Search results

Create the page "A s t r a" on this wiki! See also the search results found.

Power Mean Inequality
The '''Power Mean Inequality''' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] I ...ath>, the power mean with exponent <math>t</math>, where <math>t\in\mathbb{R}</math>, is defined by

3 KB (606 words) - 23:59, 1 July 2022
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
Combination
...generally concerned with finding the number of combinations of size <math>r</math> from an original set of size <math>n</math> ...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
...f and only if there exists a constant <math>t</math> such that <math>a_i = t b_i</math> for all <math>1 \leq i \leq n</math>, or if one list consists of ...there exists a scalar <math>t</math> such that <math>\overrightarrow{v} = t \overrightarrow{w}</math>, or if one of the vectors is zero. This formulati

13 KB (2,048 words) - 15:28, 22 February 2024
Ptolemy's theorem
...[equality condition | equality case]] of [[Ptolemy's Inequality]]. Ptolemy's theorem frequently shows up as an intermediate step in problems involving i ...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
...r theory include the [[Birch and Swinnerton-Dyer Conjecture]] and [[Fermat's Last Theorem]]. ...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

5 KB (849 words) - 16:14, 18 May 2021
Divisibility rules
...integer]]s are [[divisibility | divisible]] by particular other [[integer]]s. All of these rules apply for [[Base number| base-10]] ''only'' -- other b https://youtu.be/6xNkyDgIhEE?t=1699

8 KB (1,315 words) - 18:18, 2 March 2024
Jensen's Inequality
'''Jensen's Inequality''' is an inequality discovered by Danish mathematician Johan Jen ...[convex function]] of one real variable. Let <math>x_1,\dots,x_n\in\mathbb R</math> and let <math>a_1,\dots, a_n\ge 0</math> satisfy <math>a_1+\dots+a_n

3 KB (623 words) - 13:10, 20 February 2024
Combinatorial identity
== Pascal's Identity == Pascal's Identity states that

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
Chain Rule
...to J</math> and <math>f:J \to \mathbb{R}</math>. Let <math>h:I \to \mathbb{R}</math> such that <math>h(x) = f(g(x)) \forall x \in I</math>. If <math>x_ ...<math>h'(x_0)</math>,<math>f'(g(x_0))</math>, and <math>g'(x_0)</math> is a matrix.)

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>: ...ath> majorizes <math>(4,2)</math> (as <math>5>4, 5+1=4+2</math>), Muirhead's inequality states that for any positive <math>x,y</math>,

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 This section is for people who know what [[integral]]s are but don't know the Fundamental Theorem of Calculus yet, and would like to try to figu

11 KB (2,082 words) - 15:23, 2 January 2022
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
Logarithm
...would be a pain to have to calculate any time you wanted to use it (say in a comparison of large numbers). Its natural logarithm though (partly due to ...ly 7 digits before the decimal point. Comparing the logs of the numbers to a given precision can allow easier comparison than computing and comparing th

4 KB (680 words) - 12:54, 16 October 2023
Roots of unity
...ts of unity come up when we examine the [[complex number|complex]] [[root]]s of the [[polynomial]] <math> x^n=1 </math>. ...making <math>r^n=1\Rightarrow r=1</math> (magnitude is always expressed as a positive real number). This leaves us with <math>e^{ni\theta} = e^{2\pi ik

3 KB (558 words) - 21:36, 11 December 2011
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
Rational approximation
...n [[real number]] <math>x</math> can be approximated by [[rational number]]s. Of course, since the rationals are dense on the real line, we, surely, can ...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
Trigonometry
...s and angles of triangles through the '''trigonometric functions'''. It is a fundamental branch of mathematics, and its discovery paved the way towards ...[[Law of Sines]] and the [[Law of Cosines]]; many more, such as [[Stewart's Theorem]], are most easily proven using trigonometry. In algebra, expressio

8 KB (1,217 words) - 20:15, 7 September 2023
Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
...ric 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
...every [[even integer]] greater than two is the sum of two [[prime number]]s. The conjecture has been tested up to 400,000,000,000,000. Goldbach's conjecture is one of the oldest unsolved problems in [[number theory]] and

7 KB (1,201 words) - 16:59, 19 February 2024
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
Majorization
We say a [[nonincreasing]] [[sequence]] of [[real number]]s <math> a_1, \ldots ,a_n</math> '''majorizes''' another nonincreasing sequen ...also say that this is the case if and only if for all <math> t \in \mathbb{R} </math>,

2 KB (288 words) - 22:48, 5 July 2023
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
Modular arithmetic/Introduction
...is article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily https://youtu.be/7an5wU9Q5hk?t=777

15 KB (2,396 words) - 20:24, 21 February 2024
Modular arithmetic/Intermediate
...or <math>a \equiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. ...Z}_n</math> for short). This structure gives us a useful tool for solving a wide range of number-theoretic problems, including finding solutions to [[D

14 KB (2,317 words) - 19:01, 29 October 2021
2006 AIME I Problems
In quadrilateral <math> ABCD , \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <mat ...f <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 13
...st integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect square]]. ...ldots,</math> and <math>2^{n-1}-2^{n-2} = 2^{n-2}</math> elements of <math>S</math> that are divisible by <math>2^1</math> but not by <math>2^2</math>.

10 KB (1,702 words) - 00:45, 16 November 2023
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
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
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
Newman's Tauberian Theorem
'''Newman's Tauberian Theorem''' is a [[tauberian theorem]] Let <math>f:(0,+\infty)\to\mathbb C</math> be a bounded function. Assume that

6 KB (1,034 words) - 07:55, 12 August 2019
2005 AIME I Problems/Problem 4
...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 ...the number of students is <math>n(n + 7)</math> which must be 5 more than a perfect square, so <math>n \leq 14</math>. In fact, when <math>n = 14</mat

8 KB (1,248 words) - 11:43, 16 August 2022
2005 AIME I Problems/Problem 6
...ath>1 + ir</math> and <math>1 - ir</math>. Their product is <math>P = 1 + r^2 = 1 + \sqrt{2006}</math>. <math>44^2 = 1936 < 2006 < 2025 = 45^2</math> If you think of each part of the product as a quadratic, then <math>((x-1)^2+\sqrt{2006})</math> is bound to hold the two

4 KB (686 words) - 01:55, 5 December 2022
2005 AIME II Problems/Problem 14
pair A = origin; pair C = rotate(15,A)*(A+dir(-50));

13 KB (2,129 words) - 18:56, 1 January 2024
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
...th> 7+77+777+7+7=875 </math> could be obtained from eight <math> 7 </math>'s in this way. For how many values of <math> n </math> is it possible to inse ...</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
1990 AIME Problems
...neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. ...9}{58}</math> as large as each interior angle of <math>P_2^{}</math>. What's the largest possible value of <math>s_{}^{}</math>?

6 KB (870 words) - 10:14, 19 June 2021
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 The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught <math

8 KB (1,275 words) - 06:55, 2 September 2021
1997 AIME Problems
...math> rectangles, of which <math>s</math> are squares. The number <math>s/r</math> can be written in the form <math>m/n,</math> where <math>m</math> an ...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

7 KB (1,098 words) - 17:08, 25 June 2020
2001 AIME I Problems
...than the mean of <math>\mathcal{S}</math>. Find the mean of <math>\mathcal{S}</math>. ...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- ...gram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the r

8 KB (1,374 words) - 21:09, 27 July 2023
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
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. ...l("$P$",P,1.5*dir(80)); label("$R$",R,NE); label("12",waypoint(O1--O2,0.4),S);</asy>

13 KB (2,149 words) - 18:44, 5 February 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);

19 KB (3,221 words) - 01:05, 7 February 2023
1984 AIME Problems/Problem 8
...math> is a root of <math>z^6+z^3+1</math>, then <math>0=(r^3-1)(r^6+r^3+1)=r^9-1</math>. The polynomial <math>x^9-1</math> has all of its roots with [[a ...\circ </math>, then <math>r^6+r^3+1=3</math>. (When we multiplied by <math>r^3 - 1</math> at the beginning, we introduced some extraneous solutions, and

3 KB (430 words) - 19:05, 7 February 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
1986 AIME Problems/Problem 13
...tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by <tt>TH</tt>, <tt>HH</tt>, and etc. For ex ...tches to <tt>H</tt> four times; hence it follows that our string will have a structure of <tt>THTHTHTH</tt>.

