Create the page "Pi over two" on this wiki! See also the search results found.

• 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
  4 KB (473 words) - 16:14, 1 May 2024
• 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>
  5 KB (849 words) - 16:14, 18 May 2021
• 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.
  5 KB (892 words) - 21:52, 1 May 2021
• 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
• Mock AIME 3 Pre 2005 Problems/Problem 1
  ...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>. ...h>, so <math>r=14</math>. Thus the area of the circle is <math>\boxed{196}\pi</math>.
  795 bytes (129 words) - 10:22, 4 April 2012
• Arc
  ...the circle. More generally, an arc is a portion of a smooth curve joining two points. ..., in particular, the [[circumference]] of a circle is given by <math>C = 2\pi</math>.
  1 KB (170 words) - 00:24, 16 December 2023
• 2007 AIME II Problems/Problem 11
  ...t starts. The distance <math>x</math> can be expressed in the form <math>a\pi+b\sqrt{c},</math> where <math>a,</math> <math>b,</math> and <math>c</math> If it weren’t for the small tube, the larger tube would travel <math>144\pi</math>. Consider the distance from which the larger tube first contacts the
  2 KB (407 words) - 16:31, 29 February 2020
• 2007 AIME II Problems
  ...ers can produce <math>300</math> widgets and <math>200</math> whoosits. In two hours, <math>60</math> workers can produce <math>240</math> widgets and <ma ...t starts. The distance <math>x</math> can be expressed in the form <math>a\pi+b\sqrt{c},</math> where <math>a,</math> <math>b,</math> and <math>c</math>
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2007 AMC 12A Problems
  The larger of two consecutive odd integers is three times the smaller. What is their sum? <math>\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18</math>
  11 KB (1,750 words) - 13:35, 15 April 2022
• 2007 USAMO Problems/Problem 6
  ...i/2)--intersectionpoint(incircle(A,B,C),I--I+O-A)-abs(A-O)*expi(angle(A-O)-pi/2), q_a--q_a+O-A); ...i/2)--intersectionpoint(incircle(A,B,C),I--I+A-O)-abs(A-O)*expi(angle(A-O)-pi/2), O--A);
  7 KB (1,274 words) - 15:11, 31 August 2017
• Cylinder
  ...] from a line [[perpendicular]] to both planes. Thus, a '''cylinder''' has two circular flat surfaces--think of a can of soup. Typically, to solve for sur * Surface [[area]]: <math>2 \pi r^2 + 2 \pi r h</math>
  827 bytes (135 words) - 18:13, 27 July 2023
• 2007 Cyprus MO/Lyceum/Problem 21
  ...ent pulleys, that are connected with a strap. If the distances between any two pulley center points are <math>AB=3\,\mathrm{m}</math>, <math>AC=4\,\mathrm ...\qquad \mathrm{(B) \ } (12 + \pi)\,\mathrm{m}\qquad \mathrm{(C) \ } (12+ 4\pi)\,\mathrm{m}\qquad \mathrm{(D) \ } \left(12+\frac\pi5\right)\,\mathrm{m}\qq
  1 KB (175 words) - 22:21, 23 July 2020
• Pascal's Theorem
  ...l prove the theorem for a cyclic hexagon, using directed angles mod <math>\pi </math>. '''Lemma.''' Let <math>\omega_1, \omega_2 </math> be two circles which intersect at <math>M, N </math>, let <math>AB </math> be a ch
  4 KB (712 words) - 21:57, 25 April 2020
• 2007 AMC 12A Problems/Problem 24
  ...the [[equation]] <math>\sin{x}=\sin{(nx)}</math> on the interval <math>[0,\pi]</math>. What is <math>\sum_{n=2}^{2007} F(n)</math>? Notice that the solutions are basically reflections across <math>x = \frac{\pi}{2}</math>.
