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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
• 2006 AMC 12A Problems/Problem 24
  <math> \mathrm{(A) \ } 6018\qquad \mathrm{(B) \ } 671,676\qquad \mathrm{(C) \ } 1,007,514\qqu <cmath>\sum_{a+b+c=2006}{\frac{2006!}{a!b!c!}x^ay^bz^c}</cmath>
  8 KB (1,332 words) - 17:37, 17 September 2023
• 2005 AMC 12B Problems/Problem 11
  <math>\textbf{(A) }\ {{{\frac{1}{4}}}} \qquad \textbf{(B) }\ {{{\frac{2}{5}}}} \qquad \textb ...>14</math> ways to attain a sum of <math>20</math> or greater, so there is a total probability of <math>\dfrac{14}{28}=\boxed{\textbf{(D) }\frac{1}{2}}<
  4 KB (607 words) - 21:01, 20 May 2023
• 2005 AMC 12B Problems/Problem 12
  ...<math>m,n,</math> and <math>p</math> is zero. What is the value of <math>n/p</math>? <math>\textbf{(A) }\ {{{1}}} \qquad \textbf{(B) }\ {{{2}}} \qquad \textbf{(C) }\ {{{4}}} \qq
  2 KB (317 words) - 12:27, 16 December 2021
• Root (polynomial)
  ...<math>a</math> in the [[domain]] of the function such that <math>f(a) = (x-a) = 0</math>. ...and so on, there are in fact exactly <math>n</math> complex roots of <math>P(x)</math>.
  8 KB (1,427 words) - 21:37, 13 March 2022
• 1995 AHSME Problems/Problem 6
  The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the fa path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3);
  1 KB (168 words) - 00:49, 14 October 2013
• Set
  ...ms, see [[Zermelo-Fraenkel Axioms]]. In this article we shall present just a brief discussion of the most common properties of sets and operations relat ...g: <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
• Computer science
  ...nder certain constraints. For example, the [[P versus NP]] problem is such a problem.
  1 KB (157 words) - 23:11, 16 November 2023
• 2006 USAMO Problems
  ...at there exist integers <math>m</math> and <math>n</math> with <math>0<m<n<p</math> and ...ce\frac{sm}{p}\right\rbrace<\left\lbrace\frac{sn}{p}\right\rbrace<\frac{s}{p}</cmath>
  3 KB (520 words) - 09:24, 14 May 2021
• Prime ideal
  ...eteq P</math> we have either <math>I\subseteq P</math> or <math>J\subseteq P</math>. ...ath> if <math>ab\in P</math> then either <math>a\in P</math> or <math>b\in P</math>.
  967 bytes (176 words) - 18:08, 7 April 2012
• 2005 AIME II Problems/Problem 13
  ...ies <math> 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 <m ...it has roots at <math>17</math> and <math>24</math>. Hence <math>P(x)-x+7=A(x-17)(x-24)</math>. In particular, this means that
  4 KB (642 words) - 14:55, 12 August 2019
• 2005 AIME II Problems
  A game uses a deck of <math> n </math> different cards, where <math> n </math> is an inte ...e indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one r
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 8
  ...n </math> is not divisible by the square of any [[prime]], find <math> m+n+p. </math> pair A = OP(cir3, t), B = IP(cir3, t), T1 = IP(cir1, t), T2 = IP(cir2, t);
  4 KB (693 words) - 13:03, 28 December 2021
• 2005 AIME II Problems/Problem 12
  ...r </math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math> ...D--A);draw(E--O--F);draw(G--O); dot(A^^B^^C^^D^^E^^F^^G^^O); label("\(A\)",A,(-1,1));label("\(B\)",B,(1,1));label("\(C\)",C,(1,-1));label("\(D\)",D,(-1,
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME II Problems/Problem 11
  ...math> be a positive integer, and let <math> a_0, a_1,\ldots,a_m </math> be a sequence of reals such that <math>a_0 = 37, a_1 = 72, a_m = 0, </math> and Thus the product <math>a_{k}a_{k+1}</math> is a [[monovariant]]: it decreases by 3 each time <math>k</math> increases by 1.
  3 KB (499 words) - 18:52, 21 November 2022
• 2005 AIME I Problems
  ...ngent to two circles adjacent to it. All circles are internally tangent to a circle <math> C </math> with radius 30. Let <math> K </math> be the area of ...members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Fi
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 3
  ...p\cdot q</math> or <math>n=p^3</math> for distinct [[prime number]]s <math>p</math> and <math>q</math>. ...h>47</math>) so there are <math> {15 \choose 2} =105</math> ways to choose a pair of primes from the list and thus <math>105</math> numbers of the first
  2 KB (249 words) - 09:37, 23 January 2024
• 2005 AIME I Problems/Problem 6
  ...nonreal roots of <math> x^4-4x^3+6x^2-4x=2005. </math> Find <math> \lfloor P\rfloor. </math> ...6}</math>. <math>44^2 = 1936 < 2006 < 2025 = 45^2</math> so <math>\lfloor P \rfloor = 1 + 44 = \boxed{045}</math>.
  4 KB (686 words) - 01:55, 5 December 2022
• 2005 AIME I Problems/Problem 7
  ...p </math> and <math> q </math> are [[positive]] [[integer]]s, find <math> p+q. </math> label("$A$",(0,0),SW);
  4 KB (567 words) - 20:20, 3 March 2020
• 2005 AIME I Problems/Problem 9
  ...</math> and <math> c </math> are [[positive integer]]s, find <math> a+b+c+p+q+r. </math> ...}{6} = \frac{2}{3}</math> of all 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
  ...has [[slope]] <math> -5. </math> Find the largest possible value of <math> p+q. </math> pair A=(15,32), B=(12,19), C=(23,20), M=B/2+C/2, P=(17,22);
  5 KB (852 words) - 21:23, 4 October 2023
• 2005 AIME I Problems/Problem 15
  ...math> is not [[divisor | divisible]] by the [[perfect square | square]] of a prime, find <math> m+n. </math> pair A=(0,0),B=(26,0),C=IP(circle(A,10),circle(B,20)),D=(B+C)/2,I=incenter(A,B,C);
  5 KB (906 words) - 23:15, 6 January 2024
• 2005 AIME II Problems/Problem 15
  ...p </math> and <math> q </math> are relatively prime integers, find <math> p+q. </math> pair A = (-5, 12), B = (5, 12), C = (0, 0);
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2005 AIME II Problems/Problem 14
  ...ong \angle CAD. </math> Given that <math> BE=\frac pq </math> where <math> p </math> and <math> q </math> are relatively prime positive integers, find < pair A = origin;
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2004 AIME I Problems/Problem 13
  The polynomial <math> P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17} </math> has <math>34</math> complex roo We see that the expression for the [[polynomial]] <math>P</math> is very difficult to work with directly, but there is one obvious tr
  2 KB (298 words) - 20:02, 4 July 2013
• 2004 AIME I Problems/Problem 3
  ...that do not belong to the same face. How many space diagonals does <math> P </math> have? ...f the [[polyhedron]] determines either an [[edge]], a face [[diagonal]] or a space diagonal. We have <math>{26 \choose 2} = \frac{26\cdot25}2 = 325</ma
  1 KB (156 words) - 17:56, 1 January 2016
• 2004 AIME I Problems/Problem 4
  ...re. The [[midpoint]]s of the line segments in set <math> S </math> enclose a region whose [[area]] to the nearest hundredth is <math>k</math>. Find <ma ...midpoints lying on the sides determined by vertex <math>(0,0)</math> form a quarter-[[circle]] with [[radius]] 1.
  3 KB (532 words) - 09:22, 11 July 2023
• 2004 AIME I Problems/Problem 7
  Let our [[polynomial]] be <math>P(x)</math>. ...h>P(x)</math> is <math>-1 + 2 - 3 + \ldots + 14 - 15 = -8</math>, so <math>P(x) = 1 -8x + Cx^2 + Q(x)</math>, where <math>Q(x)</math> is some polynomial
  6 KB (941 words) - 11:37, 27 May 2024
• 2004 AIME I Problems
  The digits of a positive integer <math> n </math> are four consecutive integers in decreasi ...he absolute value of the difference between the greatest element of <math> A </math> and the greatest element of <math> B </math> is 99. Find <math> m.
  9 KB (1,434 words) - 13:34, 29 December 2021
• 2004 AIME II Problems/Problem 15
  ...eated <math>8</math> more times. After the last fold, the strip has become a stack of <math>1024</math> unit squares. How many of these squares lie belo ...number of squares below the <math>n</math> square after the final fold in a strip of length <math>2^{k}</math>.
  6 KB (899 words) - 20:58, 12 May 2022
• 2004 AIME II Problems/Problem 14
  Consider a string of <math> n </math> <math> 7 </math>'s, <math> 7777\cdots77, </math> ...</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
  Let <math> ABCDE </math> be a [[convex]] [[pentagon]] with <math> AB \parallel CE, BC \parallel AD, AC \p ...parallel CE, BC \parallel AD, </math> it follows that <math>ABCF</math> is a [[parallelogram]], and so <math>\triangle ABC \cong \triangle CFA</math>. A
  3 KB (486 words) - 22:15, 7 April 2023
• 2004 AIME II Problems/Problem 12
  ...k </math> and <math> p </math> are [[relatively prime]]. Find <math> k+m+n+p. </math> pair A=(0,0), B=(6,0), D=(1, 24^.5), C=(5,D.y), O = (3,(r^2 + 6*r)^.5);
  3 KB (431 words) - 23:21, 4 July 2013
• 2004 AIME II Problems/Problem 10
  ...> and <math> q </math> are relatively prime positive integers. Find <math> p+q. </math> A positive integer <math>n</math> has exactly two 1s in its binary representa
  8 KB (1,283 words) - 19:19, 8 May 2024
• 2004 AIME II Problems
  ...a, b, c, d, e, </math> and <math> f </math> are positive integers, <math> a </math> and <math> e </math> are relatively prime, and neither <math> c </m A jar has 10 red candies and 10 blue candies. Terry picks two candies at rand
  9 KB (1,410 words) - 05:05, 20 February 2019
• 1983 AIME Problems
  ...th> and <math>z</math> all exceed <math>1</math> and let <math>w</math> be a positive number such that <math>\log_xw=24</math>, <math>\log_y w = 40</mat ...ue taken by <math>f(x)</math> for <math>x</math> in the [[interval]] <math>p \leq x\leq15</math>.
  7 KB (1,104 words) - 03:13, 27 May 2024
• 2019 AIME I Problems/Problem 2
  ...>J</math>. The value of <math>B - J</math> is at least <math>2</math> with a probability that can be expressed in the form <math>\frac{m}{n}</math> wher ...ath>1 + 18 \cdot 2 + 1 = 38</math> ones that form <math>P</math>, so <math>P = \frac{38}{380} = \frac{1}{10}</math>. Therefore the answer is <math>\frac
  5 KB (830 words) - 22:15, 28 December 2023
• 1984 AIME Problems
  ...of <math>\triangle ABC</math> such that when lines are drawn through <math>P</math> parallel to the sides of <math>\triangle ABC</math>, the resulting s pair A=(0,0),B=(12,0),C=(4,5);
  6 KB (933 words) - 01:15, 19 June 2022
• 1986 AIME Problems
  The pages of a book are numbered <math>1_{}^{}</math> through <math>n_{}^{}</math>. When t ...r point <math>P</math> is then drawn, and segments are drawn through <math>P</math> parallel to the sides of the triangle. If these three segments are o
  5 KB (847 words) - 15:48, 21 August 2023
• 1987 AIME Problems
  ...er we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the p ...r which <math>[a,b] = 1000</math>, <math>[b,c] = 2000</math>, and <math>[c,a] = 2000</math>.
  6 KB (869 words) - 15:34, 22 August 2023
• 1988 AIME Problems
  Let <math>\frac{m}{n}</math>, in lowest terms, be the probability that a randomly chosen positive divisor of <math>10^{99}</math> is an integer mult ...C</math>, <math>\tan \angle CAB = 22/7</math>, and the altitude from <math>A</math> divides <math>BC</math> into segments of length <math>3</math> and <
  6 KB (902 words) - 08:57, 19 June 2021
• 1989 AIME Problems
  Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be dra ...ose <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
  ...neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. ...}</math> be a regular <math>r~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that ea
  6 KB (870 words) - 10:14, 19 June 2021
• 1991 AIME Problems
  Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerat Suppose <math>r^{}_{}</math> is a real number for which
  7 KB (1,106 words) - 22:05, 7 June 2021
• 1992 AIME Problems
  A positive integer is called ascending if, in its decimal representation, the ...she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math>0.500</math>. During the weekend, s
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1993 AIME Problems
  ...n for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went :(a) the winner caught <math>15</math> fish;
  8 KB (1,275 words) - 06:55, 2 September 2021
• 1994 AIME Problems
  ...\,</math> consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence ...circle of radius 20. Square <math>ABCD\,</math> is constructed with <math>A\,</math> and <math>B\,</math> on the larger circle, <math>\overline{CD}\,</
  7 KB (1,141 words) - 07:37, 7 September 2018
• 1995 AIME Problems
  ...object reaches <math>(2,2)</math> in six or fewer steps. Given that <math>p</math> can be written in the form <math>m/n,</math> where <math>m</math> an ...us <math>9</math>. The circle of radius <math>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the le
  6 KB (1,000 words) - 00:25, 27 March 2024
• 1996 AIME Problems
  ...mn, or diagonal is the same value. The figure shows four of the entries of a magic square. Find <math>x</math>. ...at <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer?
  6 KB (931 words) - 17:49, 21 December 2018
• 1997 AIME Problems
  ...he two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah sho ...ny of which could be zero. It is desired to approximate <math>r</math> by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The
  7 KB (1,098 words) - 17:08, 25 June 2020
• 1998 AIME Problems
  ...Extend <math>\overline{DA}</math> through <math>A</math> to a point <math>P,</math> and let <math>\overline{PC}</math> meet <math>\overline{AB}</math> ...egative]] term encountered. What positive integer <math>x</math> produces a sequence of maximum length?
  7 KB (1,084 words) - 02:01, 28 November 2023
• 2000 AIME I Problems
  ...Let <math>A = (u,v)</math>, let <math>B</math> be the reflection of <math>A</math> across the line <math>y = x</math>, let <math>C</math> be the reflec ...icients of <math>x^{2}</math> and <math>x^{3}</math> are equal. Find <math>a + b</math>.
  7 KB (1,204 words) - 03:40, 4 January 2023
• 2001 AIME I Problems
  A finite set <math>\mathcal{S}</math> of distinct real numbers has the follow ...and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>.
  7 KB (1,212 words) - 22:16, 17 December 2023
• 2002 AIME I Problems
  ...plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right- ...where <math>p</math> and <math>q</math> are positive integers. Find <math>p+q</math>.
  8 KB (1,374 words) - 21:09, 27 July 2023
• 2003 AIME I Problems
  ...centric circles with radii <math> 1, 2, 3, \dots, 100 </math> are drawn in a plane. The interior of the circle of radius 1 is colored red, and each regi ...et <math> \mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}. </math> Susan makes a list as follows: for each two-element subset of <math> \mathcal{S}, </math>
  6 KB (965 words) - 16:36, 8 September 2019
• 2000 AIME II Problems
  A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola <math>x^2 - y^ ...h>m/n</math> be the probability that two randomly selected cards also form a pair, where <math>m</math> and <math>n</math> are relatively prime positive
  6 KB (947 words) - 21:11, 19 February 2019
• 2001 AIME II Problems
  ...rom left to right, each pair of consecutive digits of <math>N</math> forms a perfect square. What are the leftmost three digits of <math>N</math>? Each of the 2001 students at a high school studies either Spanish or French, and some study both. The numb
  8 KB (1,282 words) - 21:12, 19 February 2019
• 2002 AIME II Problems
  Three [[vertex|vertices]] of a [[cube]] are <math>P=(7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=(11,3,9)</math>. What ...th>b - a</math> is the [[Perfect square|square]] of an integer. Find <math>a + b + c</math>.
  7 KB (1,177 words) - 15:42, 11 August 2023
• 2003 AIME II Problems
  ...<math>C</math>, and <math>C</math> is never immediately followed by <math>A</math>. How many seven-letter good words are there? In a regular tetrahedron, the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1983 AIME Problems/Problem 2
  ...ue taken by <math>f(x)</math> for <math>x</math> in the [[interval]] <math>p \leq x\leq15</math>. ...at <math>|x-p|=x-p</math>, <math>|x-15|=15-x</math>, and <math>|x-p-15|=15+p-x</math>.
