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2005 AMC 12A Problems/Problem 16
...is tangent to both axes and to the second and third circles. What is <math>r/s</math>? label("$r$",midpoint(O0--P0),NE);

2 KB (307 words) - 15:30, 30 March 2024
2015 IMO Problems
We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is

4 KB (692 words) - 22:33, 15 February 2021
Wilson's Theorem
Suppose first that <math>p</math> is composite. Then <math>p</math> has a factor <math>d > 1</math> that is less than or equal to <math>p-1</math>. ...ique, and each number is the inverse of its inverse. If one integer <math>a</math> is its own inverse, then

4 KB (639 words) - 01:53, 2 February 2023
Trivial Inequality
...(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
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
..._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
Menelaus' theorem
...earity]] of points on each of the three sides (extended when necessary) of a [[triangle]]. ...C</math>, <math>Q</math> is on the extension of <math>AC</math>, and <math>R</math> on the intersection of <math>PQ</math> and <math>AB</math>, then

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 ...}</math> is not [[divisibility|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
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
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]] ...[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
Real number
A '''real number''' is a number that falls on the real number line. It can have any value. Some exam ...<math>\mathbb{R}</math>, is a subset of [[complex number]]s(<math>\mathbb{C}</math>). Commonly used subsets of the real numbers are the [[rational numb

3 KB (496 words) - 23:22, 5 January 2022
Jensen's Inequality
...[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 If <math>{F}</math> is a concave function, we have:

3 KB (623 words) - 13:10, 20 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
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
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
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
Schur's Inequality
...ates that for all non-negative <math>a,b,c \in \mathbb{R}</math> and <math>r>0</math>: <cmath>a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0</cmath>

2 KB (398 words) - 16:57, 29 December 2021
Complex number
...n has no solutions in the real numbers. However, it is possible to define a number, <math> i </math>, such that <math> i = \sqrt{-1} </math>. If we ad ...s the set <math>\mathbb{R}</math> of the [[real number]]s, since <math>a = a + 0i</math>.

5 KB (860 words) - 15:36, 10 December 2023
Polar form
...n an [[angle]] <math>\theta</math> and a [[radius]] or [[magnitude]] <math>r</math>. ...s notation.) This represents a complex number <math>z</math> that is <math>r</math> units away from the origin, and <math>\theta</math> [[radian]]s coun

633 bytes (105 words) - 13:35, 1 April 2022
Law of Sines
The '''Law of Sines''' is a useful identity in a [[triangle]], which, along with the [[law of cosines]] and the [[law of tan ...<math>B</math>, <math>c</math> opposite to <math>C</math>, and where <math>R</math> is the circumradius:

4 KB (658 words) - 16:19, 28 April 2024
Geometric sequence
...'geometric sequence''', sometimes called a '''geometric progression''', is a [[sequence]] of numbers such that the ratio between any two consecutive ter ...e with common ratio <math>2</math> and <math>100, -50, 25, -25/2</math> is a geometric sequence with common ratio <math>-1/2</math>; however, <math>1, 3

4 KB (644 words) - 12:55, 7 March 2022
Area
...size of the region that a two-[[dimension]]al figure occupies. The size of a region in higher dimensions is referred to as [[volume]]. It is often possible to find the area of a region bounded by parts of [[circle]]s and [[line segment]]s through elemen

6 KB (1,181 words) - 22:37, 22 January 2023
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
Rational approximation
The main theme of this article is the question how well a given [[real number]] <math>x</math> can be approximated by [[rational numb ...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 ...definition of trigonometry is preferred in order to extend trigonometry to a complex domain.

8 KB (1,217 words) - 20:15, 7 September 2023
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
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 ...s, every polynomial over <math>\mathbb{C}</math> splits over <math>\mathbb{C}</math>, or decomposes into linear factors.

5 KB (832 words) - 14:22, 11 January 2024
Algebraic geometry
...ath> be a [[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 a ...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
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 ...ple, except let's replace the <math>12</math> at the top of the clock with a <math>0</math>.

15 KB (2,396 words) - 20:24, 21 February 2024
Uncountable
...example of an uncountable set is the set of [[real number]]s <math>\mathbb{R}</math>. == Proof that <math>\mathbb{R}</math> is uncountable ==

2 KB (403 words) - 20:53, 13 October 2019
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
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
Limit
...of mathematics. It took several centuries to articulate the definition of a limit and to make it rigorous. ...ue to which a function grows close when its argument is near (but not at!) a particular value. For example,

7 KB (1,325 words) - 13:51, 1 June 2015
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 10
...itive integers whose [[greatest common divisor]] is 1. Find <math> a^2+b^2+c^2. </math> ...n be easily solved to be <math>6x = 2y + 5</math>. Thus, <math>a^2 + b^2 + c^2 = \boxed{065}</math>.

4 KB (731 words) - 17:59, 4 January 2022
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), G=(0.45,-1.6), H=(3

5 KB (730 words) - 15:05, 15 January 2024
2006 AIME I Problems/Problem 5
...sqrt{2}+b\sqrt{3}+c\sqrt{5}, </math> where <math> a, b, </math> and <math> c </math> are [[positive]] [[integer]]s. Find <math>abc</math>. <cmath> a\sqrt{2}+b\sqrt{3}+c\sqrt{5} = \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}</cmath>

3 KB (439 words) - 18:24, 10 March 2015
2022 AIME I Problems/Problem 1
.../math> and <math>Q(x)</math> cancel, we conclude that <math>R(x)</math> is a linear polynomial. R(16) &= P(16)+Q(16) &&= 54+54 &&= 108, \\

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 \qquad\textbf{(E) } 13</math> <math>\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2\pi \qquad\textbf{(E) }2\pi +e</math>

12 KB (1,784 words) - 16:49, 1 April 2021
2006 AMC 12A Problems
{{AMC12 Problems|year=2006|ab=A}} <math> \mathrm{(A) \ } 31\qquad \mathrm{(B) \ } 32\qquad \mathrm{(C) \ } 33\qquad \mathrm{(D) \ } 34\qquad \mathrm{(E) \ } 35 </math>

15 KB (2,223 words) - 13:43, 28 December 2020
2005 AMC 12A Problems
{{AMC12 Problems|year=2005|ab=A}} (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 10 \qquad (\mathrm {E})\ 20

13 KB (1,971 words) - 13:03, 19 February 2020
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) \ } 2003\qquad \mathrm{(E) \ } 4006 </math>

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 \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 13 </math>

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)}\ 111 \qquad \textbf{(E)}\ 671</math> ...\textbf{(A)}\ 2000^{2001} \qquad \textbf{(B)}\ 4000^{2000} \qquad \textbf{(C)}\ 2000^{4000} \qquad \textbf{(D)}\ 4,000,000^{2000} \qquad \textbf{(E)}\ 2