  4 KB (588 words) - 14:40, 23 August 2023
• 2006 AIME II Problems/Problem 12
  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; Because the ratio of the area of two similar figures is the square of the ratio of the corresponding sides, <mat
  6 KB (1,033 words) - 02:36, 19 March 2022
• 2005 PMWC Problems
  touches its two neighbors and the larger circle. What is the radius of the ...in the list <math>1, 2, 3, 4, 5, \dots, 10000</math> which contain exactly two consecutive <math>9</math>'s such as <math>993, 1992</math> and <math>9929<
  9 KB (1,449 words) - 20:49, 2 October 2020
• 1998 PMWC Problems
  ...>\angle BOA</math> is a right angle. <math>A</math> and <math>B</math> are two points on the circumference of circle of radius <math>OA</math>. Find the a .../3),6sin(4*pi/3))--(0,0))--(-3,-3)--intersectionpoint((8cos(5*pi/6),8sin(5*pi/6))--(0,0),(-3,3)--(-3,-3))--cycle,gray);
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2000 PMWC Problems
  <math>p</math> is the product of two <math>4</math>-digit numbers formed by the digits <math>1,2,3,4,5,6,7,8</ma <math>a</math> and <math>b</math> are two numbers that have prime factors <math>3</math> and <math>5</math> only. <ma
  11 KB (1,713 words) - 22:47, 13 July 2023
• 2004 AMC 10A Problems/Problem 21
  ...e radian measure of the acute angle formed by the two lines? (Note: <math>\pi</math> radians is <math>180</math> degrees.) ...\pi}{6} \qquad \mathrm{(D) \ } \frac{\pi}{5} \qquad \mathrm{(E) \ } \frac{\pi}{4} </math>
  3 KB (476 words) - 03:50, 23 January 2023
• 2007 AMC 12B Problems
  ...t response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave the la ...lengths <math>AB = 5</math>, <math>BC = 6</math>, and <math>AC = 7</math>. Two bugs start simultaneously from <math>A</math> and crawl along the sides of
  12 KB (1,814 words) - 12:58, 19 February 2020
• 1960 IMO Problems/Problem 6
  The area of <math>T</math> may be computed in two different ways: <cmath>V_1 = \frac{1}{3}\pi R^2 h</cmath>
  7 KB (1,214 words) - 18:49, 29 January 2018
• Root (operation)
  ...t[n]{x}=\sqrt[n]{|x|}\left(\cos\frac{\theta+2\pi k}{n}+i\sin\frac{\theta+2\pi k}{n}\right)</math> , where <math>k=0,1,2,...,n-1</math> and <math>x\in\mat ...h>\sqrt[4]{16}=\sqrt[4]{16}\left(\cos\frac{0+2\pi\cdot0}{4}+i\sin\frac{0+2\pi\cdot0}{4}\right)\implies\boxed{2}</math>
  3 KB (532 words) - 16:52, 20 May 2020
• 2007 AMC 10A Problems
  The larger of two consecutive odd integers is three times the smaller. What is their sum? ...005, <math>78</math> in 2006, and is <math>85</math> in 2007. Between what two consecutive years was there the largest percentage increase?
  13 KB (2,058 words) - 17:54, 29 March 2024
• 1995 AHSME Problems
  ...oint, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point ...n attendance. A league official who knows the actual numbers attending the two games notes that:
  17 KB (2,387 words) - 22:44, 26 May 2021
• 1959 AHSME Problems
  ...is drawn parallel to the base <math>AB</math>, dividing the triangle into two equal areas. ...mean (average) of a set of <math>50</math> numbers is <math>38</math>. If two numbers, namely, <math>45</math> and <math>55</math>, are discarded, the me
  22 KB (3,345 words) - 20:12, 15 February 2023
• 2002 AMC 12B Problems/Problem 25
  ...of the area of the circle, which is <math>\frac{1}{2} (\pi \cdot 4^2) = 8\pi \approx \boxed{\textbf{(E) }25}</math>. ...e area of the circle satisfying <math>f(x)+f(y) \le 0</math>, or <math>16 \pi</math>.
