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• List of United States elementary school mathematics competitions
  * [https://pimathcontest.com/ Pi Math Contest] is a math competition for elementary, middle, and high school *[[RSM Foundation International Math Contest]] is a two-round Olympiad-style contest for students in grades 3-8. ([https://rsmfound
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• List of United States middle school mathematics competitions
  * [https://pimathcontest.org/ Pi Math Contest] is a two-round math competition for elementary and middle school students. *[[RSM Foundation International Math Contest]] is a two-round Olympiad-style contest for students in grades 3-8 ([https://rsmfounda
  7 KB (792 words) - 10:14, 23 April 2024
• Geometric inequality
  The [[Triangle Inequality]] says that the sum of the lengths of any two sides of a non[[degenerate]] triangle is greater than the length of the thi ...area]] <math>A</math> and [[perimeter]] <math>P</math>, then <math>\frac{4\pi A}{P^2} \le 1</math>. This means that given a perimeter <math>P</math> for
  7 KB (1,296 words) - 14:22, 22 October 2023
• Number theory/Advanced
  An elliptic curve is the set of points <math>(x,y)</math> satisfying some two variable third degree equation. Using certain affine transformations it can ...math> or when this line is tangent to the curve, and hence cuts it in only two points.) We define <math>P+Q</math> to be the reflection of <math>R</math>
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• Real number
  ...any value. Some examples of real numbers are:<math>1, 2, -23.25, 0, \frac{\pi}{\phi}</math>, and so on. Numbers that are not real are <math>\ 3i</math>, ...n the [[algebraic number]]s and [[transcendental number]]s, although these two classes are best understood as subsets of the [[complex number]]s.
  3 KB (496 words) - 23:22, 5 January 2022
• Circle
  ...^2</math>. Both formulas involve the mathematical constant [[pi]] (<math>\pi</math>). We shall explore two of the Greek [[mathematician]] [[Archimedes]] demonstrations of the area of
  9 KB (1,555 words) - 20:05, 2 November 2023
• Ellipse
  ...ath> to two fixed [[focus|foci]] is a constant. (The equivalence of these two definitions is a non-trivial fact.) ...endicular]] to the axis of the the cone, or (in the second definition) the two foci of the ellipse coincide.
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• Pi
  ...mber]], as proved by Lindemann in 1882) denoted by the Greek letter <math>\pi </math>. ...^2}=\frac{\pi^2}{6}</math>. Some common [[fraction]]al approximations for pi are <math>\frac{22}{7} \approx 3.14285</math> and <math>\frac{355}{113} \ap
  8 KB (1,469 words) - 21:11, 16 September 2022
• Logarithm
  ..., we have <math>\ln (-1)=i\pi</math>. Additionally, <math>\ln (-n)=\ln n+i\pi</math> for positive real <math>n</math>. Two-minute Intro to Logarithms [http://www.youtube.com/watch?v=ey7ttABX9SM]
  4 KB (680 words) - 12:54, 16 October 2023
• Irrational number
  ...n [[decimal notation]], never terminates nor repeats. Examples are <math>\pi, \sqrt{2}, e, \sqrt{32134},</math> etc. There are two types of irrational numbers: algebraic and transcendental.
  3 KB (368 words) - 19:26, 6 June 2015
• Roots of unity
  ...as a positive real number). This leaves us with <math>e^{ni\theta} = e^{2\pi ik}</math>. ...math> ni\theta = 2\pi ik</math>. Solving this gives <math> \theta=\frac{2\pi k}n </math>. Additionally, we note that for each of <math> k=0,1,2,\ldots,
  3 KB (558 words) - 21:36, 11 December 2011
• Complex analysis
  <cmath> f^{(n)}(z_0)=\frac{n!}{2\pi i} \oint_\Gamma \frac{f(z)\; dz}{(z-z_0)^{n+1}}. </cmath> ...lle's Theorem. If <math>f</math> is an entire function so that there exist two complex numbers <math>a</math> and <math>b</math> such that for every compl
  2 KB (271 words) - 22:06, 12 April 2022
• LaTeX:Symbols
  |<math>\pi</math>||\pi||<math>\varpi</math>||\varpi||<math>\rho</math>||\rho||<math>\varrho</math> | <math>\Xi</math>||\Xi||<math>\Pi</math>||\Pi||<math>\Sigma</math>||\Sigma||<math>\Upsilon</math>||\Upsilon
  16 KB (2,324 words) - 16:50, 19 February 2024
• 2021 JMC 10
  ...can <math>|\pi - |e - | e - \pi|||</math> be expressed in terms of <math>\pi</math> and <math>e?</math> ...}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 12B Problems
  ...ball game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 po ...they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's s
  13 KB (2,058 words) - 12:36, 4 July 2023
• 2006 AMC 12A Problems
  ...math> is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is <mat How many sets of two or more consecutive positive integers have a sum of <math>15</math>?