13 KB (1,948 words) - 12:26, 1 April 2022
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} \qquad \text {(D) } 1 \qquad \text {(E) } \frac{14}{3} ...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 20
\mathrm{(A)}\ \frac 18 \mathrm{(C)}\ \frac 16

3 KB (485 words) - 14:09, 21 May 2021
2006 AMC 12A Problems/Problem 17
...ngent to the circle, and <math>AF=\sqrt{9+5\sqrt{2}}</math>. What is <math>r/s</math>? real r = 50;

6 KB (958 words) - 23:29, 28 September 2023
2006 AMC 12A Problems/Problem 22
...m a randomly chosen point on the circle is <math>1/2</math>. What is <math>r</math>? ...\sqrt{2}+2\sqrt{3}\qquad \mathrm{(B) \ } 3\sqrt{3}+\sqrt{2}\qquad \mathrm{(C) \ } 2\sqrt{6}+\sqrt{3}\qquad \rm{(D) \ } 3\sqrt{2}+\sqrt{6}\qquad \mathrm{

2 KB (343 words) - 15:39, 14 June 2023
2006 AMC 12A Problems/Problem 23
...h>S=(a_1,a_2,\ldots ,a_n)</math> of <math>n</math> real numbers, let <math>A(S)</math> be the sequence ...ath>x>0</math>, and let <math>S=(1,x,x^2,\ldots ,x^{100})</math>. If <math>A^{100}(S)=(1/2^{50})</math>, then what is <math>x</math>?

3 KB (466 words) - 22:40, 29 September 2023
2005 AMC 12B Problems/Problem 14
A circle having center <math>(0,k)</math>, with <math>k>6</math>, is tangent \mathrm{(A)}\ 6\sqrt{2}-6 \qquad

2 KB (278 words) - 21:12, 24 December 2020
2005 AMC 12B Problems/Problem 18
...cute triangle. What is the closest integer to the area of the region <math>R</math>? \mathrm{(A)}\ 25 \qquad

2 KB (262 words) - 21:20, 21 December 2020
Root (polynomial)
...<math>a</math> in the [[domain]] of the function such that <math>f(a) = (x-a) = 0</math>. ...ath> with all <math>c_j \in \mathbb C</math> and <math>c_n \neq 0</math>, a general degree-<math>n</math> polynomial. The degree of <math>P(x)</math> i

8 KB (1,427 words) - 21:37, 13 March 2022
2006 AMC 10A Problems/Problem 24
Centers of adjacent faces of a unit cube are joined to form a regular [[octahedron]]. What is the volume of this octahedron? <math>\textbf{(A) } \frac{1}{8}\qquad\textbf{(B) } \frac{1}{6}\qquad\textbf{(C) } \frac{1}{4}\qquad\textbf{(D) } \frac{1}{3}\qquad\textbf{(E) } \frac{1}{2

2 KB (292 words) - 10:19, 19 December 2021
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 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 12
...math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math> ...^E^^F^^G^^O); label("$A$",A,(-1,1));label("$B$",B,(1,1));label("$C$",C,(1,-1));label("$D$",D,(-1,-1)); label("$E$",E,(0,1));label("$F$",F,(1

13 KB (2,080 words) - 21:20, 11 December 2022
2005 AIME I Problems
...us 30. Let <math> K </math> be the area of the region inside circle <math> C </math> and outside of the six circles in the ring. Find <math> \lfloor K \ ...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 1
...]] 30. Let <math> K </math> be the area of the region inside circle <math> C </math> and outside of the six circles in the ring. Find <math> \lfloor K \ ...adius of <math>C</math> has a length of <math>3r = 30</math>, and so <math>r = 10</math>. <math>K = 30^2\pi - 6(10^2\pi) = 300\pi</math>, so <math>\lflo

1 KB (213 words) - 13:17, 22 July 2017
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 I Problems/Problem 9
...ath> 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 13
A particle moves in the [[Cartesian plane]] according to the following rules: ...> the particle may only move to <math> (a+1,b), (a,b+1), </math> or <math>(a+1,b+1). </math>

5 KB (897 words) - 00:21, 29 July 2022
2005 AIME II Problems/Problem 15
...h> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally [[tangent (geometry)|tangent]] to <math> w_2 </ma Let <math>w_3</math> have center <math>(x,y)</math> and radius <math>r</math>. Now, if two circles with radii <math>r_1</math> and <math>r_2</mat

12 KB (2,000 words) - 13:17, 28 December 2020
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 I Problems/Problem 14
...[[positive]] [[integer]]s, and <math> c </math> is prime. Find <math> a+b+c. </math> triple Oxy = (0,0,0), A=(4*5^.5,-8,4), B=(0,-8,h), C=(Cxy.x,Cxy.y,0), D=(A.x,A.y,0), E=(B.x,B.y,0), O=(O.x,O.y,h);

4 KB (729 words) - 01:00, 27 November 2022
2004 AIME I Problems/Problem 11
...math> and <math> F </math> and the ratio between the [[volume]]s of <math> C </math> and <math> F </math> are both equal to <math> k</math>. Given that ...Using the [[Pythagorean Theorem]], we get <math>\ell = 5</math> and <math>A = 24\pi</math>.

5 KB (839 words) - 22:12, 16 December 2015
2004 AIME I Problems/Problem 9
...ivide rectangle <math> DEFG </math> into a triangle <math> U_2 </math> and a trapezoid <math> V_2 </math> such that <math> U_1 </math> is similar to <ma .../math> or <math>\overline{FG}</math>. <math>V_2</math> is a trapezoid with a right angle then, from which it follows that <math>V_1</math> contains one

4 KB (618 words) - 20:01, 4 July 2013
2004 AIME I Problems/Problem 7
...ath> (1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x). </math> Find <math> |C|. </math> ...and so <math>P(x)\cdot P(-x) = 1 + (2C - 64)x^2 + R(x)</math>, where <math>R(x)</math> is some polynomial divisible by <math>x^3</math>.

5 KB (833 words) - 19:43, 1 October 2023
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 12
...and circles of radius 2 centered at <math> C </math> and <math> D. </math> A circle contained within the trapezoid is [[tangent]] to all four of these c Let the radius of the center circle be <math>r</math> and its center be denoted as <math>O</math>.

3 KB (431 words) - 23:21, 4 July 2013
2004 AIME II Problems/Problem 11
...the cone is <math>125</math>, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex ...)^2}=\sqrt{360000+280000}=\sqrt{640000}=800</math>. Setting <math>\theta R=C\implies 800\theta=1200\pi\implies\theta=\frac{3\pi}{2}</math>.