  2 KB (365 words) - 14:48, 7 March 2022
• 2007 AMC 10B Problems/Problem 11
  A circle passes through the three vertices of an isosceles triangle that has two sides of length <math>3</math> and a base of length <math>2</math>. What is ...\frac{81}{32}\pi \qquad\textbf{(D) } 3\pi \qquad\textbf{(E) } \frac{7}{2}\pi</math>
  5 KB (851 words) - 22:02, 26 July 2021
• 1994 USAMO Problems
  Let <math>\, k_1 < k_2 < k_3 < \cdots \,</math> be positive integers, no two consecutive, and let <math>\, s_m = k_1 + k_2 + \cdots + k_m \,</math> for ...f the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifica
  2 KB (391 words) - 07:58, 19 July 2016
• 2008 AMC 10A Problems
  Heather compares the price of a new computer at two different stores. Store <math>A</math> offers <math>15\%</math> off the sti ...}\ \left(2\sqrt{3}+3\right)^2\pi\\\mathrm{(E)}\ 9\left(\sqrt{3}+1\right)^2\pi</math>
  14 KB (2,138 words) - 15:08, 18 February 2023
• 1976 IMO Problems/Problem 2
  ...tions, where 1 is positive and 1 is negative. The absolute values of these two solutions are also both less than 2. This proves the base case. ...l}{2^n+1}\cdot\pi</math>. As we can choose the range <math>0\leq\theta\leq\pi</math> to ensure no duplications, we get that, upon rearranging, <math>0\le
  3 KB (596 words) - 16:19, 28 July 2015
• 2008 AMC 12A Problems/Problem 25
  We can make two observations from this: ...<math>z_n = 2e^{i\pi/6}z_{n-1}</math>, so <math>z_{n-1} = \frac{1}{2}e^{-i\pi/6}z_n</math>. This is the reverse transformation. We have
  5 KB (745 words) - 10:58, 9 December 2022
• 2008 AMC 10B Problems
  ...diagonal are to be added. What will be the positive difference between the two diagonal sums? ...- 2ax + b = 0</math> has two real solutions. What is the average of these two solutions?
  12 KB (1,838 words) - 16:52, 7 October 2022
• 2008 AMC 12B Problems
  ...diagonal are to be added. What will be the positive difference between the two diagonal sums? ...is steps. The pedometer records up to <math>99999</math> steps, then flips over to <math>00000</math> on the next step. Pete plans to determine his mileage
  14 KB (2,199 words) - 13:43, 28 August 2020
• 2008 AIME I Problems
  ...ngle entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entr ...\arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.</cmath>
  9 KB (1,536 words) - 00:46, 26 August 2023
• 2008 AIME II Problems
  ...Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the Let <math>a = \pi/2008</math>. Find the smallest positive integer <math>n</math> such that
  7 KB (1,167 words) - 21:33, 12 August 2020
• 2008 AIME I Problems/Problem 8
  ...\arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.</cmath> Applying this to the first two terms, we get <math>\arctan{\dfrac{1}{3}} + \arctan{\dfrac{1}{4}} = \arctan
  3 KB (490 words) - 22:36, 28 November 2023
• 2008 AIME I Problems/Problem 10
  ...whose angle at the longer base <math>\overline{AD}</math> is <math>\dfrac{\pi}{3}</math>. The [[diagonal]]s have length <math>10\sqrt {21}</math>, and po pair A=(0,0), D=(4,0), B= A+2 expi(1/3*pi), C= D+2expi(2/3*pi), E=(-4/3,0), F=(3,0);
  4 KB (629 words) - 22:38, 28 November 2023
• 2008 AIME II Problems/Problem 5
  pair A=(0,0), D=(2*r, 0), N=(r,0), E=N+r*expi(74*pi/180); pair A=(0,0), D=(2*r, 0), N=(r,0), E=N+r*expi(74*pi/180);
  8 KB (1,338 words) - 23:15, 28 November 2023
• 2008 AIME II Problems/Problem 9
  ...efine a ''move'' for the particle as a counterclockwise rotation of <math>\pi/4</math> radians about the origin followed by a translation of <math>10</ma ...he position of the particle after a move from <math>P</math>, then we have two equations for <math>x'</math> and <math>y'</math>:
  5 KB (725 words) - 22:37, 28 January 2024
• 1974 USAMO Problems/Problem 5
  Consider the two triangles <math>\triangle ABC</math> and <math>\triangle PQR</math> shown i pair P=(7,0), Q=(14,0), R=P+7expi(pi/3), M=(10,1.2);
  7 KB (1,221 words) - 18:57, 3 July 2013
• 2005 USAMO Problems/Problem 3
  ...be an acute-angled triangle, and let <math>P</math> and <math>Q</math> be two points on side <math>BC</math>. Construct point <math>C_1 </math> in such a ...th>Q' =Q</math>. All our angles will be directed, and measured mod <math>\pi</math>.