  15 KB (2,223 words) - 13:43, 28 December 2020
• 2004 AMC 12A Problems
  ...0</math>, or <math>1</math>. How many distinct lines pass through at least two members of <math>S</math>? ...rt3}{2} \qquad \text {(D) } \frac {10\pi}{3} - \sqrt3 \qquad \text {(E) }4\pi - 2\sqrt3</math>
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2003 AMC 12A Problems
  The sum of the two 5-digit numbers <math>AMC10</math> and <math>AMC12</math> is <math>123422</ ...}{4}+\frac{1}{24}\pi\qquad \mathrm{(E) \ } \frac{\sqrt{3}}{4}+\frac{1}{12}\pi </math>
  13 KB (1,955 words) - 21:06, 19 August 2023
• 2002 AMC 12A Problems
  ...\pi \qquad \text{(C)}\ 2\pi \qquad \text{(D)}\ 3\pi \qquad \text{(E)}\ 3.5\pi</math> Two different positive numbers <math>a</math> and <math>b</math> each differ fr
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2001 AMC 12 Problems
  The sum of two numbers is <math>S</math>. Suppose <math>3</math> is added to each number a each of the resulting numbers is doubled. What is the sum of the final two
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2003 AMC 12B Problems
  ...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? ...) = 8</math> and <math>\clubsuit(123) = 1 + 2 + 3 = 6.</math> For how many two-digit values of <math>x</math> is <math>\clubsuit(\clubsuit(x)) = 3?</math>
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2004 AMC 12B Problems
  ...{C}) 75+100\pi \quad (\mathrm {D}) 100+100\pi \quad (\mathrm {E}) 100+125\pi</math> ...a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains 100 cans, how many
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  ...e for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar. What was their profit, in dollars? \mathrm{(A)}\ 80-20\pi \qquad
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2005 AMC 12B Problems/Problem 18
  ...math>\frac{14^2}{2}-\frac{4^2}{2}-(\frac{5\sqrt{2}}{2})^2\pi=98-8-\frac{25\pi}{2}</math>, which is approximately <math>51</math>. The answer is <math>\bo
  2 KB (262 words) - 21:20, 21 December 2020
• 2005 AMC 12B Problems/Problem 22
  ...i\theta_n}</math>, where <math>\theta_n=2^n(\theta_0+\frac{\pi}{2})-\frac{\pi}{2}</math>. ...\frac{iz_k}{\overline {z_k}}=i e^{i\theta_k} e^{i\theta_k}=e^{i(2\theta_k+\pi/2)}
  4 KB (660 words) - 17:40, 24 January 2021
• 2006 AMC 10A Problems
  ...math> is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is <ma How many sets of two or more consecutive positive integers have a sum of 15?