2 KB (268 words) - 22:20, 23 March 2023
2004 AIME II Problems/Problem 1
...</math> and <math> e </math> are [[relatively prime]], and neither <math> c </math> nor <math> f </math> is [[divisibility | divisible]] by the [[squar ...ath>; the rest of the area of the circle is then equal to <math>\frac{2}{3}r^2\pi</math>.

2 KB (329 words) - 23:20, 4 July 2013
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 ...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<

7 KB (1,104 words) - 12:53, 6 July 2022
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
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. ...ath> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that each interior angle of <math>P_1^{}</math> is

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 <center><math>\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\

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
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
...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
1998 AIME Problems
...rline{AB}</math> at <math>Q</math> and <math>\overline{DB}</math> at <math>R.</math> Given that <math>PQ = 735</math> and <math>QR = 112,</math> find < ...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
...>B</math> across the y-axis, let <math>D</math> be the reflection of <math>C</math> across the x-axis, and let <math>E</math> be the reflection of <math ...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 ...<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
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
2002 AIME II Problems
...a [[cube]] are <math>P=(7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=(11,3,9)</math>. What is the [[surface area]] of the cube? ...</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 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< real r=10;

11 KB (1,741 words) - 22:40, 23 November 2023
1983 AIME Problems/Problem 12
...om their intersection point <math>H</math> to the center <math>O</math> is a positive rational number. Determine the length of <math>AB</math>. label("A",(-4,0),W);

2 KB (412 words) - 18:23, 1 January 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. ...2); 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 (2,149 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 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
1995 IMO Problems/Problem 2
Let <math>a, b, c</math> be positive real numbers such that <math>abc = 1</math>. Prove that <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 9
...alpha</math>, which is a [[positive]] [[rational number]], is expressed as a [[fraction]] in lowest terms, what is the sum of its numerator and denomina real r = 8/15^0.5, a = 57.91, b = 93.135;

5 KB (763 words) - 16:20, 28 September 2019
1986 AIME Problems/Problem 15
...<math>AB</math> is <math>60</math>, and that the [[median]]s through <math>A</math> and <math>B</math> lie along the lines <math>y=x+3</math> and <math> Let <math>\theta_1</math> be the angle that the median through <math>A</math> makes with the positive <math>x</math>-axis, and let <math>\theta_2<

11 KB (1,722 words) - 09:49, 13 September 2023
1987 AIME Problems/Problem 10
...times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.) ...math>x - 75</math> steps. Since Bob and the escalator were both moving at a constant speed over the time it took Bob to climb, the [[ratio]] of their d

7 KB (1,187 words) - 16:21, 27 January 2024
1987 AIME Problems/Problem 7
...r which <math>[a,b] = 1000</math>, <math>[b,c] = 2000</math>, and <math>[c,a] = 2000</math>. ...t we must have <math>a = 2^j5^k</math>, <math>b = 2^m 5^n</math> and <math>c = 2^p5^q</math> for some [[nonnegative]] [[integer]]s <math>j, k, m, n, p,

3 KB (547 words) - 22:54, 4 April 2016
1987 AIME Problems/Problem 6
...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 ...- 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
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 We draw the [[altitude]] <math>h</math> to <math>c</math>, to get two [[right triangle]]s.

8 KB (1,401 words) - 21:41, 20 January 2024
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
2006 AMC 10B Problems/Problem 8
A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle? real r=sqrt(50), s=sqrt(10);

1 KB (176 words) - 13:49, 26 January 2022
1991 AIME Problems/Problem 12
...that [[vertex|vertices]] <math>P^{}_{}</math>, <math>Q^{}_{}</math>, <math>R^{}_{}</math>, and <math>S^{}_{}</math> are interior points on sides <math>\ ...);label("$D$",D,SW); label("$P$",P,N);label("$Q$",Q,E);label("$R$",R,SW);label("$S$",S,W); label("$15$",B/2+P/2,N);label("$20$",B/2+Q/2,E)

8 KB (1,270 words) - 23:36, 27 August 2023
1991 AIME Problems/Problem 11
...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) - 19:33, 28 December 2022
1991 AIME Problems/Problem 8
For how many real numbers <math>a</math> does the [[quadratic equation]] <math>x^2 + ax + 6a=0</math> have on ...roots of the quadratic, and since <math>x_1,x_2</math> are integers, <math>a</math> must be an integer. Applying the [[quadratic formula]],

2 KB (310 words) - 11:25, 13 June 2023
1991 AIME Problems/Problem 2
real r = 0.35; size(220); pair A=(0,0),B=(4,0),C=(4,3),D=(0,3);

4 KB (595 words) - 12:51, 17 June 2021
1992 AIME Problems/Problem 5
...h>, where the digits <math>a^{}_{}</math>, <math>b^{}_{}</math>, and <math>c^{}_{}</math> are not necessarily distinct. To write the elements of <math>S ...ultiples of <math>3</math>. We have to count these since it will reduce to a multiple of <math>3</math> which we have removed from <math>999</math>, but

2 KB (277 words) - 20:45, 4 March 2024
1993 AIME Problems/Problem 15
...verline{CH}</math> be an altitude of <math>\triangle ABC</math>. Let <math>R\,</math> and <math>S\,</math> be the points where the circles inscribed in pair A,B,C,H;

3 KB (449 words) - 21:39, 21 September 2023
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? ...B</math> be Kenny's initial and final points respectively and define <math>C</math> and <math>D</math> similarly for Jenny. Let <math>O</math> be the ce

8 KB (1,231 words) - 20:06, 26 November 2023
1993 AIME Problems/Problem 12
...e 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, then ...h> and <math>(r,s)</math> and we want to find <math>(u,v)</math> so <math>(r,s)</math> is the midpoint of <math>(u,v)</math> and <math>(p,q)</math>, the

4 KB (611 words) - 13:59, 15 July 2023
1955 AHSME Problems/Problem 27
...h> and <math>s</math> are the roots of <math>x^2-px+q=0</math>, then <math>r^2+s^2</math> equals: <math> \textbf{(A)}\ p^2+2q\qquad\textbf{(B)}\ p^2-2q\qquad\textbf{(C)}\ p^2+q^2\qquad\textbf{(D)}\ p^2-q^2\qquad\textbf{(E)}\ p^2 </math>

600 bytes (108 words) - 10:47, 15 February 2021
1993 AIME Problems/Problem 4
...ath> with <math>0 < a < b < c < d < 500\,</math> satisfy <math>a + d = b + c\,</math> and <math>bc - ad = 93\,</math>? ...c - a(k-a) = (a-c)(a+c-k) = (c-a)(d-c) = 93</math>. Hence <math>(c - a,d - c) = (1,93),(3,31),(31,3),(93,1)</math>.