  6 KB (1,117 words) - 01:17, 11 October 2021
• Cone
  <center><!-- NOTE: two different versions --><asy> size(120); ...ath> and height <math>h</math> has [[volume]] <math>V = \frac{1}{3} \cdot \pi r^2 \cdot h</math>. This is a special case of the general formula for the
  7 KB (1,128 words) - 20:12, 27 September 2022
• 2008 USAMO Problems/Problem 4
  This diagonal [[partition]]s <math>\mathcal{P}</math> into two regions <math>\mathcal{Q},\, \mathcal{R}</math>, and is the side of an isos int n = 17; real r = 1; real rad = pi/2;
  8 KB (1,318 words) - 12:37, 20 April 2022
• 2003 USAMO Problems/Problem 6
  ...by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after w label("$c$",expi(pi/3),(0,0),red);
  5 KB (739 words) - 13:39, 4 July 2013
• Millennium Problems
  ...nstitute (which also had organized the meeting), had selected the problems over the previous several months. They were led by Arthur Jaffe, the first direc ...thoroughly examine the solution. This committee would consist of at least two world-renowned mathematicians and at least one member of the SAB.
  13 KB (1,969 words) - 17:57, 22 February 2024
• 2008 iTest Problems
  ...{120}</math>. How many dollars did the family save by buying a family pass over buying single day passes for every member of the family? The difference between two prime numbers is <math>11</math>. Find their sum.
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2005 PMWC Problems/Problem T7
  ...? Calculate an approximate answer by using <math>\pi=3.14</math> or <math>\pi=22/7</math>. ...</math>. Adding, we get <math>\frac{8}{3}\pi+\frac{\pi}{6}+\frac{\pi}{6}=3\pi</math>. Then approximate.
  854 bytes (145 words) - 23:28, 21 October 2022
• 2005 AMC 8 Problems
  ...t Closet, the price of a coat is discounted 20% from its &#36;90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack brings up &#3 The Little Twelve Basketball League has two divisions, with six teams in each division. Each team plays each of the oth
  13 KB (1,821 words) - 22:18, 5 December 2023
• Van Aubel's Theorem
  ..._{AB}, P_{BC}, P_{CD}, P_{DA}</math>. Van Aubel's Theorem states that the two line segments connecting opposite centers are perpendicular and equal lengt ...by the complex number <math>x</math>. Note that <math>\angle PAB = \frac{\pi}{4}</math> and that <math>PA = \frac{\sqrt{2}}{2}AB</math>, and similarly f
  2 KB (410 words) - 14:01, 4 March 2023
• 2002 AMC 10A Problems
  ...\pi \qquad \text{(C)}\ 2\pi \qquad \text{(D)}\ 3\pi \qquad \text{(E)}\ 3.5\pi</math> Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 p
  11 KB (1,733 words) - 11:04, 12 October 2021
• 2002 AMC 10B Problems
  ...d \mathrm{(C) \ } 6\pi\qquad \mathrm{(D) \ } 9\pi\qquad \mathrm{(E) \ } 12\pi </math> ...ne}\qquad \mathrm{(C) \ } \text{two}\qquad \mathrm{(D) \ } \text{more than two, but finitely many}\qquad \mathrm{(E) \ } \text{infinitely many} </math>
  10 KB (1,540 words) - 22:53, 19 December 2023
• 2002 AMC 10A Problems/Problem 19
  ...ase that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the ...i \qquad \text{(C)}\ 5\pi/2 \qquad \text{(D)}\ 8\pi/3 \qquad \text{(E)}\ 3\pi</math>
  1 KB (204 words) - 20:45, 28 May 2023
• 2004 AMC 10B Problems
  How many two-digit positive integers have at least one <math>7</math> as a digit? ...75+100\pi \qquad \mathrm{(D) \ } 100+100\pi \qquad \mathrm{(E) \ } 100+125\pi </math>