  13 KB (2,028 words) - 16:32, 22 March 2022
• 2005 AIME II Problems/Problem 9
  ...e have that <math>\left(\sin\left(\frac\pi2 - u\right) + i \cos\left(\frac\pi 2 - u\right)\right)^n = \sin n \left(\frac\pi2 -u \right) + i\cos n \left(\ ...left(\cos u + i \sin u\right)^n = \cos nu + i\sin nu</math>. We know that two [[complex number]]s are equal if and only if both their [[real part]] and [
  6 KB (1,154 words) - 03:30, 11 January 2024
• 2005 AIME II Problems/Problem 12
  Setting these two expressions for <math>(OF)^2</math> equal and solving for <math>x</math> (i ...equation yields <math>a^2+b^2 = 290000</math>. We now finish by adding the two original stewart equations and obtaining: <cmath>2(450\sqrt{2})^2 = (500-x)
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME I Problems/Problem 1
  ...ruent]] [[circle]]s form a ring with each circle [[externally tangent]] to two circles adjacent to it. All circles are [[internally tangent]] to a circle ...ath>. <math>K = 30^2\pi - 6(10^2\pi) = 300\pi</math>, so <math>\lfloor 300\pi \rfloor = \boxed{942}</math>.
  1 KB (213 words) - 13:17, 22 July 2017
• 2005 AIME I Problems/Problem 6
  ...ation are <math>x = 1 + i^n r</math> for <math>n = 0, 1, 2, 3</math>. The two non-real members of this set are <math>1 + ir</math> and <math>1 - ir</math ...a quadratic, then <math>((x-1)^2+\sqrt{2006})</math> is bound to hold the two non-real roots since the other definitely crosses the x-axis twice since i
  4 KB (686 words) - 01:55, 5 December 2022
• 2004 AIME I Problems/Problem 11
  ...ed. A [[plane]] [[parallel]] to the base of the cone divides the cone into two solids, a smaller cone-shaped solid <math> C </math> and a [[frustum]]-shap ...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 4
  ...sides of the square determined by the vertex <math>(0,0)</math>. Let the two endpoints of the segment have coordinates <math>(x,0)</math> and <math>(0,y ...ed by all of the midpoints is <math>4-4\cdot \left(\frac{\pi}{4}\right)=4-\pi \approx .86</math> to the nearest hundredth. Thus <math>100\cdot k=\boxed{8
  3 KB (532 words) - 09:22, 11 July 2023
• 2004 AIME I Problems
  ...of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space d ...ratio was <math>\frac{300}{500} = \frac{3}{5}</math>. The largest possible two-day success ratio that Beta could achieve is <math> m/n, </math> where <mat
  9 KB (1,434 words) - 13:34, 29 December 2021
• 2004 AIME II Problems/Problem 7
  Dividing the two equations yields ...\theta</math>, so <math>\angle B'EA = \pi-2\theta</math>. Then <math>\tan(\pi-2\theta)=\frac{15}{8}</math>, or
  9 KB (1,501 words) - 05:34, 30 October 2023
• 2004 AIME II Problems/Problem 1
  ...to the smaller can be expressed in the form <math> \frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}}, </math> where <math> a, b, c, d, e, </math> and <math> f </math ...; 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
• 2004 AIME II Problems
  ...to the smaller can be expressed in the form <math> \frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}}, </math> where <math> a, b, c, d, e, </math> and <math> f </math ...es and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they ge
  9 KB (1,410 words) - 05:05, 20 February 2019
• 1989 AIME Problems
  ...llows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meetin pair A=(0,0),B=(10,0),C=6*expi(pi/3);
  7 KB (1,045 words) - 20:47, 14 December 2023
• 1991 AIME Problems
  ...th>x^{}_{}</math> satisfy the equation <math>\frac{1}{5}\log_2 x = \sin (5\pi x)</math>? Two three-letter strings, <math>aaa^{}_{}</math> and <math>bbb^{}_{}</math>, ar
  7 KB (1,106 words) - 22:05, 7 June 2021
• 1992 AIME Problems
  ...is called ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascendi In Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below.
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1994 AIME Problems
  ...the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. Th ...\,</math> determine the number of times the light beam will bounce off the two line segments. Include the first reflection at <math>C\,</math> in your co
  7 KB (1,141 words) - 07:37, 7 September 2018
• 1995 AIME Problems
  ...o sides of square <math>S_{i+1},</math> are the perpendicular bisectors of two adjacent sides of square <math>S_{i+2}.</math> The total area enclosed by ...<math>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.