8 KB (1,343 words) - 16:27, 19 December 2023
1994 AIME Problems/Problem 15
...h>s\,</math> is not divisible by the square of any prime. What is <math>q+r+s\,</math>? ...BM_2 \sim \triangle ABC</math>, we see that thse segments respectively cut a <math>120^{\circ}</math> arc in the circle with radius <math>18</math> and

4 KB (717 words) - 22:20, 3 June 2021
1994 AIME Problems/Problem 14
...l bounce off the two line segments. Include the first reflection at <math>C\,</math> in your count. ...= MP("A",expi(alpha * pi/180),N); path r = C + .4 * expi(beta * pi/180) -- C - 2*expi(beta * pi/180);

2 KB (303 words) - 00:03, 28 December 2017
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=0</ma 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 previous roo

3 KB (451 words) - 15:02, 6 September 2021
1996 AIME Problems/Problem 15
...ce as large as angle <math>DBA</math>, and angle <math>ACB</math> is <math>r</math> times as large as angle <math>AOB</math>. Find <math>\lfloor 1000r \ ...,0), A=expi(pi/4), C=IP(A--A + 2*expi(17*pi/12), B--(3,0)), D=A+C, O=IP(A--C,B--D);

5 KB (710 words) - 21:04, 14 September 2020
1996 AIME Problems/Problem 14
A <math>150\times 324\times 375</math> [[rectangle|rectangular]] [[solid]] is ...ates of the diagonally opposite corner of the rectangle, where <math>a, b, c \in \mathbb{Z_{+}}</math>.

5 KB (923 words) - 21:21, 22 September 2023
1997 AIME Problems/Problem 15
...</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
...97</math> and <math>f(f(x))=x</math> for all values except <math>\frac{-d}{c}</math>. Find the unique number that is not in the range of <math>f</math>. ...the function definition, we have <math>\frac {a\frac {ax + b}{cx + d} + b}{c\frac {ax + b}{cx + d} + d} = x</math>, which reduces to

11 KB (2,063 words) - 22:59, 21 October 2023
1997 AIME Problems/Problem 5
...is <math>\frac 27.</math> What is the number of possible values for <math>r</math>? ...> anyway. For <math>\frac 27</math> to be the best approximation for <math>r</math>, the decimal must be closer to <math>\frac 27 \approx .28571</math>

1 KB (208 words) - 11:46, 4 June 2021
1998 AIME Problems/Problem 12
...t divisible by the square of any [[prime]]. What is <math>a^{2} + b^{2} + c^{2}</math>? ...\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
...e{AD}</math> is an interior diagonal. Points <math>P, Q,</math> and <math>R</math> are on <math>\overline{AB}, \overline{BC},</math> and <math>\overlin 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 10
...math> is not divisible by the square of any [[prime]]. Find <math>a + b + c</math>. ...a ninth one; instead the circles will overlap since the middle sphere has a larger radius and will sort of “bulge” out.

3 KB (496 words) - 13:02, 5 August 2019
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 ...F\)",foot(C,A,B),NW);label("$P$",P,NW);label("$Q$",Q,NE);label("$R$",R,SE);</asy><asy>import three; defaultpen(linewidth(0.6));

7 KB (1,169 words) - 15:28, 13 May 2024
1999 AIME Problems/Problem 12
...(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 6
...f [[quadrilateral]] <math>ABCD</math> are <math>A=(900,300), B=(1800,600), C=(600,1800),</math> and <math>D=(300,900).</math> Let <math>k_{}</math> be <cmath>\begin{eqnarray*}A' = & (\sqrt {900}, \sqrt {300})\\

2 KB (354 words) - 16:42, 20 July 2021
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
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
2001 AIME I Problems/Problem 12
...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> The [[radius]] of the sphe 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
2001 AIME I Problems/Problem 9
...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 the are 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,NW));

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 ...<math>y</math>-coordinate at each of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> corresponds to the value of the roll.

11 KB (1,729 words) - 20:50, 28 November 2023
2002 AIME I Problems/Problem 15
...h>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*sqrt(19),0,12), H1=(-6-2*sqr

7 KB (1,181 words) - 20:32, 8 January 2024
2003 AIME II Problems/Problem 7
...<math>b</math>. The length of the four sides of the rhombus is <math>\sqrt{a^2+b^2}</math>. ...h other and simplifying gives <math>b=2a</math>. Substitution yields <math>a=10</math> and <math>b=20</math>, so the area of the rhombus is <math>20\cdo

2 KB (323 words) - 09:56, 16 September 2022
2002 AIME II Problems/Problem 15
...<math>a</math> and <math>c</math> are relatively prime. Find <math>a + b + c</math>. ...<math>b</math> respectively. We know that the point <math>(9,6)</math> is a point on both circles, so we have that

7 KB (1,182 words) - 09:56, 7 February 2022
2002 AIME II Problems/Problem 13
...ne{BE}</math> intersect at <math>P.</math> Points <math>Q</math> and <math>R</math> lie on <math>\overline{AB}</math> so that <math>\overline{PQ}</math> pair A,B,C,D,E,X,P,Q,R;

6 KB (935 words) - 13:23, 3 September 2021
2002 AIME II Problems/Problem 12
...h>c</math> are positive integers. Find <math>\left(p+q+r+s\right)\left(a+b+c\right)</math>. ...ne <math>y=\frac{2x}{3}</math>. We can find the number of such paths using a Pascal's Triangle type method below, computing the number of paths to each

7 KB (1,127 words) - 13:34, 19 June 2022
2002 AIME II Problems/Problem 3
...</math> is the [[Perfect square|square]] of an integer. Find <math>a + b + c.</math> ...math> that works is <math>a=27</math>, from which we can deduce that <math>c=\dfrac{36}{27}\cdot 36=\dfrac{4}{3}\cdot 36=48</math>.

2 KB (263 words) - 22:50, 5 April 2024
2001 AIME II Problems/Problem 7
Let <math>\triangle{PQR}</math> be a [[right triangle]] with <math>PQ = 90</math>, <math>PR = 120</math>, and <m pair P = (0,0), Q = (90, 0), R = (0, 120), S=(0, 60), T=(45, 60), U = (60,0), V=(60, 40), O1 = (30,30), O2

7 KB (1,112 words) - 02:15, 26 December 2022
2000 AIME II Problems/Problem 12
...ts <math>A</math>, <math>B</math> and <math>C</math> lie on the surface of a [[sphere]] with center <math>O</math> and radius <math>20</math>. It is giv ...eron's Formula]] the area of <math>\triangle ABC</math> is (alternatively, a <math>13-14-15</math> triangle may be split into <math>9-12-15</math> and <

3 KB (532 words) - 13:14, 22 August 2020
2000 AIME II Problems/Problem 10
A [[circle]] is [[inscribe]]d in [[quadrilateral]] <math>ABCD</math>, [[tange ...19}{r})+\arctan(\tfrac{26}{r}))+(\arctan(\tfrac{37}{r})+\arctan(\tfrac{23}{r}))=180</math>.