  1 KB (184 words) - 20:16, 14 January 2023
• 1983 AIME Problems/Problem 4
  ...hat of <math>BC</math> is <math>2</math> cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B< A=r*dir(45),B=(A.x,A.y-r);
  11 KB (1,741 words) - 22:40, 23 November 2023
• 1983 AIME Problems/Problem 5
  An alternate way to solve this is to let <math>x=a+bi</math> and <math>y=c+di</math>. Because we are looking for a value of <math>x+y</math> that is real, we know that <math>d=-b</math>, and
  4 KB (672 words) - 10:17, 17 March 2023
• 1983 AIME Problems/Problem 7
  ...three had been sitting next to each other. If <math>P</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? We will place <math>A</math>, <math>B</math>, and <math>C</math> with and without the restriction
  9 KB (1,392 words) - 20:37, 19 January 2024
• 1983 AIME Problems/Problem 8
  ...ppears twice in the denominator. Thus, we need <math>p</math> to appear as a factor at least three times in the numerator, so <math>3p<200</math>. The l ...>p</math> goes larger in value, the number of factors of <math>p</math> in a number decreases,)
  2 KB (249 words) - 23:25, 11 May 2024
• 1983 AIME Problems/Problem 14
  ...<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> have equal length. ...draw(O2--O1); dot(O1); dot(O2); draw(Q--R); label("$Q$",Q,NW); label("$P$",P,1.5*dir(80)); label("$R$",R,NE); label("12",waypoint(O1--O2,0.4),S);</asy>
  13 KB (2,151 words) - 17:48, 27 May 2024
• 1983 AIME Problems/Problem 15
  ...r arc <math>AB</math> is a rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <mat pair A=(-0.91,-0.41);
  20 KB (3,497 words) - 15:37, 27 May 2024
• 1984 AIME Problems/Problem 3
  ...math>\triangle ABC</math> such that when [[line]]s are drawn through <math>P</math> [[parallel]] to the sides of <math>\triangle ABC</math>, the resulti pair A=(0,0),B=(12,0),C=(4,5);
  4 KB (726 words) - 13:39, 13 August 2023
• 1984 AIME Problems/Problem 9
  ...math> is <math>12 \mbox { cm}^2</math>. These two faces meet each other at a <math>30^\circ</math> angle. Find the [[volume]] of the tetrahedron in <mat path3 rightanglemark(triple A, triple B, triple C, real s=8)
  6 KB (947 words) - 20:44, 26 November 2021
• 1984 AIME Problems/Problem 14
  Clearly, if <math>x \ge 44</math>, it can be expressed as a sum of 2 odd composites. However, if <math>x = 42</math>, it can also be ex ...prime quintuplet is <math>5,11,17,23,</math> and <math>29</math>, yielding a maximal answer of 38. Since <math>38-25=13</math>, which is prime, the answ
  8 KB (1,346 words) - 01:16, 9 January 2024
• 1995 IMO Problems/Problem 2
  Let <math>a, b, c</math> be positive real numbers such that <math>abc = 1</math>. Prove <cmath> \frac{1}{a^3(b+c)} + \frac{1}{b^3(c+a)} + \frac{1}{c^3(a+b)} \geq \frac{3}{2}. </cmath>
  6 KB (1,122 words) - 12:23, 6 January 2022
• 1985 AIME Problems/Problem 14
  ...h of the two players earned <math>\frac{1}{2}</math> point if the game was a tie. After the completion of the tournament, it was found that exactly half ...+ 90</math>. However, there was one point earned per game, and there were a total of <math>{n + 10 \choose 2} = \frac{(n + 10)(n + 9)}{2}</math> games
  5 KB (772 words) - 22:14, 18 June 2020
• 1985 AIME Problems/Problem 12
  ...= \frac{n}{729}</math> be the probability that the bug is at vertex <math>A</math> when it has crawled exactly <math>7</math> meters. Find the value of ...when it has crawled exactly <math>k</math> meters. We wish to find <math>p=P(7).</math>
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1985 AIME Problems/Problem 11
  ...of points <math>P</math> such that the sum of the distances between <math>P</math> and the two foci is constant. Let <math>F_1 = (9, 20)</math>, <math ...n our original path has shortest length. This occurs exactly when we have a straight-line path, and by the above argument, this path passes through the
  5 KB (932 words) - 17:00, 1 September 2020
• 1985 AIME Problems/Problem 6
  ...maller triangles by [[line]]s drawn from the [[vertex | vertices]] through a common interior point. The [[area]]s of four of these triangles are as indi ...two pairs of bases are actually the same, and thus in the same ratio. As a result, we have:
  5 KB (789 words) - 03:09, 23 January 2023
• 1986 AIME Problems/Problem 14
  ..., and <math>\frac{15}{\sqrt{10}}</math>. Determine the [[volume]] of <math>P</math>. We invert the above equations to get a system of linear equations in <math>\frac {1}{h^2}</math>, <math>\frac {1}{
  2 KB (346 words) - 13:13, 22 July 2020
• 1986 AIME Problems/Problem 9
  ...t]] <math>P</math> is then drawn, and [[segment]]s are drawn through <math>P</math> [[parallel]] to the sides of the [[triangle]]. If these three segmen pair C=(0,0),A=(510,0),B=IP(circle(C,450),circle(A,425));
  11 KB (1,850 words) - 18:07, 11 October 2023
• 1987 AIME Problems/Problem 13
  ...ng]] order by means of one or more "bubble passes". A bubble pass through a given sequence consists of comparing the second term with the first term, a ...ter one bubble pass, in the <math>30^{\mbox{th}}</math> place. Find <math>p + q</math>.
  3 KB (514 words) - 21:27, 31 December 2023
• 1987 AIME Problems/Problem 9
  ...ath> has [[right angle]] at <math>B</math>, and contains a [[point]] <math>P</math> for which <math>PA = 10</math>, <math>PB = 6</math>, and <math>\angl pair A, B, C, P;
  1 KB (200 words) - 18:44, 5 February 2024
• 1987 AIME Problems/Problem 7
  ...r which <math>[a,b] = 1000</math>, <math>[b,c] = 2000</math>, and <math>[c,a] = 2000</math>. ...<math>m, j</math> equal to 3. This gives 7 possible triples <math>(j, m, p)</math>: <math>(0, 3, 4), (1, 3, 4), (2, 3, 4), (3, 3, 4), (3, 2, 4), (3, 1
  3 KB (547 words) - 22:54, 4 April 2016
• 1987 AIME Problems/Problem 6
  ...th>h=53</math>. In the same way, we find that the perpendicular from <math>P</math> to <math>AD</math> is <math>53</math>. So <math>AB=53*2+87=\boxed{19 ...- a + 87</cmath> <cmath>2a = 174</cmath> <cmath>a = 87</cmath> Since <math>a + CB = XY</math>, <math>XY = 19 + 87 = 106</math>, and <math>AB = 106 + 87
  3 KB (530 words) - 07:46, 1 June 2018
• 1987 AIME Problems/Problem 3
  ...ositive]] [[integer|integral]] divisor other than 1 and the number itself. A natural number greater than 1 will be called ''nice'' if it is equal to the ...ath> denote the product of the distinct proper divisors of <math>n</math>. A number <math>n</math> is ''nice'' in one of two instances:
  3 KB (511 words) - 09:29, 9 January 2023
• 1988 AIME Problems/Problem 14
  ...rmula for <math>P' (x', y')</math> on <math>C^*</math>. Both points lie on a line that is [[perpendicular]] to <math>y=2x</math>, so the slope of <math> The matrix for a reflection about the polar line <math>\theta = \alpha, \alpha+\pi</math> is
  4 KB (700 words) - 17:21, 3 May 2021
• 1988 AIME Problems/Problem 13
  ...ath> and <math>b</math> are integers such that <math>x^2 - x - 1</math> is a factor of <math>ax^{17} + bx^{16} + 1</math>. 0 = ax^{17} + bx^{16} + 1 = a(F_{17}\cdot x + F_{16}) + b(F_{16}\cdot x + F_{15}) + 1 &\Longrightarrow (a
  10 KB (1,585 words) - 03:58, 1 May 2023
• 1988 AIME Problems/Problem 12
  ...ments indicated in the figure. Find the product <math>abc</math> if <math>a + b + c = 43</math> and <math>d = 3</math>. <math>\frac {d}{a + d} = \frac {[PBC]}{[ABC]}</math>
  4 KB (727 words) - 23:37, 7 March 2024
• 1989 AIME Problems/Problem 15
  Point <math>P</math> is inside <math>\triangle ABC</math>. Line segments <math>APD</math> ...th>RST</math>. We'll make use of the following fact: if <math>P</math> is a point in the interior of triangle <math>XYZ</math>, and line <math>XP</math
  13 KB (2,091 words) - 00:20, 26 October 2023
• 1989 AIME Problems/Problem 12
  Let <math>ABCD</math> be a [[tetrahedron]] with <math>AB=41</math>, <math>AC=7</math>, <math>AD=18</ma pair A,B,C,D,M,P,Q;
  2 KB (376 words) - 13:49, 1 August 2022
• 1989 AIME Problems/Problem 10
  ...h>\beta</math>, <math>\gamma</math>, be the angles opposite them. If <math>a^2+b^2=1989c^2</math>, find pair A = (0,0), B = (3, 0), C = (1, 4);
  8 KB (1,401 words) - 21:41, 20 January 2024
• 1989 AIME Problems/Problem 6
  ...the same time Allie leaves <math>A</math>, Billie leaves <math>B</math> at a speed of <math>7</math> meters per second and follows the [[straight]] path pair A=(0,0),B=(10,0),C=6*expi(pi/3);
  6 KB (980 words) - 15:08, 14 May 2024
• 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 ...by <math>5</math> but not <math>10</math>. The only odd number that yields a single digit <math>d</math> and 25 at the end of the three digit number is
  3 KB (499 words) - 22:17, 29 March 2024
• 1990 AIME Problems/Problem 15
  Find <math>ax^5 + by^5</math> if the real numbers <math>a,b,x,</math> and <math>y</math> satisfy the equations Set <math>S = (x + y)</math> and <math>P = xy</math>. Then the relationship
  4 KB (644 words) - 16:24, 28 May 2023
• 1990 AIME Problems/Problem 14
  ...ments <math>\overline{CP}</math> and <math>\overline{DP}</math>, we obtain a [[triangular pyramid]], all four of whose faces are [[isosceles triangle]]s pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6);
  7 KB (1,086 words) - 08:16, 29 July 2023
• 1990 AIME Problems/Problem 7
  .../math> can be written in the form <math>ax+2y+c=0_{}^{}</math>. Find <math>a+c_{}^{}</math>. pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17);
  8 KB (1,319 words) - 11:34, 22 November 2023
• 1991 AIME Problems/Problem 13
  ...s that, when two socks are selected randomly without replacement, there is a probability of exactly <math>\frac{1}{2}</math> that both are red or both a ...e socks, respectively. Also, let <math>t=r+b</math>. The probability <math>P</math> that when two socks are drawn randomly, without replacement, both ar
  7 KB (1,328 words) - 20:24, 5 February 2024
• 1991 AIME Problems/Problem 12
  ...in [[rectangle]] <math>ABCD^{}_{}</math> so that [[vertex|vertices]] <math>P^{}_{}</math>, <math>Q^{}_{}</math>, <math>R^{}_{}</math>, and <math>S^{}_{} ...el("\(Q\)",Q,E);label("\(R\)",R,SW);label("\(S\)",S,W); label("\(15\)",B/2+P/2,N);label("\(20\)",B/2+Q/2,E);label("\(O\)",O,SW); </asy></center>
  8 KB (1,270 words) - 23:36, 27 August 2023
• 1991 AIME Problems/Problem 11
  ...ath>c^{}_{}</math> is not divisible by the square of any prime. Find <math>a+b+c^{}_{}</math>. for (int a=1; a<24; a+=2)
  4 KB (740 words) - 17:46, 24 May 2024
• 1991 AIME Problems/Problem 10
  ...ath>S_b^{}</math> in alphabetical order. When <math>p</math> is written as a [[fraction]] in [[irreducible fraction|lowest terms]], what is its [[numera Let us make a chart of values in alphabetical order, where <math>P_a,\ P_b</math> are the
  5 KB (813 words) - 06:10, 25 February 2024
• 1991 AIME Problems/Problem 9
  Substituting <math>\cot x = \frac{1}{\tan x} = \frac{308}{435}</math> yields a [[quadratic equation]]: <math>0 = 435y^2 - 616y - 435 = (15y - 29)(29y + 15 Make the substitution <math>u = \tan \frac x2</math> (a substitution commonly used in calculus). By the half-angle identity for tan
  10 KB (1,590 words) - 14:04, 20 January 2023
• 1992 AIME Problems/Problem 13
  ...[[distance formula]], we see that <math>\frac{ \sqrt{a^2 + b^2} }{ \sqrt{ (a-9)^2 + b^2 } } = \frac{40}{41}</math>. Simplifying gives <math>-a^2 -\frac{3200}{9}a +1600 = b^2</math>.
  4 KB (703 words) - 02:40, 29 December 2023
• 1992 AIME Problems/Problem 9
  ...{}_{}</math> parallel to <math>CD^{}_{}</math>. A circle with center <math>P^{}_{}</math> on <math>AB^{}_{}</math> is drawn tangent to <math>BC^{}_{}</m ...0x=70\cdot 92,</math> hence <math>AP=x=\frac{161}{3}</math>. This gives us a final answer of <math>161+3=\boxed{164}</math>
  5 KB (874 words) - 10:27, 22 August 2021
• 2006 AMC 10B Problems/Problem 21
  ...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? <math> \mathrm{(A) \ } \frac{4}{63}\qquad \mathrm{(B) \ } \frac{1}{8}\qquad \mathrm{(C) \ } \
  3 KB (484 words) - 19:09, 15 October 2023
• 2006 AMC 10B Problems/Problem 23
  A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. T pair A = (0,0), B = (3,0), C = (1.4, 2), D = B + 0.4*(C-B), Ep = A + 0.3*(C-A);
  5 KB (861 words) - 00:53, 25 November 2023
• 2006 AMC 10B Problems/Problem 24
  ...d points <math>C</math> and <math>D</math> on the circle with center <math>P</math> are such that <math>AD</math> and <math>BC</math> are common externa ...nt(X, O, 2, 1), A=tangent(X, O, 2, 2), C=tangent(X, P, 4, 1), D=tangent(X, P, 4, 2);
  4 KB (558 words) - 14:38, 6 April 2024
• 1993 AIME Problems/Problem 13
  ...enny and Kenny can see each other again. If <math>t\,</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? ...the intersection of <math>AC</math> and <math>BD</math>. Finaly, let <math>P</math> and <math>Q</math> be the points of tangency of circle <math>O</math
  8 KB (1,231 words) - 20:06, 26 November 2023
• 1993 AIME Problems/Problem 12
  ...ing the rule: If the die shows label <math>L\,</math>, where <math>L \in \{A, B, C\}</math>, and <math>P_n\,</math> is the most recently obtained point, ...the midpoint of <math>(u,v)</math> and <math>(p,q)</math>, then <math>u=2r-p</math> and <math>v=2s-q</math>. So we start with the point they gave us and
  4 KB (611 words) - 13:59, 15 July 2023
• 1993 AIME Problems/Problem 11
  ...Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the fir The probability that the <math>n</math>th flip in each game occurs and is a head is <math>\frac{1}{2^n}</math>. The first person wins if the coin lands
  7 KB (1,058 words) - 20:57, 22 December 2020
• 1993 AIME Problems/Problem 10
  ...of its <math>V</math> vertices, <math>T</math> triangular faces and <math>P</math> pentagonal faces meet. What is the value of <math>100P+10T+V</math>? ...+p=F=32</math> faces. In each vertex, <math>T=2</math> triangles and <math>P=2</math> pentagons are concurrent. Now, the number of edges <math>E</math>
  4 KB (716 words) - 20:50, 17 April 2022
• 1993 AIME Problems/Problem 7
  ...he brick parallel to the sides of the box. If <math>p</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? There is a total of <math>P(1000,6)</math> possible ordered <math>6</math>-tuples <math>(a_1,a_2,a_3,b_
  5 KB (772 words) - 09:04, 7 January 2022
• 1955 AHSME Problems/Problem 27
  ...q\qquad\textbf{(C)}\ p^2+q^2\qquad\textbf{(D)}\ p^2-q^2\qquad\textbf{(E)}\ p^2 </math> <cmath>r^2 + s^2 = (r+s)^2 - 2rs = p^2 - 2q</cmath>
  600 bytes (108 words) - 10:47, 15 February 2021
• 1994 AIME Problems/Problem 15
  ...h>\triangle ABC\,</math> if these creases, which number three unless <math>P^{}_{}</math> is one of the vertices, do not intersect. Suppose that <math> ...circ}</math>; the [[locus]] of each of the respective conditions for <math>P</math> is the region inside the (semi)circles with diameters <math>\overlin
  4 KB (717 words) - 22:20, 3 June 2021
• 1994 AIME Problems/Problem 9
  ...> and <math>q\,</math> are relatively prime positive integers. Find <math>p+q.\,</math> *Case 1. We draw a pair on the first two cards. The second card is the same as the first with
  3 KB (589 words) - 14:18, 21 July 2019
• 1994 AIME Problems/Problem 7
  For certain ordered pairs <math>(a,b)\,</math> of [[real number]]s, the system of equations ...pair <math>(x,y)\,</math> of integers. How many such ordered pairs <math>(a,b)\,</math> are there?