  6 KB (1,000 words) - 00:25, 27 March 2024
• 1998 AIME Problems
  Given that <math>A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,</math> find <math>|A_{19} + A_{20} + \cdots + A_{98}|.</math> Except for the first two terms, each term of the sequence <math>1000, x, 1000 - x,\ldots</math> is o
  7 KB (1,084 words) - 02:01, 28 November 2023
• 2003 AIME I Problems
  ...ion bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green r ...math> \mathcal{S}, </math> she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.
  6 KB (965 words) - 16:36, 8 September 2019
• 2002 AIME II Problems
  ...e1,2,3,\ldots,10\rbrace</math> Let <math>n</math> be the number of sets of two non-empty disjoint subsets of <math>\mathcal{S}</math>. (Disjoint sets are ...math> radians are <math>\frac{m\pi}{n-\pi}</math> and <math>\frac{p\pi}{q+\pi}</math>, where <math>m</math>, <math>n</math>, <math>p</math>, and <math>q<
  7 KB (1,177 words) - 15:42, 11 August 2023
• 2003 AIME II Problems
  ...gers is 6 times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of <math>N</math>. ...e log. The number of cubic inches in the wedge can be expressed as <math>n\pi</math>, where n is a positive integer. Find <math>n</math>.
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1983 AIME Problems/Problem 15
  The adjoining figure shows two intersecting chords in a circle, with <math>B</math> on minor arc <math>AD< ...ally, the circle <math>P</math> can intersect the chord <math>BC</math> at two points, one point, or they may not have a point of intersection. By the pro
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1984 AIME Problems/Problem 12
  <cmath>f(x) = \sin \frac{\pi x}{10}\sin \frac{\pi (x-4)}{10}</cmath> ...tice that it starts at x=0, then it goes to x=5, x=10, etc... each f() has two possible x values, but we are only counting the total number of x values so
  3 KB (588 words) - 14:37, 22 July 2020
• 1985 AIME Problems/Problem 12
  Two solutions follow from here: ...}</math> and add them up before dividing by <math>3^7</math>. Here we have two ways to proceed:
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1985 AIME Problems/Problem 9
  ...ha + \beta</math> [[radian]]s, respectively, where <math>\alpha + \beta < \pi</math>. If <math>\cos \alpha</math>, which is a [[positive]] [[rational num pair O = (0,0), A = r*expi(pi/3);
  5 KB (763 words) - 16:20, 28 September 2019
• 1987 AIME Problems/Problem 14
  Two solutions follow from here: ...eta</math> is the argument of <math>N</math> such that <math>0\leq\theta<2\pi.</math>
  7 KB (965 words) - 10:42, 12 April 2024
• 1988 AIME Problems/Problem 14
  Solving these two equations, we find <math>x = \frac{-3x' + 4y'}{5}</math> and <math>y = \fra Now to find the equation of the hyperbola, we multiply the two expressions together to get one side of the equation: <math>(3x-4y)(4x+3y)=
  4 KB (700 words) - 17:21, 3 May 2021
• 1989 AIME Problems/Problem 6
  ...s the [[straight]] path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meetin pair A=(0,0),B=(10,0),C=6*expi(pi/3);
  5 KB (864 words) - 19:55, 2 July 2023
• 1990 AIME Problems/Problem 12
  ...method involves drawing a triangle connecting the center of the 12-gon to two vertices of the 12-gon. Since the distance from the center to a vertex of To simplify the two nested radicals, add them, and call the sum <math>x</math>:
  6 KB (906 words) - 13:23, 5 September 2021
• 1990 AIME Problems/Problem 10