4 KB (772 words) - 21:09, 7 May 2024
1987 AIME Problems/Problem 6
...XY = WZ</math>, <math>PQ = PQ</math> and the [[area]]s of the [[trapezoid]]s <math>PQZW</math> and <math>PQYX</math> are the same, then the heights of t ...is equal to <math>[PXYQ]=\frac{19}{2}* \frac{XY+87}{2}=\frac{19}{2}* \frac{a+b+106}{2}</math>. Setting these equal to each other and solving gives <math

3 KB (530 words) - 07:46, 1 June 2018
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
1989 AIME Problems/Problem 14
...lex number]] of the form <math>r+si</math>, where <math>r</math> and <math>s</math> are integers, can be uniquely expressed in the base <math>-n+i</math <center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center>

2 KB (408 words) - 17:28, 16 September 2023
1989 AIME Problems/Problem 11
...ach between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique [[mode]] (most frequent value). Let <math>D</math> be the difference ...be <math>M</math>, and the sum of the numbers <math>\neq x</math> be <math>S</math>. Then

5 KB (851 words) - 18:01, 28 December 2022
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
1992 AIME Problems/Problem 5
...^{}_{}</math> are not necessarily distinct. To write the elements of <math>S^{}_{}</math> as fractions in lowest terms, how many different numerators ar ...ere aren't any numbers which are multiples of <math>37^2</math>, so we can't get numerators which are multiples of <math>37</math>. Therefore <math>648

2 KB (277 words) - 20:45, 4 March 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? ...100).</math> Then at time <math>t</math>, they end up at points <math>(-50+t,100)</math> and <math>(-50+3t,-100).</math>

8 KB (1,231 words) - 20:06, 26 November 2023
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 Since the [[coefficient]]s of the [[polynomial]] are real, it follows that the non-real roots must com

3 KB (451 words) - 15:02, 6 September 2021
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
2000 AIME I Problems/Problem 15
A stack of <math>2000</math> cards is labelled with the integers from <math>1 ...r logic as Solution 1, we find that 1999 has position <math>1024</math> in a <math>2048</math> card stack, where the fake cards towards the front.

15 KB (2,673 words) - 19:16, 6 January 2024
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>. pair A,B,C,P,Q;

8 KB (1,275 words) - 03:04, 27 February 2022
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
2003 AIME I Problems/Problem 13
...math>2</math> representation has more <math>1</math>'s than <math>0</math>'s. Find the [[remainder]] when <math> N </math> is divided by <math>1000</mat ...</math> notation, with <math>k+1</math> of the digits being <math>1</math>'s.

4 KB (651 words) - 19:42, 7 October 2023
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>. ...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 point tha

7 KB (1,127 words) - 13:34, 19 June 2022
2001 AIME II Problems/Problem 7
...ine{ST}</math> with <math>S</math> on <math>\overline{PR}</math> and <math>T</math> on <math>\overline{QR}</math>, such that <math>\overline{ST}</math> pair P = (0,0), Q = (90, 0), R = (0, 120), S=(0, 60), T=(45, 60), U = (60,0), V=(60, 40), O1 = (30,30), O2 = (15, 75), O3 = (70, 10

7 KB (1,112 words) - 02:15, 26 December 2022
2000 AIME II Problems/Problem 13
...<math>r</math> are relatively prime, and <math>r>0</math>. Find <math>m+n+r</math>. Thus <math>r=\frac{-1+\sqrt{161}}{40}</math>, and so the final answer is <math>-1+161+40

6 KB (1,060 words) - 17:36, 26 April 2024
2006 USAMO Problems/Problem 1
...ath> be a prime number and let <math>s</math> be an integer with <math>0 < s < p </math>. Prove that there exist integers <math>m</math> and <math>n</ma <math> \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
Let <math>f(x)</math> be a non-constant polynomial in <math>x</math> of degree <math>d</math> with Let <math>g(n) = f(n^2)</math>, then <math>g(n)</math> is a polynomial of degree <math>2</math> or

9 KB (1,699 words) - 13:48, 11 April 2020
2006 USAMO Problems/Problem 6
...th>, <math>SBF</math>, <math>TCF</math>, and <math>TDE</math> pass through a common point. ...imilarity, and <math>X</math> is the center of spiral similarity for <math>A,E,B,</math> and <math>F</math>.

5 KB (986 words) - 22:46, 18 May 2015
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
Mock AIME 1 2006-2007 Problems/Problem 9
...h>m</math> and <math>n</math> are [[relatively prime]] [[positive integer]]s, compute the sum of the positive [[prime number | prime]] [[divisor | facto ..._{10}+a_{11}+a_{12}+...</math>. If the sum of all distinct values of <math>S</math> is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math>

5 KB (744 words) - 19:46, 20 October 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. ...\alpha\beta\gamma=r</math>. Then we can write <math>P(x)=x^3 - px^2 - qx - r</math>.

3 KB (568 words) - 15:50, 3 April 2012
Mock AIME 1 2006-2007 Problems/Problem 14
...</math> with a variable 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 ...ee below an attempted solution to understand why this problem doesn't have a solution:

3 KB (541 words) - 17:32, 22 November 2023
Mock AIME 1 2006-2007/Problems
...^{7}</math>. If <math>m</math> is the number of elements in <math>\mathcal{S}</math>, compute <math>\star(m)</math>. ...\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
1971 Canadian MO Problems/Problem 4
.../math> such that the two polynomials <math>x^2+ax+1</math> and <math>x^2+x+a</math> have at least one root in common. Let this root be <math>r</math>. Then we have

2 KB (346 words) - 23:20, 18 January 2023
2007 iTest Problems
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</ <math>\mathrm{(A)}\, 4</math>

30 KB (4,794 words) - 23:00, 8 May 2024
1987 IMO Problems/Problem 6
..., then we can conclude that <math>m</math> is a prime. Since there must be a factor of <math>m</math> less than <math>\sqrt{m}</math>. ...t let <math>r=\lfloor\sqrt{n/3}\rfloor</math>, then we can write <math>n=3(r+\epsilon)^2(0\leq\epsilon< 1)</math>, so <math>h=6r\epsilon+3\epsilon^2\leq

2 KB (430 words) - 13:03, 24 February 2024
Euler line
...fact of Euclidean [[geometry]]. Certain fixed orders and distance [[ratio]]s hold among these points. In particular, <math>\overline{OGNH}</math> and < ...s examine what else this transformation, which we denote as <math>\mathcal{S}</math>, will do.

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
2003 AMC 10A Problems/Problem 17
...rcumscribed circle. What is the radius, in inches, of the circle? </onlyinclude> <math> \mathrm{(A) \ } \frac{3\sqrt{2}}{\pi}\qquad \mathrm{(B) \ } \frac{3\sqrt{3}}{\pi}\qqu

2 KB (414 words) - 13:48, 4 April 2024
1985 IMO Problems/Problem 1
A [[circle]] has center on the side <math>AB</math> of the [[cyclic quadrilat ...>O</math> be the center of the circle mentioned in the problem. Let <math>T</math> be the second intersection of the circumcircle of <math>CDO </math>

4 KB (684 words) - 07:28, 3 October 2021
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
2002 IMO Shortlist Problems/A3
...x + d </math>, where <math> a, b, c, d </math> are [[integer]]s and <math> a \neq 0 </math>. Suppose that <math> xP(x) = yP(y) </math> for infinitely m 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
Asymptote: Reference
A '''point''' is associated with a cartesian coordinate pair in Asymptote. There are two useful functions tha ...ewise cubic function of a parameter <math>t</math>, parameterized as <math>t</math> ranges from <math>0</math> to the number of nodes (say <math>n</math

7 KB (1,205 words) - 21:38, 26 March 2024
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 2
Let <math>ABC </math> be a triangle such that \left( \cot \frac{A}{2} \right)^2 + \left( 2 \cot \frac{B}{2} \right)^2 + \left( 3 \cot \frac{C