  3 KB (442 words) - 19:51, 8 January 2024
• 1994 AIME Problems/Problem 5
  ...of <math>n\,</math>. (If <math>n\,</math> has only one digits, then <math>p(n)\,</math> is equal to that digit.) Let <center><math>S=p(1)+p(2)+p(3)+\cdots+p(999)</math></center>.
  2 KB (275 words) - 19:27, 4 July 2013
• 1994 AIME Problems/Problem 2
  ...a circle of radius 20. Square <math>ABCD</math> is constructed with <math>A</math> and <math>B</math> on the larger circle, <math>\overline{CD}</math> ...at point <math>E</math>), and draw <math>\overline{AO}</math>. We now have a [[right triangle]], with [[hypotenuse]] of length <math>20</math>. Since <m
  2 KB (272 words) - 03:53, 23 January 2023
• 1995 AIME Problems/Problem 12
  // p = point on line triple lineintersectplan(triple p, triple d, triple q, triple n)
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1995 AIME Problems/Problem 11
  ...1995,</math> for how many ordered triples <math>(a, b, c)</math> does such a plane exist? ...</math> be the prism similar to <math>P</math>, and let the sides of <math>P'</math> be of length <math>x,y,z</math>, such that <math>x \le y \le z</mat
  2 KB (292 words) - 19:30, 4 July 2013
• 1995 AIME Problems/Problem 5
  For certain real values of <math>a, b, c,</math> and <math>d_{},</math> the equation <math>x^4+ax^3+bx^2+cx+d Let's assume that the 2 roots multiplied together are p+qi, and r+si, and the two roots added together are the conjugates of the pr
  3 KB (451 words) - 15:02, 6 September 2021
• 1995 AIME Problems/Problem 4
  ...us <math>9</math>. The circle of radius <math>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the le ...), F=B+3*expi(acos(1/3)), P=IP(F--F+3*(D-F),CR(A,9)), Q=IP(F--F+3*(F-D),CR(A,9));
  3 KB (605 words) - 11:30, 5 May 2024
• 1995 AIME Problems/Problem 3
  ...object reaches <math>(2,2)</math> in six or fewer steps. Given that <math>p</math> can be written in the form <math>m/n,</math> where <math>m</math> an ...math>2!</math> ways to determine the order of the remaining two steps, for a total of <math>12</math> sequences that we have to exclude. This gives <mat
  3 KB (602 words) - 23:15, 16 June 2019
• 1996 AIME Problems/Problem 12
  ...ath> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. ...her way to think about this is that the total sum is 330. Because you have a total of <math>10 \cdot 9</math> possible "pairs" or "differences", the ave
  5 KB (879 words) - 11:23, 5 September 2021
• 1996 AIME Problems/Problem 11
  ...> that have a positive [[imaginary]] part, and suppose that <math>\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})</math>, where <math>0<r</mat ...e left with <math>\mathrm{cis}\ 60, 72, 144</math>; their product is <math>P = \mathrm{cis} (60 + 72 + 144) = \mathrm{cis} \boxed{276}</math>.
  6 KB (1,022 words) - 20:23, 17 April 2021
• 1996 AIME Problems/Problem 5
  ...x^3+rx^2+sx+t=0</math> are <math>a+b</math>, <math>b+c</math>, and <math>c+a</math>. Find <math>t</math>. ...al <math>P(x) = x^3+3x^2+4x-11 = (x-a)(x-b)(x-c) = 0</math>, we have <math>a + b + c = s = -3</math>, <math>ab + bc + ca = 4</math>, and <math>abc = 11<
  3 KB (585 words) - 22:08, 19 November 2022
• 1996 AIME Problems/Problem 4
  ...th>x</math> centimeters directly above an upper [[vertex]], the cube casts a shadow on the horizontal surface. The area of the shadow, which does not in triple O=(0,0,0), P=(0,0,unit+unit/(r-1)); dot(P);
  2 KB (257 words) - 17:50, 4 January 2016
• 1997 AIME Problems/Problem 15
  ...th>q</math> is not divisible by the square of any prime number. Find <math>p+q+r</math>. ...{2} - \frac {a\sqrt {3}}{2}\right) + \left(\frac {11\sqrt {3}}{2} + \frac {a}{2}\right)i = b + 10i</math>.
  4 KB (609 words) - 22:49, 17 July 2023
• 1997 AIME Problems/Problem 12
  ...math>f</math> defined by <math>f(x)= \frac{ax+b}{cx+d}</math>, where <math>a</math>,<math>b</math>,<math>c</math> and <math>d</math> are nonzero real nu ...in the domain. Substituting the function definition, we have <math>\frac {a\frac {ax + b}{cx + d} + b}{c\frac {ax + b}{cx + d} + d} = x</math>, which r
  11 KB (2,063 words) - 22:59, 21 October 2023
• 1998 AIME Problems/Problem 15
  ...rdered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after t We can draw a comparison between the domino a set of 40 points (labeled 1 through 40) in which every point is connected w
  9 KB (1,671 words) - 22:10, 15 March 2024
• 1998 AIME Problems/Problem 14
  ...and <math>m\le n\le p.</math> What is the largest possible value of <math>p</math>? <cmath>2mnp = (m+2)(n+2)(p+2)</cmath>
  2 KB (390 words) - 21:05, 29 May 2023
• 1998 AIME Problems/Problem 13
  ...,</math> where <math>p</math> and <math>q</math> are integers, find <math>|p| + |q|.</math> ...; hence we can just append a <math>9</math> to any of those subsets to get a new one.
  2 KB (384 words) - 19:02, 20 October 2023
• 1998 AIME Problems/Problem 12
  ...h>c</math> is not divisible by the square of any [[prime]]. What is <math>a^{2} + b^{2} + c^{2}</math>? ...)^{2}}{k^{2}} = \frac {4}{7 - 3\sqrt {5}} = 7 + 3\sqrt {5}</math> so <math>a^{2} + b^{2} + c^{2} = 7^{2} + 3^{2} + 5^{2} = \boxed{083}</math>.
  5 KB (876 words) - 20:27, 9 June 2022
• 1998 AIME Problems/Problem 11
  ...ath> and <math>\overline{AD}</math> is an interior diagonal. Points <math>P, Q,</math> and <math>R</math> are on <math>\overline{AB}, \overline{BC},</m triple A=(0,0,0),B=(20,0,0),C=(20,0,20),D=(20,20,20);
  7 KB (1,084 words) - 11:48, 13 August 2023
• 1998 AIME Problems/Problem 6
  ...Extend <math>\overline{DA}</math> through <math>A</math> to a point <math>P,</math> and let <math>\overline{PC}</math> meet <math>\overline{AB}</math> {{AIME box|year=1998|num-b=5|num-a=7}}
  2 KB (254 words) - 19:38, 4 July 2013
• 1999 AIME Problems/Problem 15
  ...The vertices of its midpoint triangle are the [[midpoint]]s of its sides. A triangular [[pyramid]] is formed by folding the triangle along the sides of ...;label("\(E\)",foot(B,A,C),SW);label("\(F\)",foot(C,A,B),NW);label("\(P\)",P,NW);label("\(Q\)",Q,NE);label("\(R\)",R,SE);</asy><asy>import three; defaul
  7 KB (1,169 words) - 15:28, 13 May 2024
• 1999 AIME Problems/Problem 14
  pair A=(0,0),B=(13,0),C=IP(circle(A,15),circle(B,14)); D(MP("A",A)--MP("B",B)--MP("C",C,N)--cycle);
  7 KB (1,184 words) - 13:25, 22 December 2022
• 1999 AIME Problems/Problem 12
  pair A=(0,0),B=(50,0),C=IP(circle(A,23+245/2),circle(B,27+245/2)), I=incenter(A,B,C); path P = incircle(A,B,C);
  3 KB (472 words) - 15:59, 25 February 2022
• 1999 AIME Problems/Problem 9
  ...is [[equidistant]] from that point and the [[origin]]. Given that <math>|a+bi|=8</math> and that <math>b^2=m/n,</math> where <math>m_{}</math> and <ma ...= (a-b)</math> and <math>y = (a+b)</math>, we get <math>2a = 1 \Rightarrow a = \frac 12</math>.
  6 KB (1,010 words) - 19:01, 24 May 2023
• 1999 AIME Problems/Problem 1
  ...cannot be <math>2</math> or <math>4</math>, since otherwise there would be a number in the sequence that is divisible by <math>3</math>. However, if the ...is <math>5</math>, and from a quick observation we can see that when <math>a</math> is <math>6</math>, the terms of the sequence are all prime numbers.
  2 KB (332 words) - 13:22, 3 August 2020
• 2000 AIME I Problems/Problem 14
  ...r</math> times as large as angle <math>APQ,</math> where <math>r</math> is a positive real number. Find <math>\lfloor 1000r \rfloor</math>. <center><asy>defaultpen(fontsize(10)); size(200); pen p=fontsize(8);
  8 KB (1,275 words) - 03:04, 27 February 2022
• 2000 AIME I Problems/Problem 13
  In the middle of a vast prairie, a firetruck is stationed at the intersection of two [[perpendicular]] straigh ...,0)</math>. All these circles are [[homothety|homothetic]] with respect to a center at <math>(5,0)</math>.
  3 KB (571 words) - 00:38, 13 March 2014
• 2000 AIME I Problems/Problem 8
  ...math> is not divisible by the cube of any prime number. Find <math>m + n + p</math>. ...}\right)^{3}\right)^{1/3}=12-3\left(37^{1/3}\right)</math>. Thus <math>m+n+p=\boxed{052}</math>.
  4 KB (677 words) - 16:33, 30 December 2023
• 2000 AIME I Problems/Problem 5
  If we work with the problem for a little bit, we quickly see that there is no direct combinatorics way to cal ...</math>. It follows that <math>5|a,b</math>, so there are 2 pairs of <math>a</math> and <math>b: (20,5),(15,10)</math>.
  7 KB (1,011 words) - 20:09, 4 January 2024
• 2001 AIME I Problems/Problem 12
  A [[sphere]] is inscribed in the [[tetrahedron]] whose vertices are <math>A = (6,0,0), B = (0,4,0), C = (0,0,2),</math> and <math>D = (0,0,0).</math> triple A = (6,0,0), B = (0,4,0), C = (0,0,2), D = (0,0,0);
  6 KB (1,050 words) - 18:44, 27 September 2023
• 2014 USAJMO Problems
  ...frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc. </cmath> ...riangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60^\circ</math>, and let <math>O</math> and <math>H</math> denote the circu
  3 KB (600 words) - 16:42, 5 August 2023
• 2001 AIME I Problems/Problem 9
  ...d <math>r</math> are positive and satisfy <math>p+q+r=2/3</math> and <math>p^2+q^2+r^2=2/5</math>. The ratio of the area of triangle <math>DEF</math> to real p = 0.5, q = 0.1, r = 0.05;
  4 KB (673 words) - 20:15, 21 February 2024
• 2001 AIME I Problems/Problem 7
  ...), I=incenter(A,B,C), D=IP((0,I.y)--(20,I.y),A--B), E=IP((0,I.y)--(20,I.y),A--C); D(MP("A",A,N)--MP("B",B)--MP("C",C)--cycle); D(MP("I",I,NE)); D(MP("E",E,NE)--MP("D",D
  9 KB (1,540 words) - 08:31, 1 December 2022
• 2001 AIME I Problems/Problem 6
  A fair die is rolled four times. The [[probability]] that each of the final t ...n the diagram below, the lowest <math>y</math>-coordinate at each of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> corresponds to t
  11 KB (1,729 words) - 20:50, 28 November 2023
• 2001 AIME I Problems/Problem 4
  ...and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>. First, draw a good diagram.
  3 KB (534 words) - 03:22, 23 January 2023
• 2002 AIME I Problems/Problem 15
  ...and <math>r</math> is not divisible by the square of any prime, find <math>p + q + r.</math> triple A=(-6,-6,0), B = (-6,6,0), C = (6,6,0), D = (6,-6,0), E = (2,0,12), H=(-6+2*s
  7 KB (1,181 words) - 20:32, 8 January 2024
• 2002 AIME I Problems/Problem 13
  ...,0), E=(A+B)/2, C=IP(CR(A,3*70^.5),CR(E,27)), D=(B+C)/2, F=IP(circumcircle(A,B,C),E--C+2*(E-C)); ...MP("12",(A+E)/2,SE,f);MP("12",(B+E)/2,f); MP("27",(C+E)/2,SW,f); MP("18",(A+D)/2,SE,f);
  6 KB (974 words) - 13:01, 29 September 2023
• 2002 AIME I Problems/Problem 11
  ...ht path from the time it leaves point <math>A</math> until it next reaches a vertex of the cube is given by <math>m\sqrt{n}</math>, where <math>m</math> When a light beam reflects off a surface, the path is like that of a ball bouncing. Picture that, and also imagine X, Y, and Z coordinates for t
  3 KB (591 words) - 15:11, 21 August 2019
• 2002 AIME I Problems/Problem 2
  ...where <math>p</math> and <math>q</math> are positive integers. Find <math>p+q</math>. ...ht)</math>. Thus we have <math>p=147</math> and <math>q=7</math>, so <math>p+q=\boxed{154}</math>.
  2 KB (287 words) - 19:54, 4 July 2013
• 2003 AIME I Problems/Problem 14
  ...<math>1</math>, we can rewrite this as <math>\frac{10^k m - nX}{n} = \frac{p}{q}</math>, where <math>q \le n</math>. Then the fraction <math>\frac pq = {{AIME box|year=2003|n=I|num-b=13|num-a=15}}
  3 KB (477 words) - 14:23, 4 January 2024
• 2003 AIME I Problems/Problem 12
  ...f <math> ABCD </math> is <math> 640 </math>. Find <math> \lfloor 1000 \cos A \rfloor. </math> (The notation <math> \lfloor x \rfloor </math> means the g pair A=(0,0),B=(1.8,0),D=IP(CR(A,x),CR(B,BD)),C=OP(CR(D,1.8),CR(B,2.80 - x));
  3 KB (487 words) - 22:14, 24 November 2019
• 2003 AIME I Problems/Problem 11
  ...</math> are not the lengths of the sides of a triangle. Given that <math> p = d/n, </math> where <math> d </math> is the number of degrees in <math> \t ...ssions and those three lengths not forming a [[triangle]] is equivalent to a violation of the [[triangle inequality]]
  2 KB (284 words) - 13:42, 10 October 2020
• 2003 AIME I Problems/Problem 6
  ...> m, n, </math> and <math> p </math> are [[integer]]s. Find <math> m + n + p. </math> ...e face diagonal and one space diagonal of the cube) and those which lie in a plane [[oblique]] to the edges of the cube, whose sides are three face diag
  3 KB (477 words) - 18:35, 27 December 2021
• 2003 AIME I Problems/Problem 5
  ...</math> and <math> p </math> are [[relatively prime]], find <math> m + n + p. </math> ...allelepipeds that each share a face with the large parallelepiped and have a height of <math>1</math>, the <math>1/8</math> [[sphere]]s (one centered at
  2 KB (288 words) - 19:58, 4 July 2013
• 2003 AIME II Problems/Problem 11
  ...h>n</math> is not divisible by the square of any prime, find <math>m + n + p.</math> ...>, we get <math>DM=\frac{5\sqrt{11}} {2}</math>. Since <math>ABC</math> is a right triangle, <math>M</math> is the circumcenter and thus, <math>CM=\frac
  5 KB (772 words) - 19:47, 1 August 2023
• 2003 AIME II Problems/Problem 9
  ...oots of <math>Q(x) = 0,</math> find <math>P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).</math> When we use long division to divide <math>P(x)</math> by <math>Q(x)</math>, the remainder is <math>x^2-x+1</math>.