2 KB (399 words) - 17:37, 2 January 2024
2005 AMC 10B Problems/Problem 7
...ribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the sm ...bf{(A) } \frac{\pi}{16} \qquad \textbf{(B) } \frac{\pi}{8} \qquad \textbf{(C) } \frac{3\pi}{16} \qquad \textbf{(D) } \frac{\pi}{4} \qquad \textbf{(E) }

2 KB (381 words) - 14:28, 14 December 2021
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
Incircle
...l that does have an incircle is called a [[Tangential Quadrilateral]]. For a triangle, the center of the incircle is the [[Incenter]], where the [[incir ...,c</math> and area <math>A</math> is <math>r =</math> <math>\frac{2A}{a+b+c}.</math>

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Inradius
...s [[incircle]] (assuming an incircle exists). It is commonly denoted <math>r</math>. pair A=(0,0),B=(4,0),C=(1.5,2),I=incenter(A,B,C),F=foot(I,A,B);

2 KB (336 words) - 00:44, 23 April 2024
Radius
The '''radius''' of a [[circle]] is the distance from the center to any point on the circle. Ide The radius of a circle is often denoted using R or r.

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Ring
...lgebra]], similar to a [[group]] or a [[field]]. A ring <math>R</math> is a [[set]] of elements closed under two [[operation]]s, usually called multipl * <math>(R,+)</math> is an [[abelian group]];

6 KB (994 words) - 06:16, 8 April 2015
Field
...mber]]s <math>\mathbb{R}</math>, and the [[complex number]]s <math>\mathbb{C}</math> are all fields, although there are many others, including subfields ...>k</math> stands for Körper, the German word for a mathematical field) is a [[set]] of elements with two [[operation]]s, usually called multiplication

2 KB (362 words) - 23:24, 31 December 2021
Group
...t is frequently suppressed, so <math>ab</math> is written instead of <math>a\cdot b</math>) satisfying the following conditions, known as the group axio * For all <math>a,b,c\in G</math>, <math>a(bc)=(ab)c</math> ([[associative|associativity]]).

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Holomorphic function
A '''holomorphic function''' <math>f: \mathbb{C} \to \mathbb{C}</math> is a rather than at points, for when we consider the behavior of a function

9 KB (1,537 words) - 21:04, 26 July 2017
Quadratic reciprocity
Let <math>p</math> be a [[prime number|prime]], and let <math>a</math> be any integer. Then we can define the [[Legendre symbol]] <cmath> \genfrac{(}{)}{}{}{a}{p} =\begin{cases} 1 & \text{if } a \text{ is a quadratic residue modulo } p, \\

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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
Law of Tangents
...law of sines]] and [[law of cosines]], to calculate [[angle]]s or sides in a [[triangle]]. ...>A</math> and <math>B</math> are angles in a triangle opposite sides <math>a</math> and <math>b</math> respectively, then

2 KB (306 words) - 16:11, 21 February 2023
University of South Carolina High School Math Contest/1993 Exam/Problem 29
If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the circle circums \mathrm{(A) \ } 2

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University of South Carolina High School Math Contest/1993 Exam/Problem 25
...th>x^2 + y^2 = 4</math> at <math>(0,2)</math>? (Two circles are tangent at a point <math>P</math> if they intersect at <math>P</math> and at no other po ...<math> \mathrm{(A) \ }(0,-6) \qquad \mathrm{(B) \ } (1,-9) \qquad \mathrm{(C) \ } (-1,-9) \qquad \mathrm{(D) \ } (0,-9) \qquad \mathrm{(E) \ } \rm{none

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University of South Carolina High School Math Contest/1993 Exam/Problems
If the width of a particular rectangle is doubled and the length is increased by 3, then the <cmath> \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ } 6 \qquad \mathrm{(E) \ } 9 </cmath>

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Mock AIME 1 2006-2007 Problems/Problem 9
Let <math>a_{n}</math> be a [[geometric sequence]] of [[complex number]]s with <math>a_{0}=1024</math> Let <math>a_{n}</math> be a geometric sequence for <math>n\in\mathbb{Z}</math> with <math>a_{0}=1024</m

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Mock AIME 1 2006-2007 Problems/Problem 10
...nd <math>r</math>, respectively. If <math>\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are positive in ...s <math>b=\frac{6}{11}</math>, radius <math>c=\frac{2}{5}</math> and <math>r=1</math>, see picture.

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1977 Canadian MO Problems/Problem 2
...on the circumference of the circle such that the angle <math>OPA</math> is a maximum. ...th>P</math>, then extend <math>AO</math> to meet the circle at point <math>C</math>. It is now evident that <math>O</math> is the midpoint of <math>AC</

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Mock AIME 1 2006-2007/Problems
...>\angle BAC</math> hits <math>BC</math> at <math>D</math>. If <math>\angle C=90^\circ</math>, and the maximum value of <math>\frac{[ABD]}{[ACD]}=\frac{m Let <math>\star (x)</math> be the sum of the digits of a positive integer <math>x</math>. <math>\mathcal{S}</math> is the set of pos

8 KB (1,355 words) - 14:54, 21 August 2020
Rolle's Theorem
Let <math>f:[a,b]\rightarrow\mathbb{R}</math> ...h>f</math> be continous on <math>[a,b]</math> and differentiable on <math>(a,b)</math>

1 KB (270 words) - 12:13, 30 May 2019
1971 Canadian MO Problems/Problem 1
...Join <math>OE</math> and extend <math>OE</math> to cut the circle at <math>C.</math> Given <math>EC=1,</math> find the radius of the circle. ...e at <math>P.</math> Let the radius be <math>r.</math> Applying [[power of a point]],

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1959 IMO Problems/Problem 5
(a) Prove that the points <math>N </math> and <math>N' </math> coincide. (b) Prove that the straight lines <math>MN </math> pass through a fixed point <math>S </math> independent of the choice of <math>M </math>.