  ...8</math> and <math>48</math> is <math>144</math>, so define <math>n = e^{2\pi i/144}</math>. We can write the numbers of set <math>A</math> as <math>\{n^ ...</math> are relatively prime, and by the [[Chicken McNugget Theorem]], for two relatively prime integers <math>a,b</math>, the largest number that cannot
  3 KB (564 words) - 04:47, 4 August 2023
• 1991 AIME Problems/Problem 15
  .../math> for <math>1 \le i \le n</math> and <math>0 \le \theta_{i} < \frac {\pi}{2}</math>. We then have that ...e can see that it is basically calculating distance between the origin and two points. Therefore we can assume every <math>\sqrt{(2k-1)^2+a_k^2}</math> to
  4 KB (658 words) - 16:58, 10 November 2023
• 1991 AIME Problems/Problem 11
  ...The sum of the areas of the twelve disks can be written in the from <math>\pi(a-b\sqrt{c})</math>, where <math>a,b,c^{}_{}</math> are positive integers a dot((cos(i*pi/6), sin(i*pi/6)));
  4 KB (740 words) - 19:33, 28 December 2022
• 1991 AIME Problems/Problem 4
  ...^{}_{}</math> satisfy the [[equation]] <math>\frac{1}{5}\log_2 x = \sin (5\pi x)</math>? ...h>y = 0</math>, there is exactly <math>1</math> touching point between the two functions: <math>\left(\frac{1}{5},0\right)</math>. When <math>y < 0</math>
  2 KB (300 words) - 16:01, 26 November 2019
• 1992 AIME Problems/Problem 10
  ...area of the two circles and then subtracting out their overlap. There are two methods of finding the area of overlap: ...of those three graphs is <math>40^2-(200\pi + 400) \Rightarrow 1200 - 200\pi \approx 571.68</math>
  2 KB (323 words) - 12:05, 16 July 2019
• 2006 AMC 10B Problems/Problem 19
  <math> \mathrm{(A) \ } \frac{\pi}{3}+1-\sqrt{3}\qquad \mathrm{(B) \ } \frac{\pi}{2}(2-\sqrt{3}) ...rm{(D) \ } \frac{\pi}{6}+\frac{\sqrt{3}+1}{2}\qquad \mathrm{(E) \ } \frac{\pi}{3}-1+\sqrt{3} </math>
  5 KB (873 words) - 15:39, 29 May 2023
• 1994 AIME Problems/Problem 15
  ...points of <math>\triangle ABC\,</math> can be written in the form <math>q\pi-r\sqrt{s},\,</math> where <math>q, r,\,</math> and <math>s\,</math> are pos ...of the locus of <math>P</math> (shaded region below) is simply the sum of two [[segment]]s of the circles. If we construct the midpoints of <math>M_1, M_
  4 KB (717 words) - 22:20, 3 June 2021
• 1994 AIME Problems/Problem 14
  ...\,</math> determine the number of times the light beam will bounce off the two line segments. Include the first reflection at <math>C\,</math> in your co ...a * 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
• 1994 AIME Problems/Problem 13
  Using DeMoivre, <math>13 - t = \text{cis}\left(\frac {(2k + 1)\pi}{10}\right)</math> where <math>k</math> is an integer between <math>0</math ...pi}{10}\right) \Rightarrow \bar{t} = 13 - \text{cis}\left(-\frac {(2k + 1)\pi}{10}\right)</math>.
  3 KB (375 words) - 23:46, 6 August 2021
• 1995 AIME Problems/Problem 14
  ...d the area of either of them can be expressed uniquely in the form <math>m\pi-n\sqrt{d},</math> where <math>m, n,</math> and <math>d_{}</math> are positi Let the center of the circle be <math>O</math>, and the two chords be <math>\overline{AB}, \overline{CD}</math> and intersecting at <ma
  3 KB (484 words) - 13:11, 14 January 2023
• 1996 AIME Problems/Problem 15
  pair B=(0,0), 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); Dividing the two equalities yields
  5 KB (710 words) - 21:04, 14 September 2020
• 1996 AIME Problems/Problem 11
  ...are not common to both equations/sets, or else we are overcounting a root two times, rather than once. Try out some equation to see where this might appl Now, we see the <math>z^2+z-1</math> and it reminds us of the sum of two cubes. Cleverly factoring, we obtain that..
