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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>

2 KB (297 words) - 14:54, 25 June 2023
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

10 KB (1,410 words) - 13:07, 20 February 2024
Physics
Physics 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 balanced if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is

4 KB (692 words) - 22:33, 15 February 2021
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

13 KB (2,048 words) - 15:28, 22 February 2024
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))

10 KB (809 words) - 16:40, 17 March 2024
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

3 KB (551 words) - 16:22, 13 September 2023
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]].

7 KB (1,296 words) - 14:22, 22 October 2023
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

5 KB (849 words) - 16:14, 18 May 2021
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

5 KB (898 words) - 19:12, 28 January 2024
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.

4 KB (588 words) - 12:47, 2 October 2022
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

3 KB (496 words) - 23:22, 5 January 2022
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]].

5 KB (892 words) - 21:52, 1 May 2021
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.

1 KB (178 words) - 20:34, 6 March 2022
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>

2 KB (401 words) - 09:46, 31 January 2018
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

1,007 bytes (165 words) - 00:28, 30 December 2023
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

1 KB (203 words) - 01:25, 1 March 2024
Square (geometry)
A '''square''' is a [[quadrilateral]] in which all [[edge|sides]] have equal length and all [[a pair A, B, C, D;

1 KB (169 words) - 01:12, 13 June 2022
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,

3 KB (602 words) - 09:01, 7 June 2023
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

8 KB (1,469 words) - 21:11, 16 September 2022
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>

12 KB (2,377 words) - 11:48, 22 July 2009
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.

11 KB (2,082 words) - 15:23, 2 January 2022
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.

789 bytes (115 words) - 17:08, 29 December 2021
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

641 bytes (97 words) - 00:28, 31 December 2020
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>.

3 KB (566 words) - 03:51, 12 February 2021
Vector
...y, but also most abstractly, a vector is any object which is an element of a given vector space. ...(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]].

7 KB (1,265 words) - 13:22, 14 July 2021
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

6 KB (1,003 words) - 00:02, 20 May 2024
Iff
It is also known as a [[conditional|biconditional]] statement. ...h>p\iff q</math> means <math>p\implies q</math> and <math>q\implies p</math> at the same time.

903 bytes (140 words) - 02:13, 24 December 2020
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;

3 KB (575 words) - 15:27, 19 March 2023
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

9 KB (1,434 words) - 13:10, 20 February 2024
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;

1 KB (238 words) - 22:51, 20 February 2022
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>,

4 KB (690 words) - 13:11, 20 February 2024
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

3 KB (496 words) - 13:35, 18 January 2023
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

7 KB (1,290 words) - 12:18, 30 May 2019
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

5 KB (912 words) - 20:06, 14 March 2023
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:

7 KB (1,201 words) - 16:59, 19 February 2024
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>

5 KB (778 words) - 13:10, 29 November 2017
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]]

2 KB (255 words) - 15:27, 22 February 2024
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

5 KB (832 words) - 14:22, 11 January 2024
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

2 KB (268 words) - 03:02, 3 January 2021
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

2 KB (361 words) - 01:59, 24 January 2020
Prime Number Theorem
constants <math>A</math> and <math>B</math>, <cmath> \pi(x) \sim \frac{x}{A \log x - B} . </cmath>

10 KB (1,729 words) - 19:52, 21 October 2023
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)

1,006 bytes (134 words) - 14:15, 6 March 2022
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.

3 KB (422 words) - 11:01, 25 December 2020
Algebraic topology
...of [[topology]] using methods from [[abstract algebra]]. In general, given a [[topological space]], we can associate various algebraic objects, such as ...a,1)=q(a)</math>, and <math>g(0,b)=g(1,b)=x</math>. We call <math>g</math> a [[homotopy]]. Now define <math>\pi_1(X)=L/\sim</math>. That is, we equate a

3 KB (479 words) - 15:35, 1 December 2015
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

9 KB (1,547 words) - 03:04, 13 January 2021
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:

1 KB (217 words) - 09:32, 13 August 2011
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>

7 KB (1,173 words) - 03:31, 4 January 2023
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);

6 KB (980 words) - 21:45, 31 March 2020
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),

5 KB (730 words) - 15:05, 15 January 2024
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

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