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2006 Romanian NMO Problems
...<math>2 \cdot \frac{CN}{BC} = \frac{AM}{AB}</math>. Let <math>P</math> be a point on the line <math>AC</math>. Prove that the lines <math>MN</math> and ...nimal <math>n</math>, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or

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2006 Romanian NMO Problems/Grade 7/Problem 1
...<math>2 \cdot \frac{CN}{BC} = \frac{AM}{AB}</math>. Let <math>P</math> be a point on the line <math>AC</math>. Prove that the lines <math>MN</math> and Let <math>L</math> be a point on <math>BC</math> such that <math>N</math> is the midpoint of LC, th

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2006 Romanian NMO Problems/Grade 9/Problem 3
...scribe]]d in a [[circle]] of [[radius]] <math>r</math>, for which there is a [[point]] <math>P</math> on <math>CD</math> such that <math>CB=BP=PA=AB</ma (a) Prove that there are points <math>A,B,C,D,P</math> which fulfill the above conditions.

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2005 Canadian MO Problems
...d the path never travels up (from a lower row to a higher row) or revisits a triangle. An example of one such path is illustrated below for <math>n=5</m ...n triple, ''i.e.'', a triplet of positive integers with <math>{a}^2+{b}^2={c}^2</math>.

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2005 Canadian MO Problems/Problem 3
Let <math>S</math> be a set of <math>n\ge 3</math> points in the interior of a circle. ...an any other point in <math>S</math> and <math>c</math> is closer to <math>C</math> than any other point in <math>S</math>.

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2005 Canadian MO Problems/Problem 4
...>P</math> and area <math>K</math>. Determine the maximum value of <math>KP/R^3</math>. ...GM and Jensen's to finish. (Jensen's is used for <math>\sin A+\sin B+\sin C \le \dfrac{3\sqrt3}{2}</math>.

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Addition
...ign) which combines two quantities. The result of addition is called [[sum|a sum]]. For example, the sum of 3 and 2 is 5 because <math>3+2=5</math>. ...)</math>, where <math>f</math> is a [[function]], is denoted <math>\sum_{i=a}^bf(i)</math>. (See also [[Sigma notation]])

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2005 AMC 10A Problems/Problem 10
...one solution for <math>x</math>. What is the sum of those values of <math>a</math>? <math> \textbf{(A) }-16\qquad\textbf{(B) }-8\qquad\textbf{(C) } 0\qquad\textbf{(D) }8\qquad\textbf{(E) }20 </math>

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Volume
The '''volume''' of an object is a [[measure]] of the [[amount]] of [[space]] that it occupies. Note that volu The volume of a [[prism]] of [[height]] <math>h</math> and base of [[area]] <math>b</math>

3 KB (523 words) - 20:24, 17 August 2023
Integral
...ath>c</math> is a [[real]] [[constant]]. This is because the derivative of a constant is <math>0</math>. *The integral of a function <math>f(x)</math> is written as <math>\int f(x)\,dx</math>, where

5 KB (909 words) - 14:16, 31 May 2022
1970 IMO Problems
...math> of <math>\triangle ABC</math>. Let <math>r_1, r_2</math>, and <math>r</math> be the inscribed circles of triangles <math>AMC, BMC</math>, and <ma <math>\frac{r_1}{q_1} \cdot \frac{r_2}{q_2} = \frac{r}{q}</math>.

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1970 IMO Problems/Problem 1
...math> of <math>\triangle ABC</math>. Let <math>r_1, r_2</math>, and <math>r</math> be the inscribed circles of triangles <math>AMC, BMC</math>, and <ma <math>\frac{r_1}{q_1} \cdot \frac{r_2}{q_2} = \frac{r}{q}</math>.

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Complex conjugate
...s a root of the polynomial its conjugate <math>\overline{z}</math> will be a root as well. It also interacts in simple ways with other operations on <math>\mathbb{C}</math>:

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Metric space
...es S \to \mathbb{R}_{\geq 0}</math>. The metric <math>d</math> represents a distance function between pairs of points of <math>S</math> which has the f ...B should be at least as short as a roundabout path that visits some point C first.

2 KB (324 words) - 01:19, 22 December 2012
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>

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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
1981 IMO Problems/Problem 5
...nt <math>O</math> and lie inside a given [[triangle]]. Each circle touches a pair of sides of the triangle. Prove that the [[incenter]] and the [[circum ..., and let the centers of the circles inscribed in the [[angle]]s <math>A,B,C</math> be denoted <math>O_A, O_B, O_C </math>, respectively.

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Euler line
...] <math>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain fixed orders and distan ...followed by a homothety with scale factor <math>2</math> centered at <math>A</math> brings <math>\triangle ABC \to \triangle O_AO_BO_C</math>. Let us ex

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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) \ } 2003\qquad \mathrm{(E) \ } 4006 </math>

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2003 AMC 10A Problems/Problem 17
...\sqrt{2}}{\pi}\qquad \mathrm{(B) \ } \frac{3\sqrt{3}}{\pi}\qquad \mathrm{(C) \ } \sqrt{3}\qquad \mathrm{(D) \ } \frac{6}{\pi}\qquad \mathrm{(E) \ } \sq ...s</math> be the length of a side of the equilateral triangle and let <math>r</math> be the radius of the circle.

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Matrix
...<math>F</math>, where <math>A</math> and <math>B</math> are the sets <math>A=\{1,2,\ldots,m\}</math> and <math>B=\{1,2,\ldots,n\}</math>. ...>F^m</math>, where <math>m</math> is the number of rows. If a matrix <math>A</math> has <math>m</math> rows and <math>n</math> columns, its order is sai

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2001 IMO Problems/Problem 1
...at <math>\angle C \geq \angle B+30^{\circ}</math>. Prove that <math>\angle A+\angle COP < 90^{\circ}</math>. ...of the circumcircle. But <math>AZ = YP</math> (since <math>AZYP</math> is a rectangle).

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2003 AMC 10A Problems/Problem 25
...ided by <math>100</math>. For how many values of <math>n</math> is <math>q+r</math> divisible by <math>11</math>? <math> \mathrm{(A) \ } 8180\qquad \mathrm{(B) \ } 8181\qquad \mathrm{(C) \ } 8182\qquad \mathrm{(D) \ } 9000\qquad \mathrm{(E) \ } 9090 </math>

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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) \ 10 } \qquad \mathrm{(E) \ 12 } </math>

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2003 AMC 12A Problems/Problem 18
...ided by <math>100</math>. For how many values of <math>n</math> is <math>q+r</math> divisible by <math>11</math>? <math> \mathrm{(A) \ } 8180\qquad \mathrm{(B) \ } 8181\qquad \mathrm{(C) \ } 8182\qquad \mathrm{(D) \ } 9000\qquad \mathrm{(E) \ } 9090 </math>

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2005 Alabama ARML TST Problems/Problem 3
...>300\pi</math> square units. Find the number of units in the [[length]] of a [[edge | side]] of the [[triangle]]. ...triangle | median]], so <math>O</math> trisects <math>CO</math> and <math>R = CO = 2OM = 2r</math>.)