  6 KB (1,022 words) - 20:23, 17 April 2021
• 1998 AIME Problems/Problem 5
  Given that <math>A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,</math> find <math>|A_{19} + A_{20} + \cdots + A_{98}|.</math> ...uates to an integer ([[triangular number]]), and the [[cosine]] of <math>n\pi</math> where <math>n \in \mathbb{Z}</math> is 1 if <math>n</math> is even a
  1 KB (225 words) - 02:20, 16 September 2017
• 1999 AIME Problems/Problem 11
  Let <math>x=e^{\frac{i\pi}{36}}</math>. By Euler's Formula, <math>\sin{5k^\circ}=\frac{x^k-\frac{1}{x We factor the <math>\frac{1}{2i}</math> and split into two geometric series to get <math>\frac{1}{2i}\left(\frac{-\frac{1}{x^{35}}(x^{
  4 KB (614 words) - 04:38, 8 December 2023
• 1999 AIME Problems/Problem 9
  ...re positive, <math>z</math> lies in the first quadrant and <math>\theta < \pi/2</math>; hence by right triangle trigonometry <math>\sin \theta = \frac{\s ...point of <math>OP</math> is <math>(0.5c, 0.5d)</math>. Since the slopes of two respectively nonvertical and nonhorizontal lines have a product of <math>-1
  6 KB (1,010 words) - 19:01, 24 May 2023
• 2000 AIME I Problems/Problem 8
  ...the volume of the liquid can be found by <math>\frac{\pi}{3}r^2h - \frac{\pi}{3}(r')^2h'</math>. ...rac{\pi}{3}\left(\frac{3}{4}r\right)^2 \left(\frac{3}{4}h\right) &= \frac{\pi}{3}\left(r^2h - \left(\frac{rh'}{h}\right)^2h'\right)\\
  4 KB (677 words) - 16:33, 30 December 2023
• 2003 AIME I Problems/Problem 2
  ...ion bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The [[ratio]] of the total area of the gre ...\ldots - 1^{2} \pi</math>, while the total area is given by <math>100^{2} \pi</math>, so the ratio is
  4 KB (523 words) - 15:49, 8 March 2021
• 2003 AIME II Problems/Problem 14
  <math>f + 4 i = (b + 2 i)\left(e^{i(2 \pi / 3)}\right) = (b + 2 i)\left(-1/2 + \frac{\sqrt{3}}{2} i\right) = -\frac{b The area of the hexagon can then be found as the sum of the areas of two congruent triangles (<math>EFA</math> and <math>BCD</math>, with height <ma
  9 KB (1,461 words) - 15:09, 18 August 2023
• 2003 AIME II Problems/Problem 5
  ...e log. The number of cubic inches in the wedge can be expressed as <math>n\pi</math>, where n is a positive integer. Find <math>n</math>. ...king it to the existing one). Thus, <math>V=\dfrac{6^2\cdot 12\pi}{2}=216\pi</math>, so <math>n=\boxed{216}</math>.
  1 KB (204 words) - 17:41, 30 July 2022
• 2002 AIME II Problems/Problem 10
  ...math> radians are <math>\frac{m\pi}{n-\pi}</math> and <math>\frac{p\pi}{q+\pi}</math>, where <math>m</math>, <math>n</math>, <math>p</math>, and <math>q< ...{\alpha}</math> are <math>\theta = \pi-\alpha</math>, and <math>\theta = 2\pi + \alpha</math>.
  2 KB (336 words) - 19:30, 24 June 2020
• 2006 AMC 10B Problems
  ...tball game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 po ...at any point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?