2 KB (298 words) - 22:32, 6 April 2016
2002 USAMO Problems/Problem 4
...ber]]s. Determine all functions <math> f : \mathbb{R} \rightarrow \mathbb{R} </math> such that ...math>, then we obtain <math>f(x^2) = xf(x) </math>. Therefore the problem's condition becomes

3 KB (490 words) - 22:09, 18 September 2018
Mock AIME 1 Pre 2005 Problems
...tive integers with three distinct digits. Compute the remainder when <math>S</math> is divided by <math>1000</math>. <math>A, B, C, D,</math> and <math>E</math> are collinear in that order such that <

6 KB (1,100 words) - 22:35, 9 January 2016
Mock AIME 3 Pre 2005 Problems/Problem 5
...>P's</math> is a word in Zuminglish if and only if between any two <math>O's</math> there appear at least two consonants. Let <math>N</math> denote the ...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
2007 AIME I Problems/Problem 13
...oints of <math>AE</math>, <math>BC</math>, and <math>CD</math>. The plane's intersection with the pyramid has an area that can be expressed as <math>\s Note first that the intersection is a [[pentagon]].

7 KB (1,034 words) - 21:56, 22 September 2022
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
1961 IMO Problems/Problem 2
...>b</math>, and <math>c</math> be the lengths of a triangle whose area is ''S''. Prove that <math>a^2 + b^2 + c^2 \ge 4S\sqrt{3}</math>

5 KB (860 words) - 13:12, 13 February 2024
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 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
2007 USAMO Problems/Problem 6
...ga</math>. Let <math>P_A</math> and <math>Q_A</math> denote the [[center]]s of <math>\omega_A</math> and <math>\Omega_A</math>, respectively. Define p <cmath>8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,</cmath>

7 KB (1,274 words) - 15:11, 31 August 2017
Circumradius
...very triangle is [[cyclic]], every triangle has a circumscribed circle, or a [[circumcircle]]. ==Formula for a Triangle==

4 KB (729 words) - 16:52, 19 February 2024
LaTeX:Layout
...his allows you to use the same code in many source files by just including a single line in each source file. ...font size, which is 10pt by default but can be increased to 11pt or 12pt. A reference on other options for this command can be found [http://www.nada.k

30 KB (5,171 words) - 10:16, 4 April 2021
LaTeX:Commands
| <math>a^{i+1}_3</math>||a^{i+1}_3||<math>x^{3^2}</math>||x^{3^2} Notice that we can apply both a subscript and a superscript at the same time. For subscripts or superscripts with more than

12 KB (1,898 words) - 15:31, 22 February 2024
2007 AMC 12A Problems/Problem 9
...both choices require the same amount of time. What is the [[ratio]] of Yan's distance from his home to his distance from the stadium? <math>\mathrm{(A)}\ \frac 23\qquad \mathrm{(B)}\ \frac 34\qquad \mathrm{(C)}\ \frac 45\qquad

5 KB (804 words) - 14:55, 21 August 2022
2005 AMC 12A Problems/Problem 6
...had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met? (\mathrm {A}) \ 4 \qquad (\mathrm {B}) \ 5 \qquad (\mathrm {C})\ 6 \qquad (\mathrm {D})

1,006 bytes (166 words) - 21:18, 3 July 2013
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
2007 AMC 12B Problems
Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 fee <math> \mathrm{(A) \ } 678\qquad \mathrm{(B) \ } 768\qquad \mathrm{(C) \ } 786\qquad \mathrm{

12 KB (1,814 words) - 12:58, 19 February 2020
Asymptote: Useful functions
The bundled <code>markers</code> package provides a useful function for marking angles. The exact same function exists in the < ...e functions, you can replace <code>line l1, line l2</code> with <code>pair A, pair O, pair B</code>, which would mark the angle <math>\angle AOB</math>.

4 KB (646 words) - 21:18, 26 March 2024
Ideal
...two-sided ideals of rings are similar to [[normal subgroup]]s of [[group]]s. ...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
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
1998 AHSME Problems/Problem 26
...\circ}</math>, angles <math>B</math> and <math>D</math> are [[right angle]]s, <math>AB = 13</math>, and <math>AD = 46</math>. Then <math>AC=</math> <math>\mathrm{(A)}\ 60

3 KB (391 words) - 14:30, 5 July 2013
2004 AMC 12B Problems/Problem 23
<math>\mathrm{(A)}\ 250,\!000 Let the roots be <math>r,s,r + s</math>, and let <math>t = rs</math>. Then

3 KB (533 words) - 14:52, 29 October 2023
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
2008 AMC 10A Problems
{{AMC10 Problems|year=2008|ab=A}} ...}\ {\small\text{AM}}</math> the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?

14 KB (2,138 words) - 15:08, 18 February 2023
2004 AMC 12A Problems/Problem 19
...th>. Circles <math>B</math> and <math>C</math> are congruent. Circle <math>A</math> has radius <math>1</math> and passes through the center of <math>D</ pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0);

5 KB (785 words) - 00:29, 31 July 2023
2008 AMC 12B Problems
A basketball player made <math>5</math> baskets during a game. Each basket was worth either <math>2</math> or <math>3</math> points. <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \

14 KB (2,199 words) - 13:43, 28 August 2020
2008 AMC 12B Problems/Problem 25
...B=11, BC=5, CD=19,</math> and <math>DA=7</math>. Bisectors of <math>\angle A</math> and <math>\angle D</math> meet at <math>P</math>, and bisectors of < <math>\textbf{(A)}\ 28\sqrt{3}\qquad \textbf{(B)}\ 30\sqrt{3}\qquad \textbf{(C)}\ 32\sqrt{3}

12 KB (2,015 words) - 20:54, 9 October 2022
2008 AIME I Problems
Of the students attending a school party, <math>60\%</math> of the students are girls, and <math>40\%</ ...l whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking, jogging, and swimming rates.

9 KB (1,536 words) - 00:46, 26 August 2023
2008 AIME II Problems
...te which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin ...by <math>14</math> cm. Ten slices are cut from the cheese. Each slice has a width of <math>1</math> cm and is cut parallel to one face of the cheese. T

7 KB (1,167 words) - 21:33, 12 August 2020
2008 AIME I Problems/Problem 11
...math>B</math>, and <math>AABAA</math>, while <math>BBAB</math> is not such a sequence. How many such sequences have length 14? ...math>n-2</math> ending in a <math>B</math>. Thus, we have the [[recursion]]s

7 KB (1,173 words) - 22:39, 28 November 2023
2008 AIME I Problems/Problem 14
...a</math>. Point <math>P</math> is the foot of the perpendicular from <math>A</math> to line <math>CT</math>. Suppose <math>\overline{AB} = 18</math>, an ...- (9,0), A= O + (9,0), C=A+(18,0), T = 9 * expi(-1.2309594), P = foot(A,C,T);

8 KB (1,333 words) - 00:18, 1 February 2024
Mock AIME 5 Pre 2005 Problems
&\text {ii. all ten digits, a through j are all distinct.}\\ &\text {iii.} a + b + c + d + e + f + g + h + i + j = 45\end{align*}</cmath>

6 KB (909 words) - 07:27, 12 October 2022
Mock AIME 1 2007-2008 Problems
...to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent? ...1+i)^{17} - (1-i)^{17}</math>, where <math>i=\sqrt{-1}</math>. Find <math>|S|</math>.

6 KB (992 words) - 14:15, 13 February 2018
2008 AIME II Problems/Problem 2
...mark at exactly the same time. How many minutes has it taken them?</onlyinclude> ...</math> be the time Rudolph takes disregarding breaks and <math>\frac{4}{3}r</math> be the time Jennifer takes disregarding breaks. We have the equation

3 KB (432 words) - 11:53, 19 June 2019
2008 AIME II Problems/Problem 7
Let <math>r</math>, <math>s</math>, and <math>t</math> be the three roots of the equation Find <math>(r + s)^3 + (s + t)^3 + (t + r)^3</math>.