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1987 IMO Problems/Problem 4
...ath>f(f(n)) = n + k</math> for all <math>n</math>, where <math>k</math> is a fixed positive integer, then <math>k</math> must be even. If <math>k = 2h</ ...n + kr</math>, <math>f(n + kr) = m + k(r+1)</math>. We show this leads to a contradiction. Suppose <math>m < n</math>, so <math>n = m + ks</math> for s

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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
Mock AIME 4 2006-2007 Problems/Problem 11
...d <math>r</math> are relatively prime positive integers, compute <math>p+q+r</math>. ...ath>EA = DB = b</math>. Note that we want to compute the ratio <math>\frac{a}{b}</math>.

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Mock AIME 4 2006-2007 Problems/Problem 15
We set up a trivial coordinate bash. Let A = 0,0, C = 48,0, B = 83/2, 13sqrt3/2. We find the coordinates of the circumcenter to ...be the length of the [[radius]] of <math>\omega</math>. By the [[Power of a Point Theorem]], <math>MD \cdot (2R - MD) = AM \cdot MC = 24^2</math> or <m

3 KB (532 words) - 20:29, 31 August 2020
Quaternion
...icative [[inverse with respect to an operation | inverse]]; alternatively, a non[[commutative]] [[field]]) which generalize the [[complex number]]s. ...aternions are the set <math>\{a + bi + cj + dk\}</math>, where <math>a, b, c, d</math> are any [[real number]]s and the behavior of <math>i, j, k</math>

1,007 bytes (155 words) - 20:47, 14 October 2013
Asymptote: Logical Operators and Loops
...ote uses loops and logical operators that are almost identical to those in C++. Loops are absolutely essential if you want to make diagrams that look l real r=5;

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Asymptote: Macros and Packages
...(a,b)</math> if no value of <math>r</math> is specified, so you want <math>r</math> to default to <math>0</math>. The code would be as follows: pair newfunction(pair z, real r=0)

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2006 Canadian MO Problems/Problem 2
Let <math>ABC</math> be an [[acute angle]]d [[triangle]]. Inscribe a [[rectangle]] <math>DEFG</math> in this triangle so that <math>D</math> is ...n by [[similarity]], <math>\frac{AE}{AC} = \frac{GH}{BH} = \frac{FH}{CH} = r</math>.

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2007 AIME II Problems/Problem 9
...thing for <math>F</math>, we find that <math>FD = 105</math> as well. Draw a line through <math>E,F</math> parallel to the sides of the rectangle, to in ...tor, <math>7</math>? Then we're left with much simpler numbers which saves a lot of time. In the end, we will multiply by <math>7</math>.

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2005 AMC 10A Problems/Problem 23
...ts on the circle such that <math>DC \perp AB</math> and <math>DE</math> is a second diameter. What is the ratio of the area of <math>\triangle DCE</math pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);

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2002 USAMO Problems/Problem 2
Let <math>ABC </math> be a triangle such that ...}{2} \right)^2 + \left( 2 \cot \frac{B}{2} \right)^2 + \left( 3 \cot \frac{C}{2} \right)^2 = \left( \frac{6s}{7r} \right)^2

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Mock AIME 1 Pre 2005 Problems
<math>A, B, C, D,</math> and <math>E</math> are collinear in that order such that <math>A ...th>1 + 7 + 7^2 + \cdots + 7^{2004}</math> is divided by <math>1000</math>, a remainder of <math>N</math> is obtained. Determine the value of <math>N</ma

6 KB (1,100 words) - 22:35, 9 January 2016
Mock AIME 3 Pre 2005 Problems
A function <math>f(x)</math> is defined for all real numbers <math>x</math>. ...are consonants. A string of <math>M's, O's,</math> and <math>P's</math> is a word in Zuminglish if and only if between any two <math>O's</math> there ap

7 KB (1,135 words) - 23:53, 24 March 2019
Mock AIME 3 Pre 2005 Problems/Problem 5
...are consonants. A string of <math>M's, O's,</math> and <math>P's</math> is a word in Zuminglish if and only if between any two <math>O's</math> there ap ...te the number of <math>n</math>-letter words ending in a vowel followed by a constant (<tt>VC</tt> - the only other combination, two vowels, is impossib

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Mock AIME 3 Pre 2005 Problems/Problem 6
...math> is not divisible by the square of any prime. Determine <math>p + q + r</math>. This is a [[telescoping series]]; note that when we expand the summation, all of the

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Mock AIME 3 Pre 2005 Problems/Problem 7
<math>ABCD</math> is a cyclic quadrilateral that has an inscribed circle. The diagonals of <math>A ...is a well-known fact that if <math>ABCD</math> is circumscriptable around a circle then <math>AB+CD=AD+BC</math>. Therefore <math>BC+AD=5</math>. We al

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Mock AIME 3 Pre 2005 Problems/Problem 14
...math> is not divisible by the square of any prime. Determine <math>p + q + r</math>. Let point <math>R</math> be the midpoint of <math>QP</math>. Thus, <math>O_3R \bot PQ</math>

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Arc
...en two given points on the circle. More generally, an arc is a portion of a smooth curve joining two points. ...Thus, in particular, the [[circumference]] of a circle is given by <math>C = 2\pi</math>.

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1960 IMO Problems/Problem 3
...nteger). Let <math>\alpha</math> be the acute angle subtending, from <math>A</math>, that segment which contains the midpoint of the hypotenuse. Let <ma \tan{\alpha}=\frac{4nh}{(n^2-1)a}.

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2007 AIME I Problems
...ly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the oth ...mber <math>z</math> is equal to <math>9+bi</math>, where <math>b</math> is a positive real number and <math>i^{2}=-1</math>. Given that the imaginary p

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2007 AIME I Problems/Problem 8
We can see that <math>Q_1</math> and <math>Q_2</math> must have a [[root]] in common for them to both be [[factor]]s of the same cubic. Let this root be <math>a</math>.

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2007 AIME I Problems/Problem 9
...>. Its legs <math>CA</math> and <math>CB</math> are extended beyond <math>A</math> and <math>B</math>. [[Point]]s <math>O_1</math> and <math>O_2</math ...t{\frac{1 + \frac{15}{17}}{1 - \frac{15}{17}}}</math>, and <math>x = \frac{r}{4}</math>.

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2007 AIME I Problems/Problem 12
...+ s</math>, where <math>p,q,r,s</math> are integers. Find <math>\frac{p-q+r-s}2</math>. ...ate(theta,A)*C, Bp=rotate(theta,A)*B, X=extension(A,Bp,B,C), Y=extension(B,C,Bp,Cp);

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2007 AIME I Problems/Problem 13
...math> and vertex <math>E</math> has eight edges of length <math>4</math>. A plane passes through the midpoints of <math>AE</math>, <math>BC</math>, and Note first that the intersection is a [[pentagon]].