  13 KB (1,988 words) - 23:06, 7 March 2024
• 1974 USAMO Problems
  Two boundary points of a ball of radius 1 are joined by a curve contained in th A father, mother and son hold a family tournament, playing a two person board game with no ties. The tournament rules are:
  3 KB (427 words) - 18:55, 3 July 2013
• 1962 IMO Problems/Problem 6
  ...ircle. Prove that the distance <math>d</math> between the centers of these two circles is ...cdot IL = PI\cdot QI</math> by Power of a Point. Therefore, <math>2r\rho = PI \cdot QI = (PO + OI)(QO - OI) = (r + d)(r - d)</math>, and we have <math>2r
  2 KB (308 words) - 06:29, 16 December 2023
• 1963 IMO Problems/Problem 3
  ...ath>-axis and the entire shape is above the <math>x</math>-axis. There are two cases: the number of sides is even, and the number of sides is odd: ...h>y\textrm{-coordinate} = \sum_{k = 1}^{\frac{n}{2}}a_k \cdot \sin \frac{2\pi(k-1)}{n}.\textbf{ (1)}</cmath>
  4 KB (836 words) - 17:58, 7 December 2022
• 1963 IMO Problems
  Prove that <math>\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}</math>. ...tter. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did s
  2 KB (342 words) - 21:16, 20 August 2020
• 2021 GMC 10B
  The radius of a circle that has an area of <math>\frac{\pi}{\sqrt{2}}</math> is <math>r</math>. Find <math>r^{2}</math> A three digit natural number is <math>Alternative</math> if it has two even digits and one odd digit as its number digits. Find the number of alte
  11 KB (1,695 words) - 14:33, 7 March 2022
• 1986 AJHSME Problems/Problem 23
  The large circle has diameter <math>\text{AC}</math>. The two small circles have their centers on <math>\text{AC}</math> and just touch a The small circle has radius <math>1</math>, thus its area is <math>\pi</math>.
  2 KB (277 words) - 21:32, 3 July 2013
• 2004 AMC 10B Problems/Problem 25
  ...in the picture, that is outside the smaller circle and inside each of the two larger circles? \mathrm{(A) \ } \frac{5}{3} \pi - 3\sqrt 2
  4 KB (703 words) - 22:37, 2 November 2022
• 2004 AMC 12B Problems/Problem 10
  An annulus is the region between two concentric circles. The concentric circles in the figure have radii <math>b ...hrm{(C) \ } \pi c^2 \qquad \mathrm{(D) \ } \pi d^2 \qquad \mathrm{(E) \ } \pi e^2 </math>
  2 KB (340 words) - 14:35, 23 April 2023
• 2009 AMC 12A Problems
  ...>(6m,0)</math>, and the line <math>y = mx</math> divides the triangle into two triangles of equal area. What is the sum of all possible values of <math>m< ...^3 + \cdots</math> and <math>a + ar_2 + ar_2^2 + ar_2^3 + \cdots</math> be two different infinite geometric series of positive numbers with the same first
  13 KB (2,105 words) - 13:13, 12 August 2020
• 2009 AMC 10A Problems
  \mathrm{(B)}\ \frac{\pi}{6} \mathrm{(C)}\ \frac{2}{\pi}
  14 KB (2,130 words) - 11:32, 7 November 2021
• 2009 AMC 12A Problems/Problem 25
  The first two terms of a sequence are <math>a_1 = 1</math> and <math>a_2 = \frac {1}{\sqr As the number of possible consecutive two terms is finite, we know that the sequence <math>b_n</math> is periodic. Wr
  7 KB (990 words) - 07:23, 24 October 2022
• 2009 AMC 12A Problems/Problem 21
  <center><cmath>p(2009 + 9002\pi i) = p(2009) = p(9002) = 0</cmath></center> From the three zeroes, we have <math>p(x) = (x - (2009 + 9002\pi i))(x - 2009)(x - 9002)</math>.
