands list o
  16 KB (2,324 words) - 16:50, 19 February 2024
• 2006 AIME I Problems/Problem 8
  ...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</math pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6),
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• 2006 AIME I Problems/Problem 5
  ...n be written as <math> a\sqrt{2}+b\sqrt{3}+c\sqrt{5}, </math> where <math> a, b, </math> and <math> c </math> are [[positive]] [[integer]]s. Find <math> <cmath> a\sqrt{2}+b\sqrt{3}+c\sqrt{5} = \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2
  3 KB (439 words) - 18:24, 10 March 2015
• 2022 AIME I Problems/Problem 1
  ...h the two points <math>(16,54)</math> and <math>(20,53).</math> Find <math>P(0) + Q(0).</math> .../math> and <math>Q(x)</math> cancel, we conclude that <math>R(x)</math> is a linear polynomial.
  4 KB (670 words) - 13:03, 13 November 2023
• 2021 JMC 10
  <math>\textbf{(A) } 3 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 9 \qqu <math>\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2006 AMC 12B Problems
  \text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } <math>\text {(A) } - 72 \qquad \text {(B) } - 27 \qquad \text {(C) } - 24 \qquad \text {(D)
  13 KB (2,058 words) - 12:36, 4 July 2023
• 2005 AMC 12A Problems
  {{AMC12 Problems|year=2005|ab=A}} (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D})
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2004 AMC 12A Problems
  {{AMC12 Problems|year=2004|ab=A}} <math>\text{(A) } 0.0029 \qquad \text{(B) } 0.029 \qquad \text{(C) } 0.29 \qquad \text{(D)
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2003 AMC 12A Problems
  {{AMC12 Problems|year=2003|ab=A}} <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \
  13 KB (1,955 words) - 21:06, 19 August 2023
• 2002 AMC 12A Problems
  {{AMC12 Problems|year=2002|ab=A}} <math> \mathrm{(A) \ } \frac{7}{2}\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mat
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2000 AMC 12 Problems
  <math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\ <math>\textbf{(A)}\ 2000^{2001} \qquad \textbf{(B)}\ 4000^{2000} \qquad \textbf{(C)}\ 2000^{
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2001 AMC 12 Problems
  <math>\text{(A)}\ 2S + 3\qquad \text{(B)}\ 3S + 2\qquad \text{(C)}\ 3S + 6 \qquad\text{(D) Let <math>P(n)</math> and <math>S(n)</math> denote the product and the sum, respectivel
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2002 AMC 12B Problems
  ...numbers in the set <math>\{9, 99, 999, 9999, \ldots, 999999999\}</math> is a <math>9</math>-digit number <math>M</math>, all of whose digits are distinc <math>\mathrm{(A)}\ 0
  10 KB (1,547 words) - 04:20, 9 October 2022
• 2003 AMC 12B Problems
  \text {(A) } -1 \qquad \text {(B) } -\frac{2}{3} \qquad \text {(C) } \frac{2}{3} \qqu ...ks. A green pill costs 1 dollar more than a pink pill, and Al's pills cost a total of 546 dollars for the two weeks. How much does one green pill cost?
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2005 AMC 12B Problems
  A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at t \mathrm{(A)}\ 100 \qquad
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12B Problems/Problem 12
  ...(p,p)</math> and <math>y</math>-intercept <math>(0,-p)</math>, where <math>p\ne 0</math>. What is <math>b</math>? ...t {(B) } 0 \qquad \text {(C) } 2 \qquad \text {(D) } 4 \qquad \text {(E) } p</math>
  1 KB (239 words) - 22:20, 17 October 2020
• 2006 AMC 12B Problems/Problem 13
  pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/ draw(B--C--D--A--B--F--D--E--B);
  3 KB (447 words) - 03:49, 16 January 2021
• 2006 AMC 12B Problems/Problem 15
  ...s <math> C</math> and <math> D</math> are on the circle centered at <math> P</math>, such that <math> \overline{AD}</math> and <math> \overline{BC}</mat pair A, B, C, D;
  3 KB (458 words) - 16:40, 6 October 2019
• 2006 AMC 12B Problems/Problem 23
  ...re <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>? pair A = (B.y,0);
  7 KB (1,169 words) - 14:04, 10 June 2022
• 2006 AMC 12A Problems/Problem 14
  ..."change" received in the form of goats or pigs as necessary. (For example, a <math>390</math> dollar debt could be paid with two pigs, with one goat rec <math> \textbf{(A) } 5\qquad \textbf{(B) } 10\qquad \textbf{(C) } 30\qquad \textbf{(D) } 90\q
  3 KB (442 words) - 03:13, 8 August 2022
• 2006 AMC 12A Problems/Problem 22
  ...>. The [[probability]] that three entire sides of hexagon are visible from a randomly chosen point on the circle is <math>1/2</math>. What is <math>r</m <math> \mathrm{(A) \ } 2\sqrt{2}+2\sqrt{3}\qquad \mathrm{(B) \ } 3\sqrt{3}+\sqrt{2}\qquad \ma
  2 KB (343 words) - 15:39, 14 June 2023

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