7 KB (1,251 words) - 19:18, 2 January 2024
2008 AIME II Problems/Problem 9
...h> units in the positive <math>x</math>-direction. Given that the particle's position after <math>150</math> moves is <math>(p,q)</math>, find the great ...the x-axis. If <math>(x', y')</math> is the position of the particle after a move from <math>P</math>, then we have two equations for <math>x'</math> an

5 KB (725 words) - 22:37, 28 January 2024
2008 AIME II Problems/Problem 11
real r=44-6*35^.5; pair A=(0,96),B=(-28,0),C=(28,0),X=C-(64/3,0),Y=B+(4*r/3,0),P=X+(0,16),Q=Y+(0,r),M=foot(Q,X,P);

6 KB (1,065 words) - 20:12, 9 August 2022
1974 USAMO Problems/Problem 5
pair C=(0,1), A=(1,6), B=(4,1), D=(1.5,2); draw(D--A--B--C--D--B);

7 KB (1,221 words) - 18:57, 3 July 2013
2008 AMC 10A Problems/Problem 15
<math>\mathrm{(A)}\ 120\qquad\mathrm{(B)}\ 130\qquad\mathrm{(C)}\ 140\qquad\mathrm{(D)}\ 150 ...d as <math>I</math>, and set Han's speed as <math>H</math>. Therefore, Jan's speed is <math>H+5.</math>

6 KB (1,033 words) - 15:19, 1 July 2021
2008 USAMO Problems/Problem 1
<cmath> k_0 k_1 \dotsc k_n = a^2+a+1, </cmath> ...on <math>n</math>. For our base case, <math>n=0</math>, we may let <math>a</math> be positive integer.

11 KB (1,964 words) - 03:38, 17 August 2019
2008 USAMO Problems/Problem 5
...<math>y</math> and write <math>y - x</math> in its place. Prove that after a finite number of such operations, we can end up with at least one <math>0</ ...gers <math>A = \left < a_1, a_2, a_3 \right ></math> so that <math>R \cdot A = 0</math> continues to hold. (For example, if we replace <math>r_1</math>

2 KB (407 words) - 23:37, 21 July 2020
2008 USAMO Problems/Problem 3
(''Gabriel Carroll'') Let <math>n</math> be a positive integer. Denote by <math>S_n</math> the set of points <math>(x, y) .../math> paths (a partition of <math>S_n</math> into <math>m</math> paths is a set <math>\mathcal{P}</math> of <math>m</math> nonempty paths such that eac

9 KB (1,585 words) - 01:00, 14 August 2014
Sylow Theorems
...eory of [[finite]] [[group]]s. They give a partial converse to [[Lagrange's Theorem]], and are one of the most important results in the field. They ar :'''Theorem.''' Every finite group contains a [[Sylow p-subgroup |Sylow <math>p</math>-subgroup]].

11 KB (2,071 words) - 12:25, 9 April 2019
Routh's Theorem
...be the intersection of <math>CF</math> and <math>AD</math>. Then, '''Routh's Theorem''' states that <cmath>[GHI]=\dfrac{(rst-1)^2}{(rs+r+1)(st+s+1)(tr+t+1)}[ABC]</cmath>

2 KB (267 words) - 00:02, 24 March 2021
Category (category theory)
A category, <math>\mathcal{C}</math>, is a mathematical object consisting of: * A [[class]], <math>\text{Ob}(\mathcal{C})</math> of objects.

5 KB (792 words) - 19:01, 7 April 2012
2008 iTest Problems
...s as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fue When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo.

71 KB (11,749 words) - 01:31, 2 November 2023
2002 USAMO Problems
...^{2002} </math>. Prove that it is possible to color every subset of <math>S </math> either blue or red so that the following conditions hold: (a) the union of any two red subsets is red;

3 KB (486 words) - 09:21, 14 May 2021
2002 AMC 12A Problems/Problem 7
...</math> arc of circle B. What is the ratio of circle A's area and circle B's area? <math>\textbf{(A)}\ 4/9 \qquad \textbf{(B)}\ 2/3 \qquad \textbf{(C)}\ 5/6 \qquad \textbf{(D)

3 KB (492 words) - 14:46, 31 January 2024
2002 AMC 12A Problems/Problem 11
<math>\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ ...in, we find <math>d=12</math>. From <math>d=rt</math>, we find that <math>r</math>, our answer, is <math>\boxed{\textbf{(B) }48 }</math>.

2 KB (396 words) - 00:19, 1 February 2024
2021 GMC 10B
<math>\textbf{(A)} ~99 \qquad\textbf{(B)} ~100 \qquad\textbf{(C)} ~101 \qquad\textbf{(D)} ~1 ...an area of <math>\frac{\pi}{\sqrt{2}}</math> is <math>r</math>. Find <math>r^{2}</math>

11 KB (1,695 words) - 14:33, 7 March 2022
1971 IMO Problems/Problem 6
Let <math>A = (a_{ij})(i, j = 1, 2, \cdots, n)</math> be a square matrix whose elements are non-negative integers. Suppose that whenev ...ecome smaller. So we have<cmath> T \ge (n - S)^2 + S^2 \ge 2(\frac{n - S + S}{2})^2 = \frac{n^2}{2}</cmath>by AM - QM inequality.

6 KB (1,192 words) - 14:14, 29 January 2021
2009 AMC 12A Problems/Problem 19
...ns were <math>A</math> and <math>B</math>, respectively. Each polygon had a side length of <math>2</math>. Which of the following is true? ...B\qquad \textbf{(D)}\ A </math> <math>= \frac {7}{5}B\qquad \textbf{(E)}\ A = \frac {49}{25}B</math>

4 KB (630 words) - 21:27, 30 December 2023
2009 AMC 12A Problems/Problem 16
A circle with center <math>C</math> is tangent to the positive <math>x</math> <math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \

2 KB (314 words) - 21:07, 16 January 2020
2001 AMC 12 Problems/Problem 11
...tly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white \textbf{(A) }\frac {3}{10}

2 KB (423 words) - 12:21, 11 June 2022
2002 AMC 12A Problems/Problem 24
...d pairs of real numbers <math>(a,b)</math> such that <math>(a+bi)^{2002} = a-bi</math>. \text{(A) }1001

4 KB (743 words) - 19:54, 14 March 2024
2009 AMC 10A Problems/Problem 6
...micircle but outside the circle is shaded. What fraction of the semicircle's area is shaded? \mathrm{(A)}\ \frac{1}{2}

2 KB (380 words) - 09:21, 8 June 2021
2009 AMC 12B Problems/Problem 20
...h> pyramids and a new polyhedron <math>R</math>. How many edges does <math>R</math> have? <math>\mathrm{(A)}\ 200\qquad

9 KB (1,567 words) - 13:43, 19 August 2023
2009 AMC 10B Problems
...nt muffin or a <math>75</math>-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? <math>\text{(A) } 1\qquad\text{(B) } 2\qquad\text{(C) } 3\qquad\text{(D) } 4\qquad\text{(E

15 KB (2,262 words) - 00:53, 18 June 2021
2009 AMC 10B Problems/Problem 24
real r=5, R=6; path t=r*dir(0)--r*dir(20)--R*dir(20)--R*dir(0);

3 KB (546 words) - 04:26, 16 January 2023
2009 AIME I Problems/Problem 2
...s a complex number <math>z</math> with imaginary part <math>164</math> and a positive integer <math>n</math> such that Let <math>z = a + 164i</math>.

5 KB (850 words) - 12:53, 23 December 2023
2009 AIME I Problems/Problem 12
...cle having <math>\overline{CD}</math> as a diameter. Let <math>I</math> be a point outside <math>\triangle ABC</math> such that <math>\overline{AI}</mat ...h>. Hence <math>AQ = 144, BP = 1225, AB = 1369</math> and the radius <math>r = OD = 210</math>.