  14 KB (2,059 words) - 01:17, 30 January 2024
• 2005 AMC 10B Problems
  ...f five for &#36;<math>2</math>. They sell all the candy bars at a price of two for &#36;<math>1</math>. What was the profit, in dollars? ...rac{3\pi}{16} \qquad \mathrm{(D)} \frac{\pi}{4} \qquad \mathrm{(E)} \frac{\pi}{2} </math>
  12 KB (1,874 words) - 21:20, 23 December 2020
• University of South Carolina High School Math Contest/1993 Exam/Problems
  ...athrm{(D) \ } \frac{\sqrt{3}}2 - \frac{\pi}6 \qquad \mathrm{(E) \ } \frac{\pi}6 </cmath> ...\mathrm{(C) \ }\pi/3 \qquad \mathrm{(D) \ }2\pi/3 \qquad \mathrm{(E) \ }3/\pi/4 </cmath>
  14 KB (2,102 words) - 22:03, 26 October 2018
• Mock AIME 1 2006-2007/Problems
  .../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 when <mat ...{2}+\frac{101}{100}</math> and <math>P_{2}: x=y^{2}+\frac{45}{4}</math> be two parabolas in the cartesian plane. Let <math>\mathcal{L}</math> be the commo
  8 KB (1,355 words) - 14:54, 21 August 2020
• Mock AIME 2 2006-2007 Problems/Problem 7
  ...ath> on the arc that are the farthest away from each other. Since <math>34\pi</math> is <math>1/3</math> of the circumference of a circle with radius <ma
  1 KB (231 words) - 18:10, 10 July 2014
• 1959 IMO Problems/Problem 4
  ...h that the median drawn to the hypotenuse is the [[geometric mean]] of the two legs of the triangle. ...the triangle must be <math> \frac{ \pi }{12} </math> and <math> \frac{ 5 \pi }{12}</math>, which are not difficult to construct. Q.E.D.
  6 KB (939 words) - 17:31, 15 July 2023
• 2006 Romanian NMO Problems
  ...ath> and <math>b</math>. Prove that the set <math>A</math> has ''exactly'' two elements. ...ngle with the base <math>BC</math>. We know that <math>\angle ABD = \frac{\pi}{2}</math>. Let <math>M</math> be the midpoint of <math>BC</math>. The poin
  11 KB (1,779 words) - 14:57, 7 May 2012
• Tangent (geometry)
  ...math> intersects it in 2 points and the tangent line at <math>\left(\frac{\pi}2, 1\right)</math> intersects it in [[infinite]]ly many points (and is in f ...y. Any two disjoint circles have four tangents in common, two internal and two external.
  2 KB (332 words) - 21:54, 11 March 2024
• Right angle
  ...of every right angle is 90 [[degree (geometry) | degrees]] or <math>\frac \pi 2</math> [[radian]]s. When drawing diagrams, we denote right angles with a
  673 bytes (91 words) - 23:59, 11 June 2022
• Integral
  The '''integral''' is one of the two base concepts of [[calculus]], along with the [[derivative]]. In introductory, high-school level texts, the integral is often presented in two parts, the '''indefinite integral''' and '''definite integral'''. Although
  5 KB (909 words) - 14:16, 31 May 2022
• Angle
  An '''angle''' is the [[union]] of two [[ray]]s with a common [[endpoint]]. The common endpoint of the rays is ca If two angles are [[congruent (geometry)|congruent]], they have the same angle mea
  3 KB (568 words) - 20:21, 21 June 2019
• 2007 iTest Problems
  ...>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? ...ath>4</math> years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the d
  33 KB (5,177 words) - 21:05, 4 February 2023
• Straight angle
  ...e of every straight angle is 180 [[degree (geometry) | degrees]] or <math>\pi</math> [[radian]]s.
  568 bytes (78 words) - 13:50, 12 June 2022
• 2005 AMC 10A Problems
  Three tiles are marked <math>X</math> and two other tiles are marked <math>O</math>. The five tiles are randomly arranged There are two values of <math>a</math> for which the equation <math> 4x^2 + ax + 8x + 9 =
  14 KB (2,026 words) - 11:45, 12 July 2021
• 2003 AMC 10A Problems
  The sum of the two 5-digit numbers <math>AMC10</math> and <math>AMC12</math> is <math>123422</ ...e numbers is <math>20</math>. The first is four times the sum of the other two. The second is seven times the third. What is the product of all three?