  2 KB (322 words) - 10:25, 29 July 2020
• 2009 AMC 12A Problems/Problem 20
  pair D=MP("D",(0,0)),C=MP("C",(12,0)),A=MP("A",C+14*expi(145*pi/180),N),B=MP("B",A+(9,0),N),E=IP(A--C,B--D);MP("9",(A+B)/2,N);MP("12",(C+D) ...\triangle AEB \sim \triangle EDC</math> (which should be trivial given the two equal triangles) we have that
  3 KB (543 words) - 21:09, 23 October 2023
• 2009 AMC 12A Problems/Problem 19
  ...Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were <math>A</math> and <math>B</math>, respectively. Each polygon ...isosceles triangle formed by the center of the polygon <math>S</math> and two consecutive vertices <math>X</math> and <math>Y</math>. We are given that <
  4 KB (630 words) - 21:27, 30 December 2023
• 2009 AMC 10A Problems/Problem 21
  ...e area of the large circle is <math>L = \pi R^2 = \pi r^2 (1+\sqrt 2)^2 = \pi r^2 (3+2\sqrt 2)</math>. The area of four small circles is <math>S = 4\pi r^2</math>. Hence their ratio is:
  3 KB (474 words) - 12:50, 29 September 2023
• 2009 AMC 10A Problems/Problem 19
  ...h> as it rolls once around the circumference of circle <math>A</math>. The two circles have the same points of tangency at the beginning and end of circle ...cumference of circle <math>B</math> with radius <math>r</math> is <math>2r\pi</math>. Since circle <math>B</math> makes a complete revolution and ''ends
  2 KB (276 words) - 09:57, 8 June 2021
• 2001 AMC 12 Problems/Problem 8
  ...^\circ</math> sector of a circle of radius <math>10</math> by aligning the two straight sides? The length of the red line is <math>\dfrac{252}{360}\cdot 2\pi \cdot 10 = 14\pi</math>. This is the circumference of a circle with radius <math>7</math>.
  2 KB (279 words) - 00:32, 30 December 2023
• 2009 AMC 12B Problems
  A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remain Kiana has two older twin brothers. The product of their ages is <math>128</math>. What is
  13 KB (2,030 words) - 03:04, 5 September 2021
• 2009 AMC 12B Problems/Problem 23
  ...34 + \frac34i</math> into polar form as <math>\frac{3\sqrt{2}}{4}e^{\frac{\pi}{4}i}</math>. Restated using geometric probabilities, we are trying to find ...ath>2</math> by <math>2</math> square centered at the origin. Graphing our two equations gives us the four lines <cmath>x-y=\frac{4}{3},</cmath> <cmath>x-
  2 KB (422 words) - 13:25, 20 January 2020
• 2009 AMC 12B Problems/Problem 24
  For how many values of <math>x</math> in <math>[0,\pi]</math> is <math>\sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x)</math>? ...in^{-1}(x) \leq \pi/2</math> and <math>\forall x: 0\leq \cos^{-1}(x) \leq \pi</math>.
  7 KB (1,287 words) - 07:09, 22 December 2022
• 2009 AIME I Problems/Problem 12
  ...>O</math> be center of the circle and <math>P</math>,<math>Q</math> be the two points of tangent such that <math>P</math> is on <math>BI</math> and <math> Since the ratios between corresponding lengths of two similar diagrams are equal, we can let <math>AD = 144, CD = 420</math> and
  12 KB (1,970 words) - 22:53, 22 January 2024
• 1966 AHSME Problems/Problem 4
  ...the radius of the smaller circle is <math>x</math>, so it's area is <math>\pi x^2</math>. ...be <math>2x^2 \pi</math>. Putting one over the other and dividing, you get two as the answer: or <math>\boxed{(B)}</math>.
  1 KB (191 words) - 22:09, 14 January 2018
• Liouville's Theorem (complex analysis)
  <cmath> \lvert f'(z_0) \rvert = \biggl\lvert \frac{1}{2\pi i} point <math>z \in \mathbb{C}</math>. Now for any two complex numbers <math>A</math>
  2 KB (412 words) - 20:30, 16 January 2024
• 2000 AMC 8 Problems
  ...e numbers lie in the interval between <math>\frac{5}{3}</math> and <math>2\pi?</math> Consider these two geoboard quadrilaterals. Which of the following statements is true?
  15 KB (2,165 words) - 03:32, 13 April 2024
• 2009 AIME II Problems/Problem 13
  ...ath> instead of <math>B</math>. <math>\angle AOC_1</math> = <math>\frac {\pi}{7}</math>. Using the [[Law of Cosines]], <math>\overline {AC_1}^2</math> = <math>8 - 8 \cos \frac {\pi}{7}</math>,
  8 KB (1,266 words) - 20:27, 10 December 2023
• 1965 IMO Problems
  Determine all values <math>x</math> in the interval <math>0\leq x\leq 2\pi </math> which satisfy the inequality .../math> is equal to <math>k</math>. Compute the ratio of the volumes of the two solids obtained.