12 KB (1,970 words) - 22:53, 22 January 2024
Cauchy's Integral Formula
'''Cauchy's Integral Formula''' is a fundamental result in [[complex analysis]]. It states that if <math>U</math> is a subset of

4 KB (689 words) - 17:19, 18 January 2024
2009 AIME II Problems
...ng a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finis ...ath>b</math>, and <math>c</math> are positive real numbers such that <math>a^{\log_3 7} = 27</math>, <math>b^{\log_7 11} = 49</math>, and <math>c^{\log_

8 KB (1,366 words) - 21:33, 3 January 2021
2000 AMC 8 Problems
<math>\mathrm{(A)}\ 15\qquad\mathrm{(B)}\ 16\qquad\mathrm{(C)}\ 17\qquad\mathrm{(D)}\ 21\qqu <math>\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qqua

15 KB (2,165 words) - 03:32, 13 April 2024
Radical axis
...n should peruse this material carefully, as it may contain the keys to one's future success. ...rtains to that situation. I hope after you read this text, you will become a better math student, armed with another tool to solve difficult problems. B

10 KB (1,797 words) - 02:05, 24 October 2023
2009 AIME II Problems/Problem 15
...<math>t</math> is not divisible by the square of any prime. Find <math>r+s+t</math>. ...c{3}{5}</math>, point <math>B</math> lies between <math>M</math> and <math>A</math> on the semicircular arc. We will first compute the length of <math>\

11 KB (1,849 words) - 19:43, 2 January 2023
1990 AJHSME Problems
<math>\text{(A)}\ 947 \qquad \text{(B)}\ 1037 \qquad \text{(C)}\ 1047 \qquad \text{(D)}\ 1 <math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad

15 KB (2,059 words) - 15:03, 6 October 2021
1990 AJHSME Problems/Problem 10
...ext{C}</math>. If this sum equals the sum of the dates behind <math>\text{A}</math> and <math>\text{B}</math>, then the letter is for(int a=0; a<6; ++a)

1 KB (231 words) - 10:51, 8 September 2019
Base introduction
Bases. These confuse a lot of us. Hopefully this will make it better. ...as a subscript. Every number in base 10 (our normal numbers) does not have a subscript.

3 KB (552 words) - 20:57, 9 July 2009
2009 USAMO Problems/Problem 5
...}</math> and <math>\overline{BC}</math> at points <math>R</math> and <math>S</math>, respectively. Prove that quadrilateral <math>PQRS</math> is cyclic We will use directed angles in this solution. Extend <math>QR</math> to <math>T</math> as follows:

6 KB (973 words) - 19:24, 18 October 2018
2009 USAMO Problems/Problem 6
...r <math>r</math> such that <math>(s_i - s_j)r</math> and <math>(t_i - t_j)/r</math> are integers for all <math>i</math> and <math>j</math>. ...ac{c_2}{d_2} - \frac{c_1}{d_1})</math> is an integer. Now we can set <math>r = \frac{b_1 b_2}{d_1 d_2}</math>, because the least common denominator of <

2 KB (463 words) - 12:19, 21 August 2020
Semisimple module
A '''semisimple module''' is, informally, a [[module]] that is not far removed from [[simple module]]s.

6 KB (1,183 words) - 15:02, 18 August 2009
1994 IMO Problems/Problem 3
...2k\}</math> whose base 2 representation has precisely three <math>1</math>s. * (a) Prove that, for each positive integer <math>m</math>, there exists at leas

7 KB (1,298 words) - 19:59, 9 February 2024
Asymptote: Olympiad Package Part 1
...er and over and over again. Thus, the olympiad package is a lifesaver. Let's look at this simple bit of code: (Note, the import olympiad command is not real r=2.5, a=3.5;

1 KB (209 words) - 01:48, 7 January 2010
1993 AJHSME Problems
Which pair of numbers does NOT have a product equal to <math>36</math>? <math>\text{(A)}\ \{ -4,-9\} \qquad \text{(B)}\ \{ -3,-12\} \qquad \text{(C)}\ \left\{ \df

14 KB (1,797 words) - 11:13, 28 December 2022
2010 AMC 12A Problems/Problem 7
...and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan m <math>\textbf{(A)}\ 0.04 \qquad \textbf{(B)}\ \frac{0.4}{\pi} \qquad \textbf{(C)}\ 0.4 \qqua

1 KB (198 words) - 18:08, 28 June 2021
2010 AMC 12A Problems/Problem 21
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \ ...gative, all <math>3</math> zeros must be double roots because the function's degree is <math>6</math>.

5 KB (969 words) - 21:33, 22 June 2022
2010 AMC 10A Problems
{{AMC10 Problems|year=2010|ab=A}} Mary’s top book shelf holds five books with the following widths, in centimeters:

13 KB (1,902 words) - 11:20, 5 March 2023
Characteristic polynomial
...[operator]] refers to the [[polynomial]] whose roots are the [[eigenvalue]]s of the operator. It carries much information about the operator. ...det(tI - A)</math>, which is a <math>n</math>th degree polynomial in <math>t</math>. Here, <math>I</math> refers to the <math>n\times n</math> [[identit

19 KB (3,412 words) - 14:57, 21 September 2022
2010 AIME I Problems
...h>p</math> be the probability that exactly one of the selected divisors is a perfect square. The probability <math>p</math> can be expressed in the form ...mes 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}</math> is divided by <math>1000</math>.

8 KB (1,243 words) - 21:58, 10 August 2020
2010 AIME I Problems/Problem 13
.../math>, respectively. Line <math>\ell</math> divides region <math>\mathcal{R}</math> into two regions with areas in the ratio <math>1: 2</math>. Suppose label("$A$",(-16.43287,-9.3374),NE/2);

10 KB (1,418 words) - 23:05, 20 October 2021
Mock AIME 2 2010 Answer Key
...an be easily proved through strong induction. Starting from 2010, which is a multiple of 15, we must first purge 1 lemming. We can then purge 4 lemmings ...atisfies <math>b_1+b_2+\ldots+b_{10}\equiv 0 \pmod{3}</math>, there exists a corresponding <math>(a_1, a_2, \ldots, a_{10})</math> such that <math>a_1+a

36 KB (6,214 words) - 20:22, 13 July 2023
2010 AIME II Problems
...math> denote the distance from <math>P</math> to the closest side of <math>S</math>. The probability that <math>\frac{1}{5}\le d(P)\le\frac{1}{3}</math> ...where <math>a</math> and <math>b</math> are integers satisfying <math>1\le a < b \le 20</math>. Find the greatest positive integer <math>n</math> such t

8 KB (1,246 words) - 21:58, 10 August 2020
2010 AIME II Problems/Problem 14
...<math>r</math> is not divisible by the square of any prime. Find <math>p+q+r</math>. ...o use Power of a Point on <math>BA</math> and <math>CD</math>. By Power of a Point Theorem, <math>CP\cdot PD=1\cdot 2=BP\cdot PA</math>. Since <math>BP+

10 KB (1,507 words) - 00:31, 19 November 2023
2010 AIME II Problems/Problem 7
...2w-4</math>, where <math>i^2=-1</math>. Find <math>|a+b+c|</math>.</onlyinclude> == Solution (vieta's) ==

4 KB (735 words) - 02:41, 27 June 2022
2010 AIME II Problems/Problem 10
...econd-degree [[polynomial]]s <math>f(x)</math> with integer [[coefficient]]s and integer zeros for which <math>f(0)=2010</math>. ...guishable. In one case, <math>(a,r,s) = (2010,1,1)</math>, we have <math>r=s</math>. The other <math>80</math> cases are double counting, so there are <

6 KB (972 words) - 16:01, 16 September 2023
1993 USAMO Problems/Problem 2
Let <math>ABCD</math> be a convex quadrilateral such that diagonals <math>AC</math> and <math>BD</math pair A,B,C,D;

2 KB (446 words) - 08:09, 10 April 2023
2010 USAJMO Problems/Problem 1
...Find with proof the smallest <math>n</math> such that <math>P(n)</math> is a multiple of <math>2010</math>. ...s. We claim that the relation <math>R = \{(j, k)\in [n]\times[n]\mid jk\in S\}</math> is an equivalence relation on <math>[n]</math>.