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2007 AIME I Problems/Problem 15
...an [[integer]] not divisible by the [[square]] of a [[prime]]. Find <math>r</math>. ...math>. Add up the areas of the triangles using the <math>\frac{1}{2}ab\sin C</math> formula (notice that for the three outside triangles, <math>\sin 60

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2007 AIME II Problems/Problem 15
...t]] to <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math>. If the sides of triangle <math>ABC</math> are <math>13,</math> <ma pair A,B,C,X,Y,Z,P,Q,R;

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2007 AIME II Problems/Problem 14
Let <math>f(x)</math> be a [[polynomial]] with real [[coefficient]]s such that <math>f(0) = 1,</math> ...2x^3 + x) = 2^max^{3m}</math>. Hence <math>2^ma^2 = 2^ma</math>, and <math>a = 1</math>. Because <math>f(0) = 1</math>, the product of all the roots

7 KB (1,335 words) - 17:44, 25 January 2022
2007 AIME II Problems
...rs among the four letters in AIME or the four digits in <math>2007</math>. A set of plates in which each possible sequence appears exactly once contains ...math>b</math>, <math>a</math> is a factor of <math>c</math>, and <math>a+b+c=100</math>.

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1961 IMO Problems/Problem 2
Let <math>a</math>, <math>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>

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2007 USAMO Problems/Problem 2
(''Gregory Galperin'') A [[square]] grid on the [[Cartesian plane|Euclidean plane]] consists of all ...ngent]] circles with radius greater than or equal to 5, one can always fit a circle with radius greater than <math>\frac{1}{\sqrt{2}}</math> between tho

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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.

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2007 USAMO Problems/Problem 6
...[[internally tangent|tangent internally]] to <math>\Omega</math> at <math>A</math> and tangent internally to <math>\omega</math>. Let <math>P_A</math> <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
2007 Cyprus MO/Lyceum/Problem 12
...=x+y+xy+k\ \ \ \forall x,y \in \Re</math>, where <math>k \in \Re</math> is a constant. The value of <math>f(-1)</math> is <math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } -2\qquad \mathrm{(E) \ } 3</math>

982 bytes (165 words) - 00:19, 19 January 2024
2005 IMO Shortlist Problems/A2
...mbers. Determine all functions <math> f: \mathbb{R}^+ \rightarrow \mathbb{R}^+ </math> such that or <math>f(x) = f(z) </math>, a contradiction. {{Halmos}}

7 KB (1,179 words) - 20:59, 28 March 2010
2005 IMO Shortlist Problems/A3
Four real numbers <math> \displaystyle p,q,r,s </math> satisfy <math> \displaystyle p+q+r+s = 9 </math>

3 KB (542 words) - 17:09, 19 December 2018
Circumference
...term circumference is most frequently used to refer to the distance around a [[circle]], though it may refer to the distance around any [[smooth]] curve ...ath>r</math> and [[diameter]] <math>d = 2r</math>, the circumference <math>C</math> is given by

799 bytes (130 words) - 17:27, 9 February 2020
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
Dodecagon
A '''dodecagon''' is a 12-sided [[polygon]]. The sum of its internal [[angle]]s is <math>1800^{\ci A regular dodecagon can be seen below:

1 KB (219 words) - 13:08, 15 June 2018
2005 AMC 12A Problems/Problem 24
...olynomial <math>R(x)</math> of degree 3 such that <math>P(Q(x))=P(x) \cdot R(x)</math>? <math>\mathrm {(A) } 19 \qquad \mathrm {(B) } 22 \qquad \mathrm {(C) } 24 \qquad \mathrm {(D) } 27 \qquad \mathrm {(E) } 32</math>

5 KB (833 words) - 18:17, 8 April 2024
2005 AMC 12A Problems/Problem 22
...> is 384, and the sum of the lengths of its 12 edges is 112. What is <math>r</math>? <math>\mathrm{(A)}\ 8\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 12\qquad \mathrm{(D)}\ 14\qquad \mathrm{(E)}\ 16</math>

2 KB (334 words) - 10:20, 16 September 2022
2007 AMC 12A Problems/Problem 9
<math>\mathrm{(A)}\ \frac 23\qquad \mathrm{(B)}\ \frac 34\qquad \mathrm{(C)}\ \frac 45\qquad \mathrm{(D)}\ \frac 56\qquad \mathrm{(E)}\ \frac 78</math ...=\frac{x}{7}</math>, so <math>\frac{6}{7}x = 2r</math>, and <math>\frac{x}{r} = \frac{7}{3}</math>. This minus one is the reciprocal of what we want to

5 KB (804 words) - 14:55, 21 August 2022
2002 Indonesia MO Problems/Problem 7
...S</math> is also a rhombus, show that exactly one of <math>Q</math>, <math>R</math>, and <math>S</math> is located on the vertices of rhombus <math>ABCD ...all on one side of the rhombus. Then, in order for <math>PQRS</math> to be a parallelogram, <math>P</math> should also be on that side. But this is not

4 KB (734 words) - 12:41, 18 August 2020
2005 AMC 12A Problems/Problem 6
...es apart. Yesterday, Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When the (\mathrm {A}) \ 4 \qquad (\mathrm {B}) \ 5 \qquad (\mathrm {C})\ 6 \qquad (\mathrm {D}) \ 7 \qquad (\mathrm {E})\ 8

1,006 bytes (166 words) - 21:18, 3 July 2013
2005 AMC 12A Problems/Problem 15
...\overline{DC} \perp \overline{AB}</math> and <math>\overline{DE}</math> is a second diameter. What is the [[ratio]] of the area of <math>\triangle DCE</ pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);

14 KB (1,970 words) - 17:02, 18 August 2023
Green's Theorem
'''Green's Theorem''' is a result in [[real analysis]]. It is a special case of [[Stokes' Theorem]].

2 KB (381 words) - 12:12, 30 May 2019
2006 AIME II Problems/Problem 12
...visible]] by the [[perfect square | square]] of any prime, find <math> p+q+r. </math> pair A=(0,0),B=(-3^.5,-3),C=(3^.5,-3),D=13*expi(-2*pi/3),E1=11*expi(-pi/3),F=E1+D;

6 KB (1,033 words) - 02:36, 19 March 2022
2006 AIME II Problems/Problem 9
...<math>r</math> is not divisible by the square of any prime, find <math>p+q+r.</math> ...hagorean theorem]] yields <math>r_1r = \sqrt{15}</math>. On <math>\mathcal{C}_3</math>, we can do the same thing to get <math>O_3s_1 = 4</math> and <mat

3 KB (553 words) - 10:45, 26 August 2015

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