  3 KB (475 words) - 21:53, 6 May 2024
• 2002 AIME II Problems/Problem 15
  ...me, and <math>a</math> and <math>c</math> are relatively prime. Find <math>a + b + c</math>. ...<math>b</math> respectively. We know that the point <math>(9,6)</math> is a point on both circles, so we have that
  7 KB (1,182 words) - 09:56, 7 February 2022
• 2002 AIME II Problems/Problem 14
  ...is <math>152</math>, and the angle <math>PAM</math> is a [[right angle]]. A [[circle]] of [[radius]] <math>19</math> with center <math>O</math> on <mat ...MPA</math> are similar. Also note that <math>AM = BM</math> by [[power of a point]]. Using the fact that the ratio of corresponding sides in similar tr
  4 KB (658 words) - 19:15, 19 December 2021
• 2002 AIME II Problems/Problem 13
  ...ath>\overline{AD}</math> and <math>\overline{BE}</math> intersect at <math>P.</math> Points <math>Q</math> and <math>R</math> lie on <math>\overline{AB} pair A,B,C,D,E,X,P,Q,R;
  6 KB (935 words) - 13:23, 3 September 2021
• 2002 AIME II Problems/Problem 12
  ...<math>c</math> are positive integers. Find <math>\left(p+q+r+s\right)\left(a+b+c\right)</math>. ...ne <math>y=\frac{2x}{3}</math>. We can find the number of such paths using a Pascal's Triangle type method below, computing the number of paths to each
  7 KB (1,127 words) - 13:34, 19 June 2022
• 2002 AIME II Problems/Problem 11
  ...>m</math> is not divisible by the square of any prime. Find <math>100m+10n+p</math>. ...e form is <math>x = \frac{\sqrt{5}-1}{8}</math>. Therefore, <math>100m+10n+p = \boxed{518}</math>.
  4 KB (696 words) - 16:27, 22 March 2022
• 2002 AIME II Problems/Problem 10
  ..., <math>p</math>, and <math>q</math> are positive integers. Find <math>m+n+p+q</math>. ...pi - \frac{\pi x}{180} \Rightarrow x = \frac{180\pi}{180+\pi} \Rightarrow (p,q) = (180,180)</math>.
  2 KB (336 words) - 19:30, 24 June 2020
• 2002 AIME II Problems/Problem 7
  ...ve integer <math>k</math> such that <math>1^2+2^2+3^2+\ldots+k^2</math> is a multiple of <math>200</math>. ...}</math> is a multiple of <math>200</math> if <math>k(k+1)(2k+1)</math> is a multiple of <math>1200 = 2^4 \cdot 3 \cdot 5^2</math>.
  3 KB (403 words) - 12:10, 9 September 2023
• 2002 AIME II Problems/Problem 2
  Three [[vertex|vertices]] of a [[cube]] are <math>P=(7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=(11,3,9)</math>. What ...h>PQR</math> is an equilateral triangle. Let the side of the cube be <math>a</math>.
  709 bytes (103 words) - 03:32, 6 December 2019
• 2001 AIME II Problems/Problem 15
  ...h>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>. ...ath>E</math>, where the drilling starts, is at <math>(8,8,8)</math>. Using a little visualization (involving some [[similar triangles]], because we have
  4 KB (518 words) - 15:01, 31 December 2021
• 2001 AIME II Problems/Problem 10
  ...{3} - 1)</math>, we see that <math>j-i = 6</math> works; also, <math>a-b | a^n - b^n</math> implies that <math>10^{6} - 1 | 10^{6k} - 1</math>, and so a ...an easily verify that <math>10^6 - 1 \nmid 10^a - 1</math> for <math>1 \le a \le 5</math>.
  4 KB (549 words) - 23:16, 19 January 2024
• 2001 AIME II Problems/Problem 7
  Let <math>\triangle{PQR}</math> be a [[right triangle]] with <math>PQ = 90</math>, <math>PR = 120</math>, and <m pair P = (0,0), Q = (90, 0), R = (0, 120), S=(0, 60), T=(45, 60), U = (60,0), V=(6
  7 KB (1,112 words) - 02:15, 26 December 2022
• 2001 AIME II Problems/Problem 4
  ...s are <math>8y = 15x</math> and <math>10y = 3x</math> contain points <math>P</math> and <math>Q</math>, respectively, such that <math>R</math> is the [[ pair R = (8,6), P = (32,60)/7, Q= (80,24)/7;
  2 KB (240 words) - 20:34, 4 July 2013
• 2000 AIME II Problems/Problem 10
  ...al]] <math>ABCD</math>, [[tangent]] to <math>\overline{AB}</math> at <math>P</math> and to <math>\overline{CD}</math> at <math>Q</math>. Given that <mat Take the <math>\tan</math> of both sides and use the identity for <math>\tan(A+B)</math> to get <cmath>\tan(\arctan(\tfrac{19}{r})+\arctan(\tfrac{26}{r}))
  2 KB (399 words) - 17:37, 2 January 2024
• 2005 AMC 10B Problems/Problem 14
  pair A = 2*dir(60); pair M = midpoint(A--C);
  5 KB (882 words) - 22:12, 30 April 2024
• 2005 AMC 10B Problems/Problem 21
  ...ther two bear a number <math>b \neq a</math>. What is the value of <math>q/p</math>? <math>\textbf{(A) } 162 \qquad \textbf{(B) } 180 \qquad \textbf{(C) } 324 \qquad \textbf{(D)
  3 KB (398 words) - 19:17, 17 September 2023
• 2006 USAMO Problems/Problem 1
  ...re exist integers <math>m</math> and <math>n</math> with <math>0 < m < n < p</math> and ...> \left\{ \frac{sm}{p} \right\} < \left\{ \frac{sn}{p} \right\} < \frac{s}{p} </math>
  3 KB (506 words) - 17:54, 22 June 2023
• 2006 USAMO Problems/Problem 3
  ...s <math>f</math> with integer coefficients such that the sequence <math>\{ p(f(n^2))-2n) \}_{n \in \mathbb{Z} \ge 0}</math> is bounded above. (In partic Let <math>f(x)</math> be a non-constant polynomial in <math>x</math> of degree <math>d</math> with
  9 KB (1,699 words) - 13:48, 11 April 2020
• 2006 USAMO Problems/Problem 5
  ...ow that if <math>k\ge 2</math> is a positive integer and <math>i</math> is a nonnegative integer, then the minimum number of jumps needed to reach <math ...igher power of 2). Thus we must have <math>M < i_0</math>, since otherwise a number divisible by <math>2^{i_0}</math> is visited before <math>2^{i_0} k_
  7 KB (1,280 words) - 17:23, 26 March 2016
• 2006 AMC 12A Problems/Problem 9
  ...<math>1.00</math>. A pencil costs more than an eraser, and both items cost a [[whole number]] of cents. What is the total cost, in cents, of one pencil <math>\mathrm{(A)}\ 10\qquad\mathrm{(B)}\ 12\qquad\mathrm{(C)}\ 15\qquad\mathrm{(D)}\ 18\qqu
  3 KB (429 words) - 18:14, 26 September 2020
• Diagonal
  A '''diagonal''' of a [[polygon]] is any segment joining two [[vertex|vertices]] other than an [[ The number of diagonals of a polygon with <math>n</math> vertices is given by <math>\frac{n(n-3)}{2}</m
  2 KB (374 words) - 00:37, 25 January 2015
• Locus
  ...used most frequently in [[geometry]], to denote a set of points satisfying a certain geometric condition. ...can be defined as the locus of all points that are a certain distance from a given center.
  890 bytes (160 words) - 02:58, 22 March 2011
• Radius
  The '''radius''' of a [[circle]] is the distance from the center to any point on the circle. Ide The radius of a circle is often denoted using R or r.
  929 bytes (156 words) - 22:49, 5 January 2023
• Field
  ...field''' is a structure in [[abstract algebra]], similar to a [[group]] or a [[ring]]. Informally, fields are the general structure in which the usual ...>k</math> stands for Körper, the German word for a mathematical field) is a [[set]] of elements with two [[operation]]s, usually called multiplication
  2 KB (362 words) - 23:24, 31 December 2021
• Multinomial Theorem
  Note that this is a direct generalization of the [[Binomial Theorem]], when <math>k = 2</math> For a positive integer <math>k</math> and a non-negative integer <math>n</math>,
  3 KB (476 words) - 19:37, 4 January 2023
• Quadratic reciprocity
  Let <math>p</math> be a [[prime number|prime]], and let <math>a</math> be any integer. Then we can define the [[Legendre symbol]] ...{p} =\begin{cases} 1 & \text{if } a \text{ is a quadratic residue modulo } p, \\
  7 KB (1,182 words) - 16:46, 28 April 2016
• Dedekind domain
  A '''Dedekind domain''' is a [[integral domain]] <math>R</math> satisfying the following properties: * <math>R</math> is a [[noetherian]] [[ring]].
  9 KB (1,648 words) - 16:36, 14 October 2017
• Monic polynomial
  ...+ \cdots + a_0</math>, i.e. the [[coefficient]] of the highest-[[degree of a polynomial | degree]] term (the leading coefficient) is 1.
  262 bytes (47 words) - 21:17, 23 December 2016
• 2006 AMC 10B Problems
  <math> \mathrm{(A) \ } -2006\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 0\qquad \mathrm{( <math> \mathrm{(A) \ } -72\qquad \mathrm{(B) \ } -27\qquad \mathrm{(C) \ } -24\qquad \mathrm{
  14 KB (2,059 words) - 01:17, 30 January 2024
• 2005 AMC 10B Problems
  ...t a price of five for &#36;<math>2</math>. They sell all the candy bars at a price of two for &#36;<math>1</math>. What was the profit, in dollars? <math>\mathrm{(A)} 100 \qquad \mathrm{(B)} 200 \qquad \mathrm{(C)} 300 \qquad \mathrm{(D)} 4
  12 KB (1,874 words) - 21:20, 23 December 2020
• Platonic solid
  A '''Platonic solid''' is a [[polyhedron]], or 3 dimensional figure, in which all [[face]]s are [[congr <!-- labeling: for(int i = 0; i < P.length; ++i) label("" + ((string) i), P[i], N); -->
  8 KB (1,168 words) - 22:48, 19 February 2022
• 2000 AMC 12 Problems/Problem 5
  If <math>|x - 2| = p</math>, where <math>x < 2</math>, then <math>x - p =</math> <math> \textbf{(A)} \ -2 \qquad \textbf{(B)} \ 2 \qquad \textbf{(C)} \ 2-2p \qquad \textbf{(D
  714 bytes (102 words) - 17:01, 29 July 2023
• 2000 AMC 12 Problems/Problem 8
  <math>\textbf{(A)}\ 10401 \qquad\textbf{(B)}\ 19801 \qquad\textbf{(C)}\ 20201 \qquad\textbf{ ...qual to <math>4</math>. Thus, the original sequence can be generated from a quadratic function.
  8 KB (1,202 words) - 16:17, 10 May 2024
• 2000 AMC 12 Problems/Problem 6
  <math>\textbf{(A)}\ 22 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 119 \qquad\textbf{(D)}\ 18 ...Any odd number minus an even number is an odd number, so we can eliminate A, B, and D. Since the highest two prime numbers we can pick are 13 and 17, t
  1 KB (228 words) - 19:31, 29 April 2024
• Contrapositive
  A '''contrapositive''' of a statement is always true, assuming that the conditional statement is true. A [[conditional]] statement is usually expressed as
  648 bytes (96 words) - 16:03, 8 May 2024
• Angle bisector theorem
  ...$",D,(0,-1)); dot(A^^B^^C^^D,blue);label("$b$",(A+C)/2,(1,0));label("$c$",(A+B)/2,(0,1));label("$m$",(B+D)/2,(0,-1));label("$n$",(D+C)/2,(0,-1)); </asy> # Let ABC be a triangle with angle bisector AD with D on line segment BC. If <math> BD =
  3 KB (438 words) - 14:20, 4 March 2023
• University of South Carolina High School Math Contest/1993 Exam/Problem 6
  After a <math>p\%</math> price reduction, what increase does it take to restore the origina ...mathrm{(D) \ } \frac{100p}{100+p}\% \qquad \mathrm{(E) \ } \frac{100p}{100-p}\% </math></center>
  2 KB (299 words) - 21:06, 5 July 2017
• University of South Carolina High School Math Contest/1993 Exam/Problem 25
  ...o circles are tangent at a point <math>P</math> if they intersect at <math>P</math> and at no other point.) <center><math> \mathrm{(A) \ }(0,-6) \qquad \mathrm{(B) \ } (1,-9) \qquad \mathrm{(C) \ } (-1,-9) \qq
  1 KB (172 words) - 00:11, 16 February 2016
• University of South Carolina High School Math Contest/1993 Exam/Problems
  If the width of a particular rectangle is doubled and the length is increased by 3, then the <cmath> \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D)
  14 KB (2,102 words) - 22:03, 26 October 2018
• Mock AIME 1 2006-2007 Problems/Problem 5
  For a [[prime number]] <math>p</math>, define the [[function]] <math>f_p(n)</math> as follows: If there exists <math>y</math>, <math>0 \leq y < p</math>, such that
  2 KB (340 words) - 15:52, 3 April 2012
• Mock AIME 1 2006-2007 Problems/Problem 8
  Let <math>ABCDE</math> be a convex pentagon with <math>\frac{AB}{\sqrt{2}}=BC=CD=DE</math>, <math>\angl ...0)+expi(pi/2), B=expi(pi*5/12), C=(0,0), D=expi(0), E=expi(0)+expi(pi/12), P=expi(pi*5/12)+expi(0);
  1 KB (244 words) - 14:54, 21 August 2020
• Mock AIME 1 2006-2007 Problems/Problem 13
  ...se variables - but there is no obvious way of computing them. We will show a different solution. Consider the polynomial <math>P(x)=(x-\alpha)(x-\beta)(x-\gamma)</math> with <math>\alpha</math>, <math>\be
  3 KB (568 words) - 15:50, 3 April 2012
• Mock AIME 1 2006-2007 Problems/Problem 14
  ...le radius that is tangent to <math>AB</math> at <math>T</math>. Let <math>P</math> be the point such that circle <math>O</math> is the incircle of <mat ...ee below an attempted solution to understand why this problem doesn't have a solution:
  3 KB (541 words) - 17:32, 22 November 2023
• 1977 Canadian MO Problems/Problem 2
  ...on the circumference of the circle such that the angle <math>OPA</math> is a maximum. ...B</math> is the chord perpendicular to <math>OX</math> through point <math>P</math>, then extend <math>AO</math> to meet the circle at point <math>C</ma
  2 KB (365 words) - 23:28, 21 September 2014
• Mock AIME 1 2006-2007/Problems
  Let <math>\star (x)</math> be the sum of the digits of a positive integer <math>x</math>. <math>\mathcal{S}</math> is the set of pos ...\sqrt{7}</math>, <math>CA=1</math>, and <math>AB=3</math>. If <math>\angle A=\frac{\pi}{n}</math> where <math>n</math> is an integer, find the remainder
  8 KB (1,355 words) - 14:54, 21 August 2020
• Mock AIME 2 2006-2007 Problems/Problem 7
  ...a point on the circumference of the base and the shortest path from <math>P</math> around the cone and back is drawn (see diagram). If the length of th ...ther. Since <math>34\pi</math> is <math>1/3</math> of the circumference of a circle with radius <math>51</math>, we must have that <math>\angle AOB=\fra
  1 KB (231 words) - 18:10, 10 July 2014
• Mock AIME 2 2006-2007 Problems/Problem 9
  .../math> has no perfect square divisors excluding <math>1,</math> find <math>a+b+c+d.</math> pair A=(0,6),B=(5,0),C=origin,X=(3,0),Y=A/(sqrt(3)+1);
  2 KB (358 words) - 23:22, 3 May 2014
• Mock AIME 2 2006-2007 Problems/Problem 14
  ...[[intersect]] at <math>P</math> and are an [[angle bisector]], [[median of a triangle | median]], and [[altitude]] of the triangle, respectively, comput {{Mock AIME box|year=2006-2007|n=2|num-b=13|num-a=15}}
  2 KB (284 words) - 10:53, 4 April 2012
• Mock AIME 2 2006-2007 Problems
  ...r positive integers <math>a,b,c,</math> and <math>d</math> such that <math>a+4=b-4=4c=d/4.</math> Find the smallest dragon. ...ll positive integers <math>n</math> such that <math>n^2+12n-2007</math> is a perfect square. Find the remainder when <math>S</math> is divided by <math>
  5 KB (848 words) - 23:49, 25 February 2017
• Expected value
  Given an event with a variety of different possible outcomes, the '''expected value''' is what on A video that goes over the type of Expected value, practical examples, and pr
  5 KB (789 words) - 20:56, 10 May 2024
• 1959 IMO Problems
  <math>\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A,</math> given (a) <math>A = \sqrt{2}</math>, (b) <math>A=1</math>, (c) <math>A=2</math>, where only non-negative real numbers are admitted for square root
  3 KB (480 words) - 11:57, 17 September 2012
• 1959 IMO Problems/Problem 1
  Denoting the greatest common divisor of <math>a, b </math> as <math>(a,b) </math>, we use the [[Euclidean algorithm]]: ...<math>\dfrac{14n+3}{21n+4}</math> is a [[reducible fraction]] where <math>p</math> is the [[greatest common factor]] of <math>14n+3</math> and <math>21
  5 KB (767 words) - 10:59, 23 July 2023
• 1977 Canadian MO Problems
  ...uation <math>4f(a)=f(b)</math> has no solutions in positive integers <math>a</math> and <math>b.</math> ...on the circumference of the circle such that the angle <math>OPA</math> is a maximum.