12 KB (2,338 words) - 20:30, 13 February 2024
2010 USAMO Problems/Problem 1
Let <math>AXYZB</math> be a convex pentagon inscribed in a semicircle of diameter <math>AB</math>. Denote by <math>P, Q, R, S</math> the feet of the perpendiculars from <math>Y</math> onto

13 KB (2,178 words) - 14:14, 11 September 2021
1984 USAMO Problems/Problem 2
...(\text{i})\quad</math> For which positive integers <math>n</math> is there a finite set <math>S_n</math> of <math>n</math> distinct positive integers su ...sitive integers such that the geometric mean of any finite subset of <math>S</math> is an integer?

4 KB (855 words) - 03:12, 23 October 2022
2010 AMC 12B Problems/Problem 19
A high school basketball game between the Raiders and Wildcats was tied at th <math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\

6 KB (990 words) - 19:14, 25 March 2024
2010 AMC 12B Problems/Problem 21
Let <math>a > 0</math>, and let <math>P(x)</math> be a polynomial with integer coefficients such that <math>P(1) = P(3) = P(5) = P(7) = a</math>,

7 KB (1,166 words) - 00:04, 26 February 2024
2019 IMO Problems/Problem 6
...perpendicular to <math>EF</math> meets <math>\omega</math> again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumci ...t lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.

5 KB (792 words) - 01:52, 19 November 2023
1996 AJHSME Problems
<math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad <math>\text{(A)}\ \text{Jose} \qquad \text{(B)}\ \text{Thuy} \qquad \text{(C)}\ \text{Kare

13 KB (1,880 words) - 13:35, 19 February 2020
Python reference
This page is intended to serve as a brief, beginners reference to the [[Python]] programming language. In Python, there is no need to specify a type when storing information in a variable. It is basic and straightforward to store information.

28 KB (4,762 words) - 21:20, 12 June 2023
2006 AMC 8 Problems
<math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qqu ...answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score?

16 KB (2,215 words) - 19:18, 10 April 2024
1998 AJHSME Problems
<math>\text{(A)}\ \dfrac{6}{x} \qquad \text{(B)}\ \dfrac{6}{x+1} \qquad \text{(C)}\ \dfrac ...\text{b}\cdot \text{c}</math>, what is the value of <math>\begin{tabular}{r|l}3&4 \\ \hline 1&2\end{tabular}</math>?

14 KB (1,920 words) - 19:31, 31 January 2024
2010 AMC 10B Problems/Problem 22
<math>\textbf{(A)}\ 1930 \qquad \textbf{(B)}\ 1931 \qquad \textbf{(C)}\ 1932 \qquad \textbf{ ...ions such that the red bag is empty is equal to <math>2^7</math>, since it's equivalent to distributing the <math>7</math> candies into <math>2</math> b

4 KB (750 words) - 22:59, 18 January 2024
2011 AMC 12A Problems/Problem 12
...<math>A.</math> How many hours did it take the power boat to go from <math>A</math> to <math>B</math>? \textbf{(A)}\ 3 \qquad

5 KB (928 words) - 23:01, 14 April 2022
1958 AHSME Problems
<math> \textbf{(A)}\ 5\qquad <math> \textbf{(A)}\ y - x\qquad \textbf{(B)}\ x - y\qquad \textbf{(C)}\ \frac {y - x}{xy}\qq

25 KB (3,872 words) - 14:21, 20 February 2020
2011 AMC 10A Problems
{{AMC10 Problems|year=2011|ab=A}} A cell phone plan costs <math>\textdollar 20</math> each month, plus <math>5<

13 KB (1,903 words) - 18:09, 19 April 2021
1993 AHSME Problems
...<math>c</math>, define <math>\boxed{a,b,c}</math> to mean <math>a^b-b^c+c^a</math>. Then <math>\boxed{1,-1,2}</math> equals <math>\text{(A)} \ -4 \qquad \text{(B)} \ -2 \qquad \text{(C)} \ 0 \qquad \text{(D)} \ 2 \

20 KB (2,814 words) - 08:15, 27 June 2021
2012 IMO Problems/Problem 1
...tersection of the lines <math>AF</math> and <math>BC</math>, and let <math>T</math> be the point of intersection of the lines <math>AG</math> and <math> .../math>. Since <math>AK</math> and <math>AL</math> are tangents from <math>A</math> to the circle <math>J</math>, <math>AK = AL</math>. Then, we have <

7 KB (1,189 words) - 01:22, 19 November 2023
2011 AIME I Problems
...olution in jar <math>C</math> is added to jar B. At the end both jar <math>A</math> and jar <math>B</math> contain solutions that are <math>50\%</math> .../math> the <math>y</math>-axis. In the new coordinate system, point <math>A</math> is on the positive <math>x</math>-axis, and point <math>B</math> is

10 KB (1,634 words) - 22:21, 28 December 2023
2011 AIME I Problems/Problem 11
...e the sum of the elements in <math>R</math>. Find the remainder when <math>S</math> is divided by 1000. ...<math>2^i \equiv 2^j \pmod{1000}</math>. All that is left is to find <math>S</math> in mod <math>1000</math>. After some computation:

11 KB (1,668 words) - 22:10, 24 February 2023
2011 AIME I Problems/Problem 13
..., <math>s</math>, and <math>t</math> are positive integers. Find <math>r+s+t</math>. ...h>a^2+b^2+c^2=1</math>. The distance from a point <math>(X,Y,Z)</math> to a plane with equation <math>Ax+By+Cz+D=0</math> is

4 KB (613 words) - 11:37, 8 September 2022
2011 AIME II Problems/Problem 10
...14 intersect at point <math>P</math>. The distance between the [[midpoint]]s of the two chords is 12. The quantity <math>OP^2</math> can be represented The line through the midpoint of a chord of a circle and the center of that circle is perpendicular to that chord, so <ma

11 KB (1,720 words) - 03:12, 18 December 2023
1999 AMC 8 Problems
<math>\text{(A)} \div \qquad \text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \ What is the degree measure of the smaller angle formed by the hands of a clock at 10 o clock?

17 KB (2,394 words) - 19:51, 8 May 2023
2011 USAMO Problems/Problem 3
...r of opposite sides are parallel. The internal angles satisfy <math>\angle A = 3\angle D</math>, <math>\angle C = 3\angle F</math>, and <math>\angle E = ...B = \beta</math>, <math>AB=DE=p</math>, <math>BC=EF=q</math>, <math>CD=FA=r</math>. Define the vectors: <cmath>\vec{u} = \vec{AB} + \vec{DE}</cmath> <

11 KB (1,925 words) - 12:07, 31 August 2023
2001 AMC 8 Problems/Problem 23
...front page--><onlyinclude>Points <math>R</math>, <math>S</math> and <math>T</math> are vertices of an equilateral triangle, and points <math>X</math>, drawn using any three of these six points as vertices?</onlyinclude>

3 KB (549 words) - 16:35, 23 July 2023
2007 AMC 10B Problems
Isabella's house has <math>3</math> bedrooms. Each bedroom is <math>12</math> feet lon <math>\textbf{(A) } 678 \qquad\textbf{(B) } 768 \qquad\textbf{(C) } 786 \qquad\textbf{(D) }

15 KB (2,297 words) - 12:57, 19 February 2020
Olympiad books
Here is a list of '''Olympiad Books''' that have Olympiad-level problems used to trai ...om/community/c142h405377 here] about these books or request new books. Let's categorize books into '''Theory''' books, '''Problem''' books, and '''Both'

17 KB (2,261 words) - 00:30, 22 April 2024
Mock Amc 12
...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 ...r more information on Mock AMCs, as well as tips on how to write them, and a complete list of all Mock AMC 10s and Mock AMC 12s.