  3 KB (497 words) - 12:39, 29 January 2021
• Fundamental group
  ...example, the fundamental group of a figure eight is the [[free group]] on two [[generator]]s, which is not abelian. However, the fundamental group of a c We say that two binary operations <math>\circ, \cdot</math> on a
  8 KB (1,518 words) - 20:11, 23 January 2017
• 2003 AMC 12B Problems/Problem 25
  ...ave to place the third point is a <math>60</math> degrees arc(if the first two are <math>60</math> degrees apart), with a <math>\frac{1}{6}</math> probabi ...ave to place the third point is a <math>120</math> degree arc(if the first two are the same point), with a <math>\frac{1}{3}</math> probability.
  4 KB (635 words) - 11:46, 1 September 2022
• 2010 AMC 12A Problems
  ...}\ \frac{0.4}{\pi} \qquad \textbf{(C)}\ 0.4 \qquad \textbf{(D)}\ \frac{4}{\pi} \qquad \textbf{(E)}\ 4</math> In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and
  12 KB (1,817 words) - 15:00, 12 August 2020
• 2010 AMC 12A Problems/Problem 13
  The image below shows the two curves for <math>k=4</math>. The blue curve is <math>x^2+y^2=k^2</math>, wh Clearly the only such integer is <math>k=1</math>, hence the two curves are only disjoint for <math>k=1</math> and <math>k=-1</math>.
  9 KB (1,622 words) - 20:53, 11 September 2023
• 2010 AMC 12A Problems/Problem 17
  ...math>\triangle ABC</math>, we get that <math>AC^2 = r^2+1^2-2r\cos{\frac{2\pi}{3}} = r^2+r+1</math>. Therefore, the area of <math>\triangle ACE</math> is Note: To verify that the quadratic <math>r^2-6r+1</math> has two positive roots, we can either solve for the roots directly or note that dis
  4 KB (690 words) - 10:13, 14 October 2022
• 2010 AMC 12A Problems/Problem 24
  ...\left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)</math>. The intersection of the domain of <math>f(x)</math> with ...align*}\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x) &= 0\\
  9 KB (1,434 words) - 17:54, 17 August 2022
• 2010 AMC 10A Problems
  The area of a circle whose circumference is <math>24\pi</math> is <math>k\pi</math>. What is the value of <math>k</math>? ...}\ \frac{0.4}{\pi} \qquad \textbf{(C)}\ 0.4 \qquad \textbf{(D)}\ \frac{4}{\pi} \qquad \textbf{(E)}\ 4</math>
  13 KB (1,902 words) - 11:20, 5 March 2023
• 2010 AMC 10B Problems
  Markala attended two meetings during her <math>9</math>-hour work day. The first meeting took <m \textbf{(A)}\ \dfrac{\pi}{3}-1
  12 KB (1,817 words) - 22:44, 22 December 2020
• Proofs without words
  ...4)+shiftR--(n-1,-n+1)+r*expi(pi/4)+shiftR^^r*expi(5*pi/4)+shiftR--r*expi(5*pi/4)+(n-1,-n+1)+shiftR,linetype("4 4")); fill((n-1.5,-1.5) -- (n-1.5,-1.5)+r*expi(5.2*pi/6) -- (n-1.5,-1.5)+r*expi(3.3*pi/6) -- cycle); /* manual arrowhead? avoid resizing */
  55 KB (7,986 words) - 17:04, 20 December 2018
• 2010 AIME I Problems
  ...ll the positive divisors of <math>2010^2</math>. She then randomly selects two distinct divisors from this list. Let <math>p</math> be the probability tha Jackie and Phil have two fair coins and a third coin that comes up heads with probability <math>\fra
  8 KB (1,243 words) - 21:58, 10 August 2020
• 2010 AIME I Problems/Problem 11
  ...x = 15</math>, the [[volume]] of the resulting [[solid]] is <math>\frac {m\pi}{n\sqrt {p}}</math>, where <math>m</math>, <math>n</math>, and <math>p</mat ...about the line <math>y = x/3+5</math>, the resulting solid is the union of two right [[cone]]s that share the same base and axis.