  3 KB (560 words) - 19:23, 10 March 2015
• 1971 Canadian MO Problems/Problem 1
  <math>DEB</math> is a chord of a circle such that <math>DE=3</math> and <math>EB=5 .</math> Let <math>O</mat ...e at <math>P.</math> Let the radius be <math>r.</math> Applying [[power of a point]],
  680 bytes (114 words) - 21:38, 9 July 2019
• 1959 IMO Problems/Problem 5
  ...ive bases. The circles about these squares, with respective centers <math>P </math> and <math>Q </math>, intersect at <math>M </math> and also at anoth (a) Prove that the points <math>N </math> and <math>N' </math> coincide.
  4 KB (729 words) - 08:23, 23 May 2024
• 1959 IMO Problems/Problem 6
  ...| vertices]] <math>B </math> and <math>D </math> lying in the planes <math>P </math> and <math>Q </math>, respectively. ...st both intersect <math>p </math> (since they are both coplanar with <math>p </math>), making them [[skew lines | skew]].
  2 KB (383 words) - 05:58, 11 February 2024
• 2006 Romanian NMO Problems
  ...<math>2 \cdot \frac{CN}{BC} = \frac{AM}{AB}</math>. Let <math>P</math> be a point on the line <math>AC</math>. Prove that the lines <math>MN</math> and ...nimal <math>n</math>, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or
  11 KB (1,779 words) - 14:57, 7 May 2012
• 2006 Romanian NMO Problems/Grade 7/Problem 1
  ...<math>2 \cdot \frac{CN}{BC} = \frac{AM}{AB}</math>. Let <math>P</math> be a point on the line <math>AC</math>. Prove that the lines <math>MN</math> and Let <math>L</math> be a point on <math>BC</math> such that <math>N</math> is the midpoint of LC, th
  2 KB (288 words) - 21:17, 11 October 2013
• 2006 Romanian NMO Problems/Grade 7/Problem 4
  ...on multiple of <math>a</math> and <math>b</math>. Prove that the set <math>A</math> has ''exactly'' two elements. ...\frac{a^{2}}{n-a} < 1</math> if <math>n > a^{2}+a</math>. Therefore <math>A</math> is finite.
  3 KB (531 words) - 23:27, 7 July 2011
• 2006 Romanian NMO Problems/Grade 8/Problem 1
  We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove t ...es in space that intersect at two points, there exists a point P such that P is equidistant from any point on one circle and any point on the other circ
  3 KB (509 words) - 23:22, 15 August 2012
• 2006 Romanian NMO Problems/Grade 9/Problem 2
  ...>, and <math>\angle BDE = \angle ADP = \angle CDF</math>. Prove that <math>P</math> is the midpoint of <math>EF</math> and <math>DP \perp EF</math>. ...P</math>. Since P and P' are on the same ray (<math>DP</math>), P = P' and P is the midpoint of <math>EF</math>.
  3 KB (488 words) - 14:05, 15 December 2022
• 2006 Romanian NMO Problems/Grade 9/Problem 3
  ...ircle]] of [[radius]] <math>r</math>, for which there is a [[point]] <math>P</math> on <math>CD</math> such that <math>CB=BP=PA=AB</math>. (a) Prove that there are points <math>A,B,C,D,P</math> which fulfill the above conditions.
  6 KB (1,080 words) - 19:28, 21 September 2014
• 1969 Canadian MO Problems/Problem 4
  ...drawn to the three sides of the triangle. Show that, no matter where <math>P</math> is chosen, <math>\frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}</math ...<math>PD+PE+PF=\frac{\sqrt3}{2}s.</math> (It is independant of point <math>P</math>) Because the sum of the sides is <math>3s</math>, the ratio is alway
  1 KB (215 words) - 00:23, 9 October 2020
• 2005 Canadian MO Problems
  ...d the path never travels up (from a lower row to a higher row) or revisits a triangle. An example of one such path is illustrated below for <math>n=5</m ...a Pythagorean triple, ''i.e.'', a triplet of positive integers with <math>{a}^2+{b}^2={c}^2</math>.
  2 KB (436 words) - 11:45, 26 December 2018
• 1969 Canadian MO Problems/Problem 10
  ...of the [[rectangle]] <math>QCRP</math>. Prove that regardless of how <math>P</math> is chosen, the largest of these three areas is at least <math>2/9</m ...s <math>APQ</math> and <math>BPR</math> both contain a [[right angle]] and a <math>45^\circ</math> angle, they are [[isosceles triangle | isosceles]] [[
  1 KB (211 words) - 13:13, 30 December 2015
• 2005 Canadian MO Problems/Problem 3
  Let <math>S</math> be a set of <math>n\ge 3</math> points in the interior of a circle. ...math> on the circle such that <math>a</math> is (strictly) closer to <math>A</math> than any other point in <math>S</math>, <math>b</math> is closer to
  2 KB (460 words) - 13:35, 9 June 2011
• 2005 Canadian MO Problems/Problem 4
  ...ABC</math> be a triangle with circumradius <math>R</math>, perimeter <math>P</math> and area <math>K</math>. Determine the maximum value of <math>KP/R^3 ...s, and use AM-GM and Jensen's to finish. (Jensen's is used for <math>\sin A+\sin B+\sin C \le \dfrac{3\sqrt3}{2}</math>.
  491 bytes (89 words) - 08:13, 1 November 2022
• 2005 Canadian MO Problems/Problem 5
  ...a,b,c) = 1</math>, and <math>a^n + b^n + c^n</math> is divisible by <math>a+b+c</math>. For example, <math>(1,2,2)</math> is 5-powerful. Consider <math>P(x)=(x-a)(x-b)(x-c)</math>.
  984 bytes (177 words) - 18:21, 28 November 2023
• P versus NP
  ...me, or whether there are problems in <math>NP</math> that are not in <math>P</math>. Since all modern computers (with the exception of a few quantum computers) are deterministic, non-deterministic algorithms are
  6 KB (1,104 words) - 15:11, 25 October 2017
• Navier-Stokes Equation
  ...ate how sticky (viscous) a fluid is. Thus, the Navier-Stokes equations are a dynamical statement of the balance of forces acting at any given region of ...odel weather, ocean currents, water flow in a pipe, motion of stars inside a galaxy, and flow around an airfoil (wing). They are also used in the design
  3 KB (553 words) - 22:08, 2 May 2022
• Tetrahedron
  ...ce will be the vertices of a tetrahedron as long as they do not all lie on a single [[plane]]. triple[] P = {(0,0,(2/3)^.5),(3^(-0.5),0,0),(-1/2/3^.5,1/2,0),(-1/2/3^.5,-1/2,0)};
  1 KB (205 words) - 12:54, 20 February 2024
• 2005 AMC 10A Problems/Problem 24
  ...th>n</math> is it true that both <math> P(n) = \sqrt{n} </math> and <math> P(n+48) = \sqrt{n+48} </math>? <math> \textbf{(A) } 0\qquad \textbf{(B) } 1\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qqua
  6 KB (895 words) - 13:52, 4 April 2024
• Graph of a function
  A '''graph''' is a visual representation of a [[function]]. If <math>f(x) = y</math> then the [[point]] <math>(x,y)</mat ...is the simplest thing to graph. The graph of <math>(2,5)</math> would be a dot 2 units to the right of <math>y</math>-axis and 5 units above the <math
  3 KB (497 words) - 05:11, 8 June 2023
• Linear congruence
  A '''Linear Congruence''' is a [[congruence]] [[modular arithmetic|mod]] p of the form <cmath>ax+b\equiv c\pmod{p}</cmath>
  1 KB (225 words) - 15:52, 3 August 2022
• Integral
  ...ath>c</math> is a [[real]] [[constant]]. This is because the derivative of a constant is <math>0</math>. *The integral of a function <math>f(x)</math> is written as <math>\int f(x)\,dx</math>, where
  5 KB (909 words) - 14:16, 31 May 2022
• Divisibility rules/Rule for 2 and powers of 2 proof
  A number <math>N</math> is divisible by <math>2^n</math> if the last <math>{n ...</math>. Since <math>(10^n)k</math> has <math>n</math> trailing 0's, <math>p</math> is the last <math>n</math> digits of the number <math>n</math>.
  2 KB (406 words) - 16:46, 3 May 2020
• Power set
  ...''power set''' of a given [[set]] <math>S</math> is the set <math>\mathcal{P}(S)</math> of all [[subset]]s of that set. It is also sometimes denoted by The [[empty set]] has only one subset, itself. Thus <math>\mathcal{P}(\emptyset) = \{\emptyset\}</math>.
  4 KB (757 words) - 11:44, 8 March 2018
• Perpendicular bisector
  ...the '''perpendicular bisector''' of a [[line segment]] <math>AB</math> is a [[line]] <math>l</math> such that <math>AB</math> and <math>l</math> are [[ ...hat plane. The [[set]] of lines which are perpendicular bisectors of form a plane which is the plane perpendicularly bisecting <math>AB</math>.
  2 KB (367 words) - 15:20, 1 January 2014
• Orthocenter
  The '''orthocenter''' of a [[triangle]] is the point of intersection of its [[altitude|altitudes]]. I ''Note: The orthocenter's existence is a trivial consequence of the trigonometric version of [[Ceva's Theorem]]; how
  5 KB (829 words) - 13:11, 20 February 2024
• Perfect number
  A [[positive integer]] <math>n</math> is called a '''perfect number''' if it is the sum of its [[proper divisor]]s. ...er if and only if <math>n=\frac{p(p+1)}{2}</math>, where <math>p</math> is a [[prime number]] of the form <math>2^k-1</math> for some <math>k</math>.
  1 KB (193 words) - 19:09, 3 March 2022
• 2007 iTest Problems
  ...p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin <math>\mathrm{(A)}\, 4</math>
  30 KB (4,794 words) - 23:00, 8 May 2024
• 1981 IMO Problems/Problem 1
  .../math> to the lines <math>BC, CA, AB</math>, respectively. Find all <math>P</math> for which ...equality exactly when <math>PD = PE = PF </math>, which occurs when <math>P </math> is the triangle's incenter, Q.E.D.
  913 bytes (150 words) - 20:08, 1 February 2021
• 1981 IMO Problems/Problem 4
  ...nsecutive [[positive integer]]s such that the largest number in the set is a [[divisor]] of the [[least common multiple]] of the remaining <math>n-1</ma ...3 must be divisible by a number that is greater than two and is a power of a prime.
  3 KB (516 words) - 09:43, 28 March 2012
• 1981 IMO Problems/Problem 5
  ...nt <math>O</math> and lie inside a given [[triangle]]. Each circle touches a pair of sides of the triangle. Prove that the [[incenter]] and the [[circum ...vely, and let the centers of the circles inscribed in the [[angle]]s <math>A,B,C</math> be denoted <math>O_A, O_B, O_C </math>, respectively.
  2 KB (373 words) - 23:09, 29 January 2021
• Apothem
  The '''apothem''' of a [[regular polygon]] is the [[line segment]] drawn from the [[center]] of th ...of the apothem, <math>a</math>, and the [[perimeter]], <math>p</math>, of a regular polygon, the [[area]] of the polygon is <math>\frac{ap}{2}</math>,
  1 KB (169 words) - 18:22, 9 March 2014
• 2005 AMC 10A Problems
  {{AMC10 Problems|year=2005|ab=A}} <math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \
  14 KB (2,026 words) - 11:45, 12 July 2021
• Euler line
  ...] <math>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain fixed orders and distan ...followed by a homothety with scale factor <math>2</math> centered at <math>A</math> brings <math>\triangle ABC \to \triangle O_AO_BO_C</math>. Let us ex
  59 KB (10,203 words) - 04:47, 30 August 2023
• 2003 AMC 10A Problems
  {{AMC10 Problems|year=2003|ab=A}} <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \
  13 KB (1,900 words) - 22:27, 6 January 2021
• Fermat point
  ...</math> (with no angle more than <math>120^{\circ}</math> is a point <math>P</math> which has the minimum total distance to three [[vertices]] (i.e., <m pair A=(2,4), B=(1,1), C=(6,1);
  4 KB (769 words) - 16:07, 29 December 2019
• Balkan MO Problems and Solutions, with authors
  Triangle ABC has a right angle at C. The internal bisectors of angles BAC and ABC meet ...Q respectively. The points M and N are the feet of the perpendiculars from P and Q
  4 KB (371 words) - 16:41, 1 January 2024
• 1985 IMO Problems
  A circle has center on the side <math>AB</math> of the cyclic quadrilateral < ...efficients, the number of coefficients which are odd is denoted by <math>w(P) </math>. For <math> i = 0, 1, \ldots </math>, let <math>Q_i (x) = (1+x)^i
  3 KB (465 words) - 03:00, 29 March 2021
• 1985 IMO Problems/Problem 1
  A [[circle]] has center on the side <math>AB</math> of the [[cyclic quadrilat ...\frac{ \pi - \angle DAB}{2} = \angle DCO</math>, so <math>DCOT </math> is a cyclic quadrilateral and <math>T </math> is in fact the <math>T</math> of t
  4 KB (684 words) - 07:28, 3 October 2021
• 2003 AMC 10A Problems/Problem 23
  ...iangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of <mat for(pair p : points)
  5 KB (686 words) - 18:01, 28 January 2021
• 1985 IMO Problems/Problem 3
  ...efficients, the number of coefficients which are odd is denoted by <math>w(P) </math>. For <math> i = 0, 1, \ldots </math>, let <math>Q_i (x) = (1+x)^i ...ath> of degree less than <math>2^m </math>, <math> w(P\cdot Q_{2^m}) = 2w (P)</math>.
  2 KB (354 words) - 04:56, 11 March 2023
• 2001 IMO Problems/Problem 1
  ...at <math>\angle C \geq \angle B+30^{\circ}</math>. Prove that <math>\angle A+\angle COP < 90^{\circ}</math>. ...of the circumcircle. But <math>AZ = YP</math> (since <math>AZYP</math> is a rectangle).