18 KB (2,206 words) - 19:41, 24 December 2020
1967 AHSME Problems
...umber <math>5b9</math>. If <math>5b9</math> is divisible by 9, then <math>a+b</math> equals <math> \text{(A)}\ 2\qquad\text{(B)}\ 4\qquad\text{(C)}\ 6\qquad\text{(D)}\ 8\qquad\text{(E

20 KB (3,108 words) - 14:14, 20 February 2020
2003 AMC 8 Problems
Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of <math>\mathrm{(A)}\ 12 \qquad\mathrm{(B)}\ 16 \qquad\mathrm{(C)}\ 20 \qquad\mathrm{(D)}\ 22

16 KB (2,236 words) - 12:02, 19 February 2024
2005 CEMC Gauss (Grade 7) Problems
= Part A: Each correct answer is worth 5 points = <math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad

16 KB (2,317 words) - 03:54, 24 October 2014
2009 AMC 8 Problems
Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. The <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 11\qquad\

18 KB (2,551 words) - 18:46, 27 February 2024
1987 USAMO Problems
...sector of <math>\angle A</math>, find all possible values for <math>\angle A</math>. ...<math>X</math>, then <math>x\cdot r(x)</math> and <math>(x + (1 - x) \cdot r(x) )</math> both belong to <math>X</math>.

2 KB (356 words) - 18:42, 18 July 2016
1988 USAMO Problems
...t</math>. Show that <math>k=a^2-3b\ge 0</math> and that <math>\sqrt k\le r-t</math>. ...ath>X</math>. Show that for any map <math>f: P\mapsto X</math> we can find a 10-element subset <math>Y</math> of <math>X</math>, such that <math>f(Y-\{k

2 KB (319 words) - 16:33, 5 February 2018
1996 AJHSME Problems/Problem 5
...<math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math>, and <math>T</math> represent numbers located on the number line as shown. label("$-3$",(-3,-0.1),S);

3 KB (506 words) - 00:24, 5 July 2013
1996 AJHSME Problems/Problem 17
...<math>Q</math> has coordinates (2,2). What are the coordinates for <math>T</math> so that the area of triangle <math>PQT</math> equals the area of squ pair O,P,Q,R,T;

2 KB (257 words) - 11:20, 22 March 2015
2010 PUMaC Problems
== Division A == ...math> such that <math>4^x = x^4</math>. Find the nearest integer to <math>S</math>.

22 KB (3,694 words) - 23:58, 3 June 2022
2016 AMC 10A Problems/Problem 19
...<math>r,s,</math> and <math>t</math> is <math>1.</math> What is <math>r+s+t</math>? <math>\textbf{(A) } 7 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 15

8 KB (1,386 words) - 15:10, 8 October 2023
1993 AJHSME Problems/Problem 23
...math> beats <math>R</math>, <math>Q</math> beats <math>S</math>, and <math>T</math> finishes after <math>P</math> and before <math>Q</math>. Who could ...and }S \qquad \text{(D)}\ P\text{ and }T \qquad \text{(E)}\ P,S\text{ and }T</math>

1 KB (228 words) - 00:12, 5 July 2013
2003 AMC 8 Problems/Problem 14
In this addition problem, each letter stands for a different digit. ...sep}{0.5mm}\begin{array}{cccc}&T & W & O\\ +&T & W & O\\ \hline F& O & U & R\end{array} </math>

1 KB (177 words) - 19:56, 15 April 2023
2003 AMC 8 Problems/Problem 18
...east one of her friends. How many classmates will not be invited to Sarah's party? pair a=(102,256), b=(68,131), c=(162,101), d=(134,150);

2 KB (250 words) - 22:17, 5 January 2024
1950 AHSME Problems
<math> \textbf{(A)}\ 5\frac{1}{3}\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 10\frac{2}{3}\qqua ...>. When <math>S=8</math>, <math>R=16</math>. When <math>S=10</math>, <math>R</math> is equal to:

22 KB (3,306 words) - 19:50, 3 May 2023
2009 PUMaC Problems
'''Algebra A''' Let <math>x_1,x_2,\dots,x_n</math> be a sequence of integers, such that <math>-1\leq x_i\leq 2</math>, for <math>i=

25 KB (4,154 words) - 16:27, 2 September 2011
2010 AMC 8 Problems
...students in Mr. Newton's class, and <math>9</math> students in Mrs. Young's class taking the AMC 8 this year. How many mathematics students at Euclid M <math> \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qq

18 KB (2,768 words) - 21:05, 9 January 2024
2011 AMC 8 Problems
...math> apples at a cost of <math> 50 </math> cents per apple. She paid with a 5-dollar bill. How much change did Margie receive? <math>\textbf{(A) }\ \textdollar 1.50 \qquad \textbf{(B) }\ \textdollar 2.00 \qquad \textbf{

16 KB (2,371 words) - 17:34, 9 January 2024
1985 IMO Problems/Problem 6
...ution, one can write <math>x_n=P_n(x_1)</math> , where <math>P_n</math> is a polynomial with non-negative coefficients and zero constant term. Thus, <ma (a) <math>\displaystyle x_n<x_{n+1} \Leftrightarrow x_n>1-\frac{1}{n} \Leftrig

9 KB (1,788 words) - 00:02, 30 January 2021
Mathtime Version 1 Issue 2
...e the positive integer <math>n</math> such that <math>n^2+20n+40</math> is a perfect square. (Proposed by djmathman) ...umber of ordered integer pairs <math>(a,b)</math> such that <math>a^2+b^2|(a+b)^2</math>. (Proposed by djmathman)

15 KB (2,444 words) - 21:46, 1 January 2012
2012 AMC 10A Problems/Problem 25
<math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\ ...0</math>. Then the set of points <math>(x,y,z)</math> is a tetrahedron, or a triangular pyramid. The point <math>(x,y,z)</math> distributes uniformly in

13 KB (2,133 words) - 01:22, 6 February 2024
2012 AMC 10A Problems/Problem 18
...each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve? <math>\textbf{(A)}\ 2\pi+6\qquad\textbf{(B)}\ 2\pi+4\sqrt{3}\qquad\textbf{(C)}\ 3\pi+4\qquad

5 KB (775 words) - 22:33, 22 October 2023
2012 AMC 12A Problems/Problem 16
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf label("$Z$",Z,S);

9 KB (1,496 words) - 02:40, 2 October 2022
2012 AMC 12A Problems/Problem 22
...igcup_{j=1}^{k}p_{j} </math>. The intersection of <math>P</math> and <math>S</math> consists of the union of all segments joining the midpoints of every <math> \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qqua

5 KB (940 words) - 17:13, 4 April 2020
1989 AHSME Problems/Problem 19
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of l <math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \frac{18}{\pi^2} } \qquad \mathrm{(C) \frac{9}{\p

4 KB (649 words) - 10:04, 20 May 2021
2012 AIME I Problems
...s with three not necessarily distinct digits, <math>abc</math>, with <math>a \neq 0</math> and <math>c \neq 0</math> such that both <math>abc</math> and ...e, and they have been traveling for <math>t</math> minutes. Find <math>n + t</math>.

10 KB (1,617 words) - 14:49, 2 June 2023
2012 AIME I Problems/Problem 14
...math>b,</math> and <math>c</math> in the complex plane are the vertices of a right triangle with hypotenuse <math>h.</math> Find <math>h^2.</math> ...9}\cdot\frac{x^2+y^2}{4}=\frac{x^2}{9}+\frac{y^2}{9}</math>. Hence, <math>|a|^2+|b|^2+|c|^2=\frac{6x^2+6y^2}{9}=\frac{2x^2+2y^2}{3}=250</math>. Therefor

12 KB (2,228 words) - 23:59, 20 January 2024
2012 AIME II Problems/Problem 15
...les <math>\omega</math> and <math>\gamma</math> meet at <math>E</math> and a second point <math>F</math>. Then <math>AF^2 = \frac mn</math>, where <math ...math>. Solving gives <math>x^2=\frac{49}{19}</math>, from which by Ptolemy's we see <math>AF=\frac{30}{\sqrt{19}}</math>. We conclude the answer is <mat

13 KB (2,298 words) - 12:56, 10 September 2023
2017 AMC 8 Problems
<math>\textbf{(A) }2+0+1+7\qquad\textbf{(B) }2 \times 0 +1+7\qquad\textbf{(C) }2+0 \times 1 Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the vo

12 KB (1,771 words) - 21:13, 20 January 2024
Mock AIME I 2012 Problems
...<math>q</math> and <math>r</math> are relatively prime. What is <math>p+q+r</math>? A permutation of the numbers <math>\{2</math>, <math>3</math>, <math>4</math>

7 KB (1,309 words) - 11:13, 8 April 2012
Mock AIME I 2012 Problems/Problem 8
...e expressed in the form <math>\dfrac{r+\sqrt{s}}{t}</math>. Find <math>r+s+t</math>. ...rth power, the desired answer is <math>\dfrac{7+\sqrt{45}}{2} \implies r+s+t=\boxed{054}</math>.