  13 KB (1,900 words) - 22:27, 6 January 2021
• 1997 AJHSME Problems/Problem 24
  ...CE</math> is divided at <math>C</math> in the ratio <math>2:3</math>. The two semicircles, <math>ABC</math> and <math>CDE</math>, divide the circular reg ...{2}\pi\cdot3^2 - \frac{1}{2}\pi\cdot2^2 =\frac{\pi}{2}(5^2 + 3^2 - 2^2)=15\pi</math>
  2 KB (394 words) - 17:05, 20 October 2023
• 1985 IMO Problems/Problem 1
  ...es of arcs, <math> \angle DTA = \angle DCO = \frac{\angle DCB}{2} = \frac{\pi}{2} - \frac{\angle DAB}{2} </math>. It follows that <math>AT = AD </math>. ...</math> such that <math>AT = AD </math>. Then <math>\angle DTA = \frac{ \pi - \angle DAB}{2} = \angle DCO</math>, so <math>DCOT </math> is a cyclic qua
  4 KB (684 words) - 07:28, 3 October 2021
• 2004 AMC 10A Problems
  ...hat is the probability that the number of heads obtained from flipping the two fair coins is the same? ...tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, or three meat patties, and any collection of condiments. How many differe
  15 KB (2,092 words) - 20:32, 15 April 2024
• 1985 AIME Problems
  ...rotated about the other leg, the volume of the cone produced is <math>1920\pi \;\textrm{cm}^3</math>. What is the length (in cm) of the hypotenuse of the ...alpha + \beta </math> radians, respectively, where <math>\alpha + \beta < \pi</math>. If <math>\cos \alpha</math>, which is a positive rational number, i
  7 KB (1,071 words) - 19:24, 23 February 2024
• 1985 AIME Problems/Problem 2
  ...rotated about the other leg, the volume of the cone produced is <math>1920\pi \;\textrm{ cm}^3</math>. What is the length (in cm) of the [[hypotenuse]] o ...}b</math>. Then <math>\frac{1}{3} \pi \left(\frac{12}{5}b\right)b^2 = 800\pi</math> so <math>b^3 = 1000</math> and <math>b = 10</math> so <math>a = 24</
  3 KB (463 words) - 15:10, 4 September 2020
• 2003 AMC 12A Problems/Problem 15
  draw(2*expi(2*pi/6)--2*expi(4*pi/6)); ...}{4}+\frac{1}{24}\pi\qquad \mathrm{(E) \ } \frac{\sqrt{3}}{4}+\frac{1}{12}\pi </math>
  3 KB (380 words) - 21:53, 19 March 2022
• Sector
  A '''sector''' of a [[circle]] is a region bounded by two [[radius|radii]] of the circle and an [[arc]]. If the [[central angle]] of the sector is <math>\pi</math> (or <math>180^{\circ}</math>), then the sector is a [[semicircle]].
  1 KB (178 words) - 21:12, 24 April 2008
• Georgia Tech High School Math Competition
  ...annual [[math tournament]] hosted and run by the School of Mathematics and Pi Mu Epsilon of the Georgia Institute of Technology (better known as Georgia After lunch, there is a two-hour, six-question proof test. These papers are graded by School of Mathema
  3 KB (475 words) - 21:51, 31 December 2013
• Asymptote: Advanced Configuration
  There are two things that Asymptote users commonly would like to do with their figures: m \(\int_0^{\pi}{\sin{x}}\,dx=2\)
  12 KB (1,931 words) - 13:53, 26 January 2020
• Asymptote: Reference
  ...'' is associated with a cartesian coordinate pair in Asymptote. There are two useful functions that allow one to use polar coordinates as well: ...joined smoothly in an curve that can be twisted. Keep in mind that if only two things are connected with <tt>..</tt>, the curve will be a straight line.
  7 KB (1,205 words) - 21:38, 26 March 2024
• Asymptote (geometry)
  .../math>; thus the functions is undefined at <math>x=\frac{\pi}{6} + \frac{n\pi}{3}</math>. ...ymptote is at the quotient of the leading coefficient of <math>P(x)</math> over the leading coefficient of <math>Q(x)</math>.
  4 KB (664 words) - 11:44, 8 May 2020
• Mock AIME 3 Pre 2005 Problems
  ...r centers is <math>84</math>, then the area of the third circle is <math>k\pi</math> for some integer <math>k</math>. Determine <math>k</math>. ...lish if and only if between any two <math>O's</math> there appear at least two consonants. Let <math>N</math> denote the number of <math>10</math>-letter
  7 KB (1,135 words) - 23:53, 24 March 2019

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