  2 KB (417 words) - 17:24, 21 July 2018
• 2001 AMC 8 Problems
  ...class is making a golf trophy. He has to paint <math>300</math> dimples on a golf ball. If it takes him <math>2</math> seconds to paint one dimple, how <math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D)
  13 KB (1,994 words) - 13:04, 18 February 2024
• 1985 AIME Problems
  When a right triangle is rotated about one leg, the volume of the cone produced is ...>b</math>, and <math>c</math> are positive integers which satisfy <math>c=(a + bi)^3 - 107i</math>, where <math>i^2 = -1</math>.
  7 KB (1,071 words) - 19:24, 23 February 2024
• Euclid's Lemma
  '''Euclid's Lemma''' is a result in [[number theory]] attributed to [[Euclid]]. It states that: ...<math>p \mid a</math> or <math>p\mid b</math>, for all [[integer]]s <math>a</math> and <math>b</math>.
  1 KB (188 words) - 04:18, 20 November 2023
• 2002 IMO Shortlist Problems/N3
  ...ry divisor <math>a </math> of <math>m </math>, <math>ka </math> is clearly a divisor of <math>km </math>, but not <math>m </math>. ...</math>, <math> 2^{p_1} +1 </math> is clearly greater than 3 and cannot be a larger power of three, since this would require <math> p_1 \equiv 3 \pmod{6
  10 KB (1,739 words) - 06:38, 12 November 2019
• 2002 IMO Shortlist Problems/A3
  ...of integers with <math> x \neq y </math>. Prove that the equation <math> P(x) = 0 </math> has an integer root. a(x^4 - y^4) + b(x^3 - y^3) + c(x^2 - y^2) + d(x - y) = 0
  4 KB (662 words) - 19:51, 8 September 2009
• Mock AIME 4 2006-2007 Problems/Problem 11
  ..., and <math>r</math> are relatively prime positive integers, compute <math>p+q+r</math>. ...ath>EA = DB = b</math>. Note that we want to compute the ratio <math>\frac{a}{b}</math>.
  2 KB (325 words) - 19:33, 9 February 2017
• Fundamental Theorem of Arithmetic
  ...'' states that every [[positive integer]] <math>n</math> can be written as a product <math>n = p_1 \cdot p_2 \cdot \cdots \cdot p_k</math> where the <ma ...mber such that <math>p \mid ab</math>, then <math>p \mid a</math> or <math>p \mid b</math>. This proof is not terribly interesting, but it does prove t
  2 KB (376 words) - 23:28, 4 August 2022
• 2020 AMC 10A Problems
  {{AMC10 Problems|year=2020|ab=A}} <math>\textbf{(A)}\ {-}\frac{2}{3}\qquad\textbf{(B)}\ \frac{7}{36}\qquad\textbf{(C)}\ \frac{
  13 KB (1,968 words) - 18:32, 29 February 2024
• Asymptote: Getting Started/Windows/Interactive Mode
  ...run the program asy.exe. You will see a black window, prompting you with a <tt>></tt> sign. # Press Enter again, and GSview should pop up, displaying a diagonal line about 1.3 inches in length. (The command drew the path conne
  4 KB (604 words) - 17:36, 10 September 2016
• Asymptote: Advanced Configuration
  ...rs commonly would like to do with their figures: make the images output in a different format, and include Asymptote code directly into LaTeX files. ...ly make your images in pdf format rather than eps format, with the help of a program called '''GhostScript'''.
  12 KB (1,931 words) - 13:53, 26 January 2020
• Asymptote: Reference
  A '''point''' is associated with a cartesian coordinate pair in Asymptote. There are two useful functions tha ...ining points (which can be thought of as paths with length 0) or paths <tt>p</tt>, <tt>q</tt> together with one of the following operators:
  7 KB (1,205 words) - 21:38, 26 March 2024
• Asymptote: Macros and Packages
  ...+r,b+r)</math>. In addition, you want it to simply return the pair <math>(a,b)</math> if no value of <math>r</math> is specified, so you want <math>r</ real a,b;
  3 KB (515 words) - 00:43, 24 April 2019
• Asymptote: Useful commands and their Output
  For each of the following, we have put a blue dot at the origin in order to indicate relative location of the output In addition, a comment after a line such as <tt>//math - extension</tt> indicates that the command (in thi
  6 KB (871 words) - 21:14, 12 June 2023
• Asymptote: Help
  ...ick Program Files, then Asymptote, then the file asy with the stylized red A next to it. ...er, but when I hit the hotkey to compile my code, all that happens is that a window flashes briefly then disappears. Nothing else seems to happen.
  6 KB (1,055 words) - 23:30, 6 October 2019
• Carmichael function
  ...ely prime]] to <math>n</math>. The [[Order_(group theory)|order]] of <math>a\pmod {n}</math> always divides <math>\lambda(n)</math>. <center><p><math>\lambda(n) = \begin{cases}
  2 KB (258 words) - 11:56, 1 August 2022
• Asymptote (geometry)
  An '''asymptote''' is a [[line]] or [[curve]] that a certain [[function]] approaches. ...unction <math>y=\tfrac{2x}{x-2}</math> has a vertical asymptote at x=2 and a horizontal asymptote at y=2]]
  4 KB (664 words) - 11:44, 8 May 2020
• Pigeonhole Principle/Solutions
  The Martian must pull 5 socks out of the drawer to guarantee he has a pair. Also, note that it is possible to pull out 4 socks without obtaining a pair.
  10 KB (1,617 words) - 01:34, 26 October 2021
• 2005 IMO Shortlist Problems/A1
  ...splaystyle q(x) </math> such that <math> \displaystyle p(x)q(x) </math> is a polynomial having all coefficients <math> \pm 1 </math>. ...q(x) </math> both have integral constant terms, the constant term of <math>p(x) </math> must be <math> \pm 1 </math>.
  2 KB (346 words) - 08:56, 14 May 2024
• 2002 AMC 12P Problems/Problem 3
  The dimensions of a rectangular box in inches are all positive integers and the volume of the b <math>\text{(A) }36 \qquad \text{(B) }38 \qquad \text{(C) }42 \qquad \text{(D) }44 \qquad
  1 KB (197 words) - 22:07, 30 December 2023
• 2007 AIME II Problems/Problem 9
  ...iangle]] <math>BEF</math> is [[tangent]] to <math>EF</math> at point <math>P,</math> and the inscribed circle of triangle <math>DEF</math> is tangent to ...thing for <math>F</math>, we find that <math>FD = 105</math> as well. Draw a line through <math>E,F</math> parallel to the sides of the rectangle, to in
  5 KB (818 words) - 11:05, 7 June 2022
• 2007 AIME II Problems/Problem 10
  .../math> is the set of all elements of <math>S</math> which are not in <math>A.</math>) **<math>A</math> must have either 0 or 6 elements, probability: <math>\frac{2}{2^6} =
  8 KB (1,367 words) - 11:48, 23 October 2022
• 2002 USAMO Problems/Problem 3
  Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree <math>n </math> with real ...l polynomial <math>P(x) </math> of degree <math>n </math> such that <math>P(x_i) = y_i </math> for all integers <math> i \in [1,n] </math>.
  4 KB (688 words) - 13:38, 4 July 2013
• Lagrange Interpolation Formula
  .../math> such that for all [[integer]]s <math> 0 \le i \le n </math>, <math> P(x_i) = y_i </math>, and this polynomial is P(x) = \sum_{i=0}^{n}y_i \frac{(x-x_0) \cdots (x-x_{i-1}) (x-x_{i+1}) \cdots
  2 KB (398 words) - 03:50, 20 November 2023
• 2002 USAMO Problems/Problem 5
  ...h>n_1, n_2, \ldots, n_k </math> of positive integers such that <math>n_1 = a</math>, <math>n_k = b </math>, and <math>n_in_{i+1} </math> is divisible by ...ive]] (<math> a \leftrightarrow b \leftrightarrow c </math> implies <math> a \leftrightarrow c </math> ).
  7 KB (1,194 words) - 15:39, 28 March 2015
• Euclidean domain
  A '''Euclidean domain''' (or '''Euclidean ring''') is a type of [[ring]] in which the [[Euclidean algorithm]] can be used. Formally we say that a ring <math>R</math> is a Euclidean domain if:
  2 KB (357 words) - 15:28, 22 August 2009
• Mock AIME 1 Pre 2005 Problems
  ...h that <math>AB = BC = 1, CD = 2,</math> and <math>DE = 9</math>. If <math>P</math> can be any point in space, what is the smallest possible value of <m ...th>1 + 7 + 7^2 + \cdots + 7^{2004}</math> is divided by <math>1000</math>, a remainder of <math>N</math> is obtained. Determine the value of <math>N</ma
  6 KB (1,100 words) - 22:35, 9 January 2016
• Mock AIME 2 Pre 2005/Problems
  <math>x</math> is a real number with the property that <math>x + \frac{1}{x} = 3</math>. Let < ...l, and a black ball. These balls are randomly drawn out of the box one at a time (without replacement) until two of the same color have been removed.
  6 KB (990 words) - 15:23, 11 November 2009
• Mock AIME 3 Pre 2005 Problems
  A function <math>f(x)</math> is defined for all real numbers <math>x</math>. ...are consonants. A string of <math>M's, O's,</math> and <math>P's</math> is a word in Zuminglish if and only if between any two <math>O's</math> there ap
  7 KB (1,135 words) - 23:53, 24 March 2019
• Mock AIME 3 Pre 2005 Problems/Problem 5
  ...are consonants. A string of <math>M's, O's,</math> and <math>P's</math> is a word in Zuminglish if and only if between any two <math>O's</math> there ap ...te the number of <math>n</math>-letter words ending in a vowel followed by a constant (<tt>VC</tt> - the only other combination, two vowels, is impossib
  5 KB (795 words) - 16:03, 17 October 2021
• Mock AIME 3 Pre 2005 Problems/Problem 6
  ...math>r</math> is not divisible by the square of any prime. Determine <math>p + q + r</math>. This is a [[telescoping series]]; note that when we expand the summation, all of the
  3 KB (501 words) - 14:48, 29 November 2019
• Mock AIME 3 Pre 2005 Problems/Problem 7
  ...and <math>q</math> are relatively prime positive integers. Determine <math>p + q</math>. ...is a well-known fact that if <math>ABCD</math> is circumscriptable around a circle then <math>AB+CD=AD+BC</math>. Therefore <math>BC+AD=5</math>. We al
  2 KB (330 words) - 10:23, 4 April 2012
• Mock AIME 3 Pre 2005 Problems/Problem 9
  ...isosceles triangle with base <math>\overline{AB}</math>. <math>D</math> is a point on <math>\overline{AC}</math> and <math>E</math> is the point on the ...at <math>H.</math> Let <math>CH</math> intersect <math>BD</math> at <math>P.</math> Then let <math>AP</math> intersect <math>BC</math> at <math>F.</mat
  2 KB (278 words) - 16:32, 27 December 2019
• Mock AIME 3 Pre 2005 Problems/Problem 14
  ...math>q</math> is not divisible by the square of any prime. Determine <math>p + q + r</math>. <math>A_1=\sqrt{s(s-a)(s-b)(s-c)}</math>, where <math>s=\frac{a+b+c}{2}=\frac{3+6+4}{2}=\frac{13}{2}</math> we have:
  3 KB (563 words) - 02:05, 25 November 2023
• Periodic function
  ...> such that <math>f(x + p) = f(x)</math>. The smallest positive such <math>p</math> is called the period. The most common examples of periodic function
  515 bytes (80 words) - 15:47, 21 August 2011
• 1960 IMO Problems/Problem 3
  ...nteger). Let <math>\alpha</math> be the acute angle subtending, from <math>A</math>, that segment which contains the midpoint of the hypotenuse. Let <ma \tan{\alpha}=\frac{4nh}{(n^2-1)a}.
  3 KB (501 words) - 00:14, 17 May 2015
• 1960 IMO Problems
  ...nteger). Let <math>\alpha</math> be the acute angle subtending, from <math>A</math>, that segment which contains the midpoint of the hypotenuse. Let <ma \tan{\alpha}=\frac{4nh}{(n^2-1)a}.
  3 KB (511 words) - 21:21, 20 August 2020
• 2007 AIME I Problems
  ...ly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the oth ...mber <math>z</math> is equal to <math>9+bi</math>, where <math>b</math> is a positive real number and <math>i^{2}=-1</math>. Given that the imaginary p
  7 KB (1,218 words) - 15:28, 11 July 2022
• 2007 AIME I Problems/Problem 8
  ...and <math>Q_2(x) = 2x^2+ (2k-43)x + k</math> are both [[factor]]s of <math>P(x)</math>? We can see that <math>Q_1</math> and <math>Q_2</math> must have a [[root]] in common for them to both be [[factor]]s of the same cubic.
  4 KB (728 words) - 00:11, 29 November 2023
• 2007 AIME I Problems/Problem 10
  In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so th ...nother row that is complementary to the first. We remove those two and use a similar argument again to show that every group of <math>6</math> rows can
  13 KB (2,328 words) - 00:12, 29 November 2023
• 2007 AIME I Problems/Problem 9
  ...h>q</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>p+q</math>. ...= \frac{57r}{20} = 34 \Longrightarrow r = \frac{680}{57}</math>, and <math>p + q = 737</math>.
  11 KB (1,851 words) - 12:31, 21 December 2021
• 2007 AIME I Problems/Problem 11
  ...) = 2</math> and <math>b(23) = 5</math>. If <math>S = \sum_{p=1}^{2007} b(p),</math> find the [[remainder]] when <math>S</math> is divided by 1000. ...h>. There are <math>2k</math> numbers in this range, so the sum of <math>b(p)</math> over this range is <math>(2k)k=2k^2</math>. <math>44<\sqrt{2007}<45
  3 KB (562 words) - 20:02, 30 December 2023
• 2007 AIME I Problems/Problem 12
  ...{6} + s</math>, where <math>p,q,r,s</math> are integers. Find <math>\frac{p-q+r-s}2</math>. ...sectorpoint(A,B)), Cp=rotate(theta,A)*C, Bp=rotate(theta,A)*B, X=extension(A,Bp,B,C), Y=extension(B,C,Bp,Cp);
  10 KB (1,458 words) - 20:50, 3 November 2023
• 2007 AIME I Problems/Problem 13
  ...id has an area that can be expressed as <math>\sqrt{p}</math>. Find <math>p</math>. Note first that the intersection is a [[pentagon]].
  7 KB (1,034 words) - 21:56, 22 September 2022
• 2007 AIME I Problems/Problem 15
  ...l, and <math>r</math> is an [[integer]] not divisible by the [[square]] of a [[prime]]. Find <math>r</math>. Denote the length of a side of the triangle <math>x</math>, and of <math>\overline{AE}</math> as <
  4 KB (673 words) - 22:14, 6 August 2022
• 2007 AIME II Problems/Problem 3
  pair A, B, C, D, E, F; A = (0,13);
  6 KB (933 words) - 00:05, 8 July 2023
• 2007 AIME II Problems/Problem 15
  .../math>, and <math>\omega</math> is [[externally tangent]] to <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math>. If the sides pair A,B,C,X,Y,Z,P,Q,R;
  11 KB (2,099 words) - 17:51, 4 January 2024
• 2007 AIME II Problems/Problem 14
  Let <math>f(x)</math> be a [[polynomial]] with real [[coefficient]]s such that <math>f(0) = 1,</math> ...2x^3 + x) = 2^max^{3m}</math>. Hence <math>2^ma^2 = 2^ma</math>, and <math>a = 1</math>. Because <math>f(0) = 1</math>, the product of all the roots
  7 KB (1,335 words) - 17:44, 25 January 2022
• 2007 AIME II Problems
  ...rs among the four letters in AIME or the four digits in <math>2007</math>. A set of plates in which each possible sequence appears exactly once contains ...of <math>b</math>, <math>a</math> is a factor of <math>c</math>, and <math>a+b+c=100</math>.
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2005 USAMO Problems
  ...sible to arrange all divisors of <math>n</math> that are greater than 1 in a circle so that no two adjacent divisors are relatively prime. ...th>. Prove that points <math>B_1, C_1,P</math>, and <math>Q</math> lie on a circle.
  4 KB (609 words) - 09:24, 14 May 2021
• 2002 USA TST Problems
  Let <math> \displaystyle ABC </math> be a triangle. Prove that \cos\frac{A-B}{2} + \cos\frac{B-C}{2} + \cos\frac{C-A}{2}.