3 KB (543 words) - 18:51, 7 May 2020
2015 IMO Problems/Problem 1
We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> ...any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is

4 KB (773 words) - 08:14, 19 July 2016
2002 Romanian NMO Problems
Eight card players are seated around a table. One remarks that at some moment, any player and his two neighbours h Prove that any real number <math>0<x<1</math> can be written as a difference of two positive and less than <math>1</math> irrational numbers.

10 KB (1,695 words) - 10:03, 10 May 2012
Install Eclipse
Note that, first, you need to make sure you have a version of JVM(Java Virtual Machine) installed. See [[Install Java]] page f You can read if you have a 32- or 64-bit system under System Type:

8 KB (1,222 words) - 17:47, 9 October 2014
1988 USAMO Problems/Problem 2
...t</math>. Show that <math>k=a^2-3b\ge 0</math> and that <math>\sqrt k\le r-t</math>. By Vieta's Formulas, <math>a=-r-s-t</math>, <math>b=rs+st+rt</math>, and <math>c=-rst</math>.

3 KB (674 words) - 18:17, 13 June 2017
1988 USAMO Problems/Problem 4
...ta IAB</math>, <math>\Delta IBC</math>, and <math>\Delta ICA</math> lie on a circle whose center is the circumcenter of <math>\Delta ABC</math>. ...>, and <math>O_b</math>, respectively. It then suffices to show that <math>A</math>, <math>B</math>, <math>C</math>, <math>O_a</math>, <math>O_b</math>,

3 KB (504 words) - 19:25, 14 October 2021
1974 AHSME Problems
<math> \mathrm{(A)\ } 4x+3y=xy \qquad \mathrm{(B) \ }y=\frac{4x}{6-y} \qquad \mathrm{(C) \ } <math> \mathrm{(A)\ } -\frac{h}{3} \qquad \mathrm{(B) \ }\frac{h}{3} \qquad \mathrm{(C) \ }

15 KB (2,151 words) - 14:04, 19 February 2020
1974 AHSME Problems/Problem 23
...t of tangency. If <math> TP=4 </math> and <math> T'Q=9 </math> then <math> r </math> is pair O = (0,0); pair T = dir(90);

2 KB (337 words) - 12:44, 5 July 2013
2006 SMT/Geometry Problems
Given a cube, determine the ratio of the volume of the octahedron formed by connect ...ath> and <math> X </math> is on the same side of line <math> AB </math> as a given point <math> C </math>.

5 KB (837 words) - 12:22, 27 May 2012
2006 SMT/Calculus Problems
...equation <math> 4y''+3y'-y=0 </math> and its solution <math> y=e^{\lambda t} </math>, what are the values of <math> \lambda </math>? ...orn is a point on the ground, and that the elven rope is so light it makes a straight line from the unicorn to the tower.

3 KB (525 words) - 13:59, 27 May 2012
2010 AMC 8 Problems/Problem 3
....4), M=(1,-4.2), N=(1,-4), P=(1,-3.8), Q=(1,-3.6), R=(1,-3.4), S=(1,-3.2), T=(1,-3), U=(1,-2.8), V=(1,-2.6), W=(1,-2.4), Z=(1,-2.2), E_1=(1.4,-2.6), F_1 ...D(B--(8,-2),linewidth(0.4)); D(B--S_1); D(T--R_1); D(N--Q_1); D(H--P_1); D(A--O_1); D(C_1--E_1); D(E_1--F_1); D(F_1--D_1); D(D_1--C_1); D(G_1--L_2); D(L

5 KB (847 words) - 19:59, 5 February 2023
1952 AHSME Problems
If the radius of a circle is a rational number, its area is given by a number which is: ...ational} \qquad \textbf{(C)\ } \text{integral} \qquad \textbf{(D)\ } \text{a perfect square }\qquad \textbf{(E)\ } \text{none of these} </math>

23 KB (3,556 words) - 15:35, 30 December 2023
1954 AHSME Problems
<math>\textbf{(A)}\ y^2-5\sqrt{y^2-25} \qquad \textbf{(B)}\ -y^2 \qquad \textbf{(C)}\ y^2 \\ <math>\textbf{(A)}\ 4 \text{ and }1 \qquad \textbf{(B)}\ \text{only }1 \qquad \textbf{(C)}\

23 KB (3,535 words) - 16:29, 24 April 2020
1955 AHSME Problems
<math> \textbf{(A)}\ 3.75\times 10^{-7}\qquad\textbf{(B)}\ 3\frac{3}{4}\times 10^{-7}\qquad\t The smaller angle between the hands of a clock at <math>12:25</math> p.m. is:

22 KB (3,509 words) - 21:29, 31 December 2023
1998 USAMO Problems/Problem 6
...er <math>k</math> (as a function of <math>n</math>) such that there exists a convex <math>n</math>-gon <math>A_{1}A_{2}\dots A_{n}</math> for which exac pair A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U;

5 KB (871 words) - 18:59, 10 May 2023
1999 USAMO Problems/Problem 1
(a) every square that does not contain a checker shares a side with one that does; ...ven squares, such that every two consecutive squares of the sequence share a side.

4 KB (747 words) - 05:23, 13 May 2023
2020 Mock Combo AMC 10
Here are the problems from the 2020 Mock Combo AMC 10 II, a mock contest created by the AoPS user fidgetboss_4000. ...friend, Sara, along with eight other classmates, are randomly seated along a row of ten chairs. What is the probability that Fred sits next to Sara?

15 KB (2,452 words) - 03:03, 4 July 2020
2013 AMC 12A Problems
{{AMC12 Problems|year=2013|ab=A}} <math>\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \

14 KB (2,206 words) - 19:31, 15 May 2024
2013 AMC 10A Problems
{{AMC10 Problems|year=2013|ab=A}} A taxi ride costs \$1.50 plus \$0.25 per mile traveled. How much does a 5-mile taxi ride cost?

12 KB (1,894 words) - 15:59, 3 January 2024
2013 AIME I Problems/Problem 5
...ath>, <math>b</math>, and <math>c</math> are positive integers. Find <math>a+b+c</math>. ...3]{9}+1</math> to rationalize the denominator, and we therefore have <math>r=\frac{\sqrt[3]{81}+\sqrt[3]{9}+1}{8}</math>, and the answer is <math>\boxed

3 KB (573 words) - 21:11, 27 August 2023
2013 AIME II Problems
...\text{B}</math>, and <math>\text{C}</math> are digits. Find <math>100\text{A}+10\text{B}+\text{C}</math>. Positive integers <math>a</math> and <math>b</math> satisfy the condition

8 KB (1,402 words) - 12:17, 13 March 2020
2013 AIME II Problems/Problem 12
..., and <math>c</math> are integers. Find the number of polynomials in <math>S</math> such that each of its roots <math>z</math> satisfies either <math>|z ...real and two nonreal roots which are conjugates. This follows from [[Vieta's formulas]].

8 KB (1,393 words) - 19:00, 24 May 2023
2013 USAMO Problems/Problem 1
In triangle <math>ABC</math>, points <math>P,Q,R</math> lie on sides <math>BC,CA,AB</math> respectively. Let <math>\omega_A pair A = (-1.0, 3.0);

14 KB (1,830 words) - 18:22, 10 May 2023
Mock AIME 6 2006-2007 Problems
...integers that do not exceed <math>4</math>. Find the remainder when <math>T</math> is divided by <math>1000</math>. Draw in the diagonals of a regular octagon. What is the sum of all distinct angle measures, in degree

7 KB (1,173 words) - 21:04, 7 December 2018

Page

Toolbox

Search

Search results