  3 KB (544 words) - 06:58, 3 August 2017
• 1961 IMO Problems/Problem 4
  In the interior of [[triangle]] <math>P_1P_2P_3</math> a [[point]] <math>P</math> is given. Let <math>Q_1,Q_2,Q_3</math> be the [[intersection]]s of {{IMO box|year=1961|num-b=3|num-a=5}}
  2 KB (399 words) - 00:34, 4 September 2023
• 1961 IMO Problems/Problem 5
  ...h> where ''M'' is the midpoint of ''BC''. Prove that the construction has a solution if and only if ...arc BD we have <math>\angle BPD = \omega</math>. Draw a circle with center A and radius AC, and the point of intersection of this circle and the major a
  1 KB (205 words) - 04:12, 7 June 2021
• Newton's Inequality
  ...e a root of <math>P' </math> <math>k </math> times. It follows that <math>P' </math> must have <math>m-1 </math> non-positive, real roots, i.e., for so P'(t) = m \prod_{i=1}^{m-1}(t+x'_i)
  5 KB (830 words) - 04:05, 28 January 2023
• 1989 USAMO Problems
  ...tegers <math>0 < a,\ b,\ c,\ d < 1000000</math> such that <math>T_{1988} = a S_{1989} - b</math> and <math>U_{1988} = c S_{1989} - d</math>. ...laying in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.
  2 KB (326 words) - 18:52, 18 July 2016
• 2003 USAMO Problems
  ...The lengths of all sides and all diagonals of the polygon <math> \mathcal{P} </math> are rational numbers. Prove that the lengths of all sides of all p A = a_0,a_1,a_2,\dots, a_n
  3 KB (487 words) - 09:21, 14 May 2021
• 2003 USAMO Problems/Problem 2
  ...The lengths of all sides and all diagonals of the polygon <math> \mathcal{P} </math> are rational numbers. Prove that the lengths of all sides of all p ...gon be <math>AC </math> and <math>BD </math>. Since <math>ABCD </math> is a convex quadrilateral with sides and diagonals of rational length, we consid
  2 KB (335 words) - 20:50, 29 April 2014
• 2007 USAMO Problems/Problem 1
  ...m Vandervelde'') Let <math>n</math> be a [[positive]] [[integer]]. Define a [[sequence]] by setting <math>a_1 = n</math> and, for each <math>k>1</math> ...integers, so <math>b_{k+1} \le b_k</math>. As the <math>b_k</math>'s form a [[non-increasing]] sequence of positive integers, they must eventually beco
  6 KB (1,204 words) - 20:06, 23 August 2023
• 2007 USAMO Problems/Problem 2
  (''Gregory Galperin'') A [[square]] grid on the [[Cartesian plane|Euclidean plane]] consists of all ...ngent]] circles with radius greater than or equal to 5, one can always fit a circle with radius greater than <math>\frac{1}{\sqrt{2}}</math> between tho
  5 KB (754 words) - 03:41, 7 August 2014
• 2007 USAMO Problems/Problem 3
  (''András Gyárfás'') Let <math>S</math> be a [[set]] containing <math>n^2+n-1</math> [[element]]s, for some [[positive]] ...ad <math>k(n+1)-1</math> elements, then we must have k disjoint subsets in a class.
  6 KB (1,071 words) - 08:40, 21 October 2020
• 2007 USAMO Problems/Problem 4
  (''Reid Barton'') An ''animal'' with <math>n</math> ''cells'' is a connected figure consisting of <math>n</math> equal-sized [[Square (geometr ...two or more dinosaurs. Find with proof the [[maximum]] number of cells in a primitive dinosaur.
  10 KB (1,878 words) - 14:56, 30 June 2021
• 2004 IMO Shortlist Problems/N2
  a) Prove that <math> \displaystyle \psi(mn) = \psi(m)\psi(n) </math> for ever ...thbf{N} </math> the equation <math> \displaystyle \psi(x) = ax </math> has a solution. <br>
  6 KB (1,007 words) - 09:10, 29 August 2011
• Dirichlet convolution
  ...</math>, <math>d_md_n = d </math>. Thus <math>p </math> is bijective. As a result of our claim, we have the identity <math> \sum_{d_m \mid m, d_n \mid n}a(d_m)b(d_n) = \sum_{d_m \mid m} a(d_m) \cdot \sum_{d_n \mid n}b(d_n) </math>,
  3 KB (613 words) - 21:40, 21 June 2009
• 2005 IMO Shortlist Problems/A3
  Four real numbers <math> \displaystyle p,q,r,s </math> satisfy <math> \displaystyle p+q+r+s = 9 </math>
  3 KB (542 words) - 17:09, 19 December 2018
• Sum and difference of powers
  ...s''' are powerful factoring techniques that, respectively, factor a sum or a difference of certain powers. <math>a^{2n+1}+b^{2n+1}=(a+b)(a^{2n}-a^{2n-1}b+a^{2n-2}b^2-\ldots-ab^{2n-1}+b^{2n})</math>
  3 KB (450 words) - 06:25, 6 May 2024
• Chicken McNugget Theorem
  ...ritten in the form <math>am + bn</math> for [[nonnegative]] integers <math>a, b</math> is <math>mn-m-n</math>. A consequence of the theorem is that there are exactly <math>\frac{(m - 1)(n
  17 KB (2,748 words) - 19:22, 24 February 2024
• Zermelo-Fraenkel Axioms
  ...ven property describable by the language of [[set theory]] could be called a [[set]]. As shown by paradoxes such as [[Russell's Paradox]], some restrict The language of set theory consists of a single binary relation <math>\in</math>. As such, all axioms can be written
  4 KB (732 words) - 20:49, 13 October 2019
• Russell's Paradox
  ...cannot be in itself, in spite of satisfying its own membership criterion, a contradiction. Assuming it doesn't, it must be in itself, in spite of not s
  580 bytes (100 words) - 17:57, 12 May 2023
• 2005 AMC 12A Problems/Problem 24
  ...e exist a polynomial <math>R(x)</math> of degree 3 such that <math>P(Q(x))=P(x) \cdot R(x)</math>? <math>\mathrm {(A) } 19 \qquad \mathrm {(B) } 22 \qquad \mathrm {(C) } 24 \qquad \mathrm {(D)
  5 KB (833 words) - 18:17, 8 April 2024
• 2005 AMC 12A Problems/Problem 22
  ...inscribed in a sphere of radius <math>r</math>. The surface area of <math>P</math> is 384, and the sum of the lengths of its 12 edges is 112. What is < <math>\mathrm{(A)}\ 8\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 12\qquad \mathrm{(D)}\ 14\q
  2 KB (334 words) - 10:20, 16 September 2022
• 2007 AMC 12A Problems/Problem 9
  <math>\mathrm{(A)}\ \frac 23\qquad \mathrm{(B)}\ \frac 34\qquad \mathrm{(C)}\ \frac 45\qquad ...r}</math>. If Yan walks directly to the stadium, then assuming he walks at a rate of <math>1</math>, it will take him <math>x-r</math> units of time. Si
  5 KB (804 words) - 14:55, 21 August 2022
• 2002 Indonesia MO Problems/Problem 7
  ...> are three points on the rhombus' perimeter. If <math>PQRS</math> is also a rhombus, show that exactly one of <math>Q</math>, <math>R</math>, and <math Firstly, all rhombi are [[parallelogram|parallelograms]], so that <math>P</math> is the [[centroid]] of <math>ABCD</math>.
  4 KB (734 words) - 12:41, 18 August 2020
• 2006 AIME II Problems/Problem 3
  ...[[odd integer]]s. Find the largest integer <math>k </math> such that <math>P </math> is divisible by <math>3^k .</math> Therefore, we have a total of <math>97-48=\boxed{049}</math> threes.
  4 KB (562 words) - 18:37, 30 October 2020
• 2006 AIME II Problems/Problem 12
  ...| divisible]] by the [[perfect square | square]] of any prime, find <math> p+q+r. </math> pair A=(0,0),B=(-3^.5,-3),C=(3^.5,-3),D=13*expi(-2*pi/3),E1=11*expi(-pi/3),F=E1+D;
  6 KB (1,033 words) - 02:36, 19 March 2022
• 2006 AIME II Problems/Problem 5
  ...te faces is 7. When two such dice are rolled, the probability of obtaining a sum of 7 is <math>47/288</math>. Given that the [[probability]] of obtainin ...th>A</math> and a 6 on die <math>B</math> or a 6 on die <math>A</math> and a 1 on die <math>B</math>:
  5 KB (712 words) - 12:10, 5 November 2023
• 2006 AIME II Problems/Problem 9
  ...and <math>r</math> is not divisible by the square of any prime, find <math>p+q+r.</math> ...>\triangle O_1r_1r \sim \triangle O_2r_2r</math> since both triangles have a [[right angle]] and have vertical angles, and the same goes for <math>\tria
  3 KB (553 words) - 10:45, 26 August 2015
• 2006 AIME II Problems/Problem 13
  ...c{(2n+1) + (2(n+j)-1)}{2}\right) = j(2n+j)</math>. Thus, <math>j</math> is a factor of <math>N</math>. ...qual <math>0</math>. We then perform casework based on the parity of <math>p-q</math>.
  4 KB (675 words) - 10:40, 14 July 2022
• 2006 AIME II Problems/Problem 11
  A [[sequence]] is defined as follows <math> a_1=a_2=a_3=1, </math> and, for a ...sums, and with a little bit of substitution and algebra we see that <math>(p, q, r) = (\frac{1}{2}, 0, \frac{1}{2})</math>, at least for the first few t
  3 KB (417 words) - 10:07, 12 October 2023
• 2006 AIME II Problems/Problem 10
  ...</math> beats team <math> B. </math> The [[probability]] that team <math> A </math> finishes with more points than team <math> B </math> is <math> m/n, ...We let this probability be <math>p</math>; then the probability that <math>A</math> and <math>B</math> end with the same score in these five games is <m
  6 KB (983 words) - 13:42, 8 December 2021
• 2006 AIME II Problems
  ...vex hexagon <math>ABCDEF</math>, all six sides are congruent, <math>\angle A</math> and <math>\angle D</math> are right angles, and <math>\angle B, \ang ...\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>.
  8 KB (1,350 words) - 12:00, 4 December 2022
• 1988 IMO Problems/Problem 6
  ...>b</math> be positive integers such that <math>ab + 1</math> divides <math>a^{2} + b^{2}</math>. Show that \frac {a^{2} + b^{2}}{ab + 1}
  4 KB (720 words) - 12:26, 7 April 2024
• 1998 PMWC Problems
  Find a pair of consecutive Triangular Numbers and the difference between a pair of consecutive Square Numbers whose difference are both <math>11</math ...<math>OB = OA=3</math> cm. <math>\angle BOA</math> is a right angle. <math>A</math> and <math>B</math> are two points on the circumference of circle of
  11 KB (1,738 words) - 19:25, 10 March 2015
• 1999 PMWC Problems
  label("$A$", (3.9,6.75), NE * labelscalefactor); pair P,Q,R,S;
  6 KB (703 words) - 21:21, 21 April 2014
• 2000 PMWC Problems
  <math>y</math> is a number that has <math>8</math> different factors (including the number <mat Given that <math>A^4=75600\times B</math>. If <math>A</math> and <math>B</math> are positive integers, find the smallest value of
  11 KB (1,713 words) - 22:47, 13 July 2023
• 2001 PMWC Problems
  void wdot(pair p) fill(circle(p,.025),white);
  4 KB (641 words) - 21:24, 21 April 2014
• 2002 PMWC Problems
  label("$\textbf{A}$",(-1,0)); pair A,B,C,D,E,F,G;
  5 KB (725 words) - 16:07, 23 April 2014
• 2003 PMWC Problems
  pair M=(-1,0), N=(1,0),a=4/5*expi(pi/10),b=expi(37pi/100); draw((M--N)^^(origin--a)^^(origin--b));
  7 KB (918 words) - 16:15, 22 April 2014
• Logic
  ...le of a statement is "A duck is a bird" which is true. Another example is "A pencil does not exist" which is false. ...ath>P</math> then <math>Q</math>. For example, "If it is a duck then it is a bird."
  3 KB (437 words) - 17:22, 31 July 2020
• British Flag Theorem
  ...British flag theorem due to the similarities between the British flag and a diagram of the points (shown below): pair A,B,C,D,P;
  3 KB (405 words) - 01:14, 3 December 2023
• 1989 USAMO Problems/Problem 3
  ...</math> and <math>b</math> such that <math>P(a + bi) = 0</math> and <math>(a^2 + b^2 + 1)^2 < 4 b^2 + 1</math>. ...h>z_1, \dotsc, z_n</math> be the (not necessarily distinct) roots of <math>P</math>, so that
  2 KB (340 words) - 19:11, 18 July 2016
• 1971 Canadian MO Problems
  <math>DEB</math> is a chord of a circle such that <math>DE=3</math> and <math>EB=5 .</math> Let <math>O</mat <math>ABCD</math> is a quadrilateral with <math>AD=BC</math>. If <math>\angle ADC</math> is greate
  3 KB (519 words) - 08:58, 13 September 2012
• 1961 IMO Problems
  \quad x + y + z \!\!\! &= a \; \, \\ ...</math> and <math>b </math> are constants. Give the conditions that <math>a </math> and <math>b </math> must satisfy so that <math>x, y, z </math> (the
  3 KB (425 words) - 21:18, 20 August 2020
• 1990 USAMO Problems
  A certain state issues license plates consisting of six digits (from 0 throug A sequence of functions <math>\, \{f_n(x) \} \,</math> is defined recursively
  3 KB (386 words) - 20:47, 3 July 2013
• 2007 AMC 12B Problems
  <math> \mathrm{(A) \ } 678\qquad \mathrm{(B) \ } 768\qquad \mathrm{(C) \ } 786\qquad \mathrm{ A college student drove his compact car 120 miles home for the weekend and av
  12 KB (1,814 words) - 12:58, 19 February 2020
• 1999 AHSME Problems
  <math> \mathrm{(A) \ -50 } \qquad \mathrm{(B) \ -49 } \qquad \mathrm{(C) \ 0 } \qquad \mathrm <math>\text{(A) All equilateral triangles are congruent to each other.}</math>
  13 KB (1,945 words) - 18:28, 19 June 2023
• 1960 IMO Problems/Problem 5
  ...ABCDA'B'C'D'</math> (with face <math>ABCD</math> directly above face <math>A'B'C'D'</math>). a) Find the locus of the midpoints of the segments <math>XY</math>, where <ma
  2 KB (402 words) - 23:28, 18 July 2016
• Asymptote: Useful functions
  The bundled <code>markers</code> package provides a useful function for marking angles. The exact same function exists in the < arrowbar ''arrow''=None, pen ''p''=currentpen,<br/>
  4 KB (646 words) - 21:18, 26 March 2024
• Asymptote: CSE5
  ...of two paths without common points, which may be useful if you try to draw a picture just from its verbal description and do not want to make any calcul CSE5 contains doubles of each function; a function with a full name and a function with an abbreviated name.
  4 KB (607 words) - 15:09, 30 June 2020
• Excircle
  ...is a [[circle]] [[Tangent line|tangent]] to the extensions of two sides of a [[triangle]] and the third side. pair excenter(pair A, pair B, pair C){
  5 KB (843 words) - 03:02, 1 July 2020
• 2001 USAMO Problems/Problem 2
  ...t <math>AD_2</math> at two points, the closer of which to the vertex <math>A</math> is denoted by <math>Q</math>. Prove that <math>AQ = D_2P</math>. It is well known that the excircle opposite <math>A</math> is tangent to <math>\overline{BC}</math> at the point <math>D_2</mat
  11 KB (2,091 words) - 08:35, 16 November 2017
• Factor Theorem
  ...theorem''' is a theorem regarding the relationships between the factors of a polynomial and its roots. ...y, you can determine whether a number in the form <math>f(a)</math> (<math>a</math> is constant, <math>f</math> is polynomial) is <math>0</math> using p
  3 KB (508 words) - 12:31, 2 May 2024
• Analysis
  ...itions and theorems concerning continuity and convergence. Analysis covers a wide variety of topics which appear quite different to each other, so there ...lon delta definition of continuity and Cauchy who tried to put calculus on a "firm logical basis."
  2 KB (352 words) - 18:12, 8 February 2015
• Zorn's Lemma
  '''Zorn's Lemma''' is a [[set theory | set theoretic]] result which is equivalent to the [[Axiom of Let <math>A</math> be a [[partially ordered set]].
  9 KB (1,669 words) - 19:02, 1 August 2018
• Ideal
  In [[ring theory]], an '''ideal''' is a special kind of [[subset]] of a [[ring]]. Two-sided ideals in rings are the [[kernel]]s of ring [[homomorp ...xa \in \alpha</math>, for all <math>x\in R</math> and <math>a\in \mathfrak{a}</math>. Symbolically, this can be written as
  8 KB (1,389 words) - 23:44, 17 February 2020
• 2004 AMC 12A Problems/Problem 23
  A [[polynomial]] <cmath>P(x) = c_{2004}x^{2004} + c_{2003}x^{2003} + ... + c_1x + c_0</cmath>
  3 KB (517 words) - 14:13, 5 September 2021
• Binet's Formula
  To derive a general formula for the Fibonacci numbers, we can look at the interesting q ...{1-\sqrt{5}}{2}.</math> We get that <cmath>F_n = \lambda_1 v^n + \lambda_2 p^n.</cmath>Plugging in our initial conditions, we get
  3 KB (550 words) - 16:12, 24 February 2024
• 2002 AMC 12B Problems/Problem 24
  ...ral]] <math>ABCD</math> with area <math>2002</math> contains a point <math>P</math> in its interior such that <math>PA = 24, PB = 32, PC = 28, PD = 45</ <math>\mathrm{(A)}\ 4\sqrt{2002} \qquad \mathrm{(B)}\ 2\sqrt{8465} \qquad \mathrm{(C)}\ 2</m
  2 KB (313 words) - 10:23, 4 July 2013
• 1995 AHSME Problems/Problem 4
  ...math> of <math>P</math>, and <math>N</math> is <math>50 \%</math> of <math>P</math>, then <math>\frac {M}{N} =</math> <math> \mathrm{(A) \ \frac {3}{250} } \qquad \mathrm{(B) \ \frac {3}{25} } \qquad \mathrm{(C)
  2 KB (232 words) - 22:30, 16 February 2018
• 2007 iTest Problems/Problem 1
  ...p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin <math>(2,2+2)\equiv (2,4)</math>. <math>4</math> isn't a prime, so this isn't a set of twin primes.
  702 bytes (113 words) - 12:33, 29 January 2021
• 1976 USAMO Problems/Problem 5
  If <math>P(x)</math>, <math>Q(x)</math>, <math>R(x)</math>, and <math>S(x)</math> are <cmath> P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x +1) S(x), </cmath>
  3 KB (572 words) - 17:14, 16 August 2015
• 2003 Polish Mathematical Olympiad Problems/Problem 3
  ..."natural number" is ambiguous, but this is irrelevant, as every integer is a divisor of <math>2^0 -1 = 0</math>.'' ..., <math>2^q \equiv 2 \pmod{q}</math>, so <math>2 \equiv 1 \pmod{q}</math>, a contradiction.
  5 KB (919 words) - 23:29, 20 January 2016
• 2000 AMC 12 Problems/Problem 10
  The point <math>P = (1,2,3)</math> is reflected in the <math>xy</math>-plane, then its image \text {(A) } (1,7, - 3) \qquad \text {(B) } ( - 1,7, - 3) \qquad \text {(C) } ( - 1,
  1 KB (173 words) - 08:56, 21 August 2023
• 2008 AMC 12B Problems/Problem 23
  <math>\text{(A)}\ 11\qquad \text{(B)}\ 12\qquad \text{(C)}\ 13\qquad \text{(D)}\ 14\qquad ...h> is a perfect square. When these are multiplied, they equal <math>2^{a+n-a} \times 5^{b+n-b} = 10^n</math>. <math>\log 10^n=n</math> so the number of
  5 KB (814 words) - 18:02, 17 January 2023
• Vieta's formulas
  ...elementary symmetric polynomials]] of its roots can be easily expressed as a ratio between two of the polynomial's coefficients. It is among the most ubiquitous results to circumvent finding a polynomial's roots in competition math and sees widespread usage in many ma
  3 KB (515 words) - 19:04, 19 May 2024
• 2000 AMC 12 Problems/Problem 13
  ...ut were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family? <math>\text {(A)}\ 3 \qquad \text {(B)}\ 4 \qquad \text {(C)}\ 5 \qquad \text {(D)}\ 6 \qqu
  6 KB (996 words) - 18:14, 29 June 2023
• 2000 AMC 12 Problems/Problem 15
  Let <math>f</math> be a function for which <math>f\left(\dfrac{x}{3}\right) = x^2 + x + 1</math>. F <cmath>\text {(A)}\ -1/3 \qquad \text {(B)}\ -1/9 \qquad \text {(C)}\ 0 \qquad \text {(D)}\
  3 KB (441 words) - 21:11, 29 April 2023
• 2000 AMC 12 Problems/Problem 22
  ...ows a portion of the [[curve]] defined by the quartic [[polynomial]] <math>P(x) = x^4 + ax^3 + bx^2 + cx + d</math>. Which of the following is the small <math>\textbf{(A)}\ P(-1)\\
  1 KB (212 words) - 14:37, 5 June 2022
• 2000 AMC 12 Problems/Problem 25
  ...[equilateral triangle]]s, each of a different color, are used to construct a regular [[octahedron]]. How many distinguishable ways are there to construc triple A=(0,0,1);
  7 KB (998 words) - 21:04, 23 December 2023
• 2007 AMC 10A Problems
  {{AMC10 Problems|year=2007|ab=A}} ...ath>25\%</math> discount. Pam buys 5 tickets using a coupon that gives her a <math>30\%</math> discount. How many more dollars does Pam pay than Susan?
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• 2007 AMC 10A Problems/Problem 5
  <math>\text{(A)}\mathdollar 1.76 \qquad \text{(B)}\mathdollar 5.84 \qquad \text{(C)}\mathd We let <math>p =</math> cost of one pencil in dollars, <math>n = </math> cost of one note
  1 KB (133 words) - 12:10, 3 June 2021
• 1995 AHSME Problems
  ...7, 83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will <math> \mathrm{(A) \ \text{remain the same} } \qquad \mathrm{(B) \ \text{increase by 1} } \qq
  17 KB (2,387 words) - 22:44, 26 May 2021
• 1997 PMWC Problems
  ...represent a single digit. Different letters represent different digits but a box can represent any digit. What does the five-digit number <math>\mathrm{ ...\hline &\Box &\Box & 9 &\Box\\ \Box &\Box &\Box & 7 &\\ \hline H & A & P & P & Y\end{array} </cmath>
  15 KB (2,057 words) - 19:13, 10 March 2015
• 1951 AHSME Problems
  <math> \textbf{(A) \ } \frac {100(M - N)}{M} \qquad \textbf{(B) \ } \frac {100(M - N)}{N} \qq A rectangular field is half as wide as it is long and is completely enclosed
  23 KB (3,641 words) - 22:23, 3 November 2023
• 1959 AHSME Problems
  Each edge of a cube is increased by <math>50</math>%. The percent of increase of the surfa <math>\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300
  22 KB (3,345 words) - 20:12, 15 February 2023
• 1966 AHSME Problems
  <math>\text{(A)} \ \frac 18 \qquad \text{(B)} \ \frac 73 \qquad \text{(C)} \ \frac78 \qqua When the base of a triangle is increased 10% and the altitude to this base is decreased 10%, t
  19 KB (3,159 words) - 22:10, 11 March 2024
• 1991 USAMO Problems/Problem 5
  ...B \,</math>, prove that the point <math>\, E \, </math> traces the arc of a circle. pair A=(0,0), B=(220,0), C=(18.7723,118.523);
  5 KB (953 words) - 12:18, 4 September 2018
• 1991 USAMO Problems/Problem 2
  where "<math>\Sigma</math>" denotes a sum involving all nonempty subsets <math>S</math> of <math>\{1,2,3, \ldots, ...when <math>n >0</math>. We will prove this equation, but first, we prove a lemma.
  3 KB (490 words) - 07:38, 19 July 2016
• 2002 AMC 12B Problems/Problem 16
  ...ided die. What is the [[probability]] that the product of the two rolls is a multiple of 3? <math>\mathrm{(A)}\ \frac 1{12}
  2 KB (317 words) - 10:26, 5 November 2023
• 2002 AMC 12B Problems/Problem 18
  ...ath>(0,0),(2,0),(2,1),(0,1)</math>. What is the [[probability]] that <math>P</math> is closer to the [[origin]] than it is to the point <math>(3,1)</mat <math>\mathrm{(A)}\ \frac 12
  3 KB (376 words) - 19:16, 20 August 2019
• Brocard point
  ...is a minimum. These points are named after Henri Brocard (1845 – 1922), a French mathematician. https://en.wikipedia.org/wiki/Brocard_points and here
  626 bytes (98 words) - 20:13, 27 January 2024
• 2006 Alabama ARML TST Problems/Problem 10
  Let <math>p</math> be the probability that Scooby Doo solves any given mystery. The pro ...m{2006}{1800}\cdot p^{1800}(1-p)^{206}=\binom{2006}{1801}\cdot p^{1801} (1-p)^{205}</cmath>
  1,003 bytes (140 words) - 20:29, 20 May 2012
• Construction
  ...to mark off segments, draw circles and arcs, and draw straight lines) are a branch of [[geometry]] that rely on the use of basic geometrical [[axiom]]s A '''compass''' is a tool that can draw circles and arcs of circles.
  3 KB (443 words) - 20:52, 28 August 2014
• 2003 AMC 12B Problems/Problem 22
  ...Let <math>N</math> be a point on <math>\overline{AB}</math>, and let <math>P</math> and <math>Q</math> be the feet of the perpendiculars from <math>N</m pair O = (15*15/17,8*15/17), C = (17,0), D = (0,0), P = (25.6,19.2), Q = (25.6, 18.5);
  4 KB (551 words) - 14:17, 23 June 2022
• 2003 AMC 12B Problems/Problem 24
  ...ive integers <math>a,b,</math> and <math>c</math> are chosen so that <math>a<b<c</math>, and the system of [[equation]]s <center><math>2x + y = 2003 \quad</math> and <math>\quad y = |x-a| + |x-b| + |x-c|</math></center>
  7 KB (1,183 words) - 11:47, 15 February 2016
• 1998 AHSME Problems/Problem 26
  In [[quadrilateral]] <math>ABCD</math>, it is given that <math>\angle A = 120^{\circ}</math>, angles <math>B</math> and <math>D</math> are [[right <math>\mathrm{(A)}\ 60
  3 KB (391 words) - 14:30, 5 July 2013
• 1974 USAMO Problems/Problem 1
  ...s impossible that <math>P(a)=b</math>, <math>P(b)=c</math>, and <math>P(c)=a</math>. ...th> is a polynomial with integral coefficients, then <cmath>a - b | P(a) - P(b).</cmath> (Why?)
  7 KB (1,291 words) - 20:30, 27 April 2020
• 1995 USAMO Problems
  ...th> in base <math>\, p-1 \,</math> and reading the result in base <math>\, p</math>. A calculator is broken so that the only keys that still work are the <math>\,
  3 KB (540 words) - 13:31, 4 July 2013
• 2004 AMC 12B Problems/Problem 19
  ...d cone has horizontal bases with radii <math>18</math> and <math>2</math>. A sphere is tangent to the top, bottom, and lateral surface of the truncated <math>\mathrm{(A)}\ 6
  3 KB (520 words) - 19:12, 20 November 2023
• 1990 USAMO Problems/Problem 5
  ...\, Q \,</math>. Prove that the points <math>\, M, N, P, Q \,</math> lie on a common circle. ...of the altitude from <math>A</math>: <math>A</math>, <math>H</math>, <math>A'</math> are concurrent, where <math>H</math> is the orthocentre.
  3 KB (604 words) - 20:52, 24 October 2018
• 1994 USAMO Problems
  ...the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in ...>AD</math>, <math>BE</math>, and <math>CF</math> are concurrent. Let <math>P</math> be the intersection of <math>AD</math> and <math>CE</math>. Prove th
  2 KB (391 words) - 07:58, 19 July 2016
• 2002 AMC 10A Problems/Problem 22
  ...fied repeatedly by the following operation: remove all tiles numbered with a [[perfect square]], and renumber the remaining tiles consecutively starting <math>\text{(A)}\ 10 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qq
  3 KB (430 words) - 23:13, 13 September 2023
• Isogonal conjugate
  ...nal conjugates''' are pairs of [[point]]s in the [[plane]] with respect to a certain [[triangle]]. ...ll</math> and a point <math>O</math> lying on <math>\ell</math> be given. A pair of lines symmetric with respect to <math>\ell</math> and containing t
  54 KB (9,416 words) - 08:40, 18 April 2024
• 2008 AMC 10A Problems
  {{AMC10 Problems|year=2008|ab=A}} A bakery owner turns on his doughnut machine at <math>\text{8:30}\ {\small\te
  14 KB (2,138 words) - 15:08, 18 February 2023
• 1976 IMO Problems/Problem 5
  ...om the set <math>\{ - 1,0,1\}</math><math>.</math> Prove that there exists a solution <math>x_{1}, \ldots,x_{q}</math> for the system with the propertie '''a.)''' all <math>x_{j}, j = 1,\ldots,q</math> are integers<math>;</math>
  2 KB (377 words) - 16:28, 29 January 2021
• Partition (combinatorics)
  ...partitions of 3: <math>3 = 2+1 =1+1+1</math>. Each of the [[summand]]s is a ''part'' of the partition. ...ence of a formula! No simpler formula is known, and the existence of such a formula is doubtful.
  10 KB (1,508 words) - 14:24, 17 September 2017
• 2008 AMC 12A Problems
  {{AMC12 Problems|year=2008|ab=A}} A bakery owner turns on his doughnut machine at <math>\text{8:30}\ {\small\te
  13 KB (2,025 words) - 13:56, 2 February 2021
• 2008 AMC 12A Problems/Problem 19
  <math>\mathrm{(A)}\ 195\qquad\mathrm{(B)}\ 196\qquad\mathrm{(C)}\ 224\qquad\mathrm{(D)}\ 378 ...+ x + x^2 + \cdots + x^{27}\right)</math>. We are expanding <math>A \cdot A \cdot B</math>.
  5 KB (895 words) - 22:54, 9 January 2021
• 2008 AMC 12A Problems/Problem 13
  ...rcle]] centered at <math>O</math>, and <math>\angle AOB = 60^\circ</math>. A second circle is internally [[tangent]] to the first and tangent to both <m <math>\mathrm{(A)}\ \frac{1}{16}\qquad\mathrm{(B)}\ \frac{1}{9}\qquad\mathrm{(C)}\ \frac{1}{
  4 KB (630 words) - 20:32, 4 June 2021
• 2008 AMC 12A Problems/Problem 20
  <math>\mathrm{(A)}\ \frac{1}{28}\left(10-\sqrt{2}\right)\qquad\mathrm{(B)}\ \frac{3}{56}\lef pair C=(0,0),A=(0,3),B=(4,0),D=(4-2.28571,1.71429);
  6 KB (951 words) - 16:31, 2 August 2019
• 2008 AMC 12A Problems/Problem 11
  Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the <math>13</math> visible numbers hav pen p = linewidth(1);
  2 KB (295 words) - 00:11, 19 April 2020
• 2007 AMC 12B Problems/Problem 14
  ...R</math>, and <math>S</math> are the feet of the perpendiculars from <math>P</math> to <math>\overline{AB}</math>, <math>\overline{BC}</math>, and <math <math>\mathrm{(A)}\ 4 \qquad \mathrm{(B)}\ 3\sqrt{3} \qquad \mathrm{(C)}\ 6 \qquad \mathrm{(
  3 KB (401 words) - 22:58, 8 May 2023
• 1992 USAMO Problems/Problem 3
  ...integer <math>n \le 1500</math>, there is a subset <math>S</math> of <math>A</math> for which <math>\sigma(S) = n</math>. What is the smallest possible ...ts Y\subseteq Z</math>, <math>\sigma(Y)=x</math>, <math>\nexists \sigma(Y)=p+1</math>.
  5 KB (858 words) - 07:52, 19 July 2016

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