  4 KB (636 words) - 16:46, 25 November 2023
• 2010 AIME I Problems/Problem 13
  ...ively. Line <math>\ell</math> divides region <math>\mathcal{R}</math> into two regions with areas in the ratio <math>1: 2</math>. Suppose that <math>AU = <math>Y = \frac {1}{3}\pi(3)^2 = 3\pi</math>
  10 KB (1,418 words) - 23:05, 20 October 2021
• Mock AIME 2 2010 Answer Key
  ...hooses one of the other four to shoot. The probability that there are some two people shooting each other can be expressed in the form <math>\frac{a}{b}</ ...r of arrangements, <math>4^5</math>. For any pair to shoot each other, the two shoot each other in exactly one way, while the other three people shoot any
  36 KB (6,214 words) - 20:22, 13 July 2023
• LaTeX:LaTeX on AoPS
  LaTeX uses a special "math mode" to display mathematics. There are two types of this "math mode": Besides displaying in-line vs. displaying centered and on a new line, the two modes render differently in other ways. Note that <code>$\sum_{k=1}^n k^2$<
  8 KB (1,356 words) - 22:35, 26 June 2020
• 2010 USAMO Problems/Problem 3
  multiplying any two and dividing by the third, we get: &= \frac{\Gamma^4(\frac{1}{4})}{8\pi^2}
  11 KB (1,889 words) - 13:45, 4 July 2013
• 2010 USAMO Problems/Problem 4
  We know that angle <math>BIC = 135^{\circ}</math>, as the other two angles in triangle <math>BIC</math> add to <math>45^{\circ}</math>. Assume markangle(Label("$\scriptstyle{\frac{\pi}{4} - \frac{\theta}{2}}$"), radius=40, I, C, B);
  7 KB (1,230 words) - 19:47, 31 January 2024
• 2007 AMC 8 Problems/Problem 25
  ...rt will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values of the regio ...> or <math>9\pi</math>. Note that the area of the whole circle is <math>36\pi</math>.
  3 KB (516 words) - 13:40, 23 January 2024
• 2010 AMC 12B Problems/Problem 18
  ...and either circle's center (symmetry, you chose!). The intersection of the two circles should form a geometrical lens shape. By sectors, <cmath>f(\theta)=\frac{\theta}{2\pi}\cdot2\pi=\theta</cmath>
  6 KB (1,105 words) - 13:39, 9 January 2024
• 2004 AMC 8 Problems
  Find the number of two-digit positive integers whose digits total <math>7</math>. ...of the five numbers in a list is <math>54</math>. The average of the first two
  13 KB (1,860 words) - 19:58, 8 May 2023
• 2006 AMC 8 Problems
  ...{4}</math> revolutions. In what direction does the spinner point after the two moves? The letter T is formed by placing two <math> 2 \times 4 </math> inch rectangles next to each other, as shown. Wha
  16 KB (2,215 words) - 19:18, 10 April 2024
• 2010 AMC 10B Problems/Problem 20
  Two circles lie outside regular hexagon <math>ABCDEF</math>. The first is tange ...area of the first circle is <math>(\frac{\sqrt{3}}{6})^2 \cdot \pi =\frac{\pi}{12}</math>.
  3 KB (455 words) - 19:28, 5 September 2022
• 2004 JBMO Problems/Problem 4
  ...have interior points in common. We paint in black the triangles that have two sides that are also sides of the polygon, in red if only one side of the tr Prove that there are two more black triangles that white ones.
  2 KB (451 words) - 01:03, 23 December 2018
• 2010 AMC 10B Problems/Problem 19
  A circle with center <math>O</math> has area <math>156\pi</math>. Triangle <math>ABC</math> is equilateral, <math>\overline{BC}</math The formula for the area of a circle is <math>\pi r^2</math> so the radius of this circle is <math>\sqrt{156}.</math>
  3 KB (439 words) - 13:21, 23 September 2023
• 2010 AMC 10B Problems/Problem 16
  ...c{\pi}{18} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{2\pi}{9}</math> ...f by the two radii - area of the triangle with sides on the square and the two radii).
  3 KB (544 words) - 20:54, 24 March 2024
• 2011 AMC 12A Problems
  ...nts per text message sent, plus <math>10</math> cents for each minute used over <math>30</math> hours. In January Michelle sent <math>100</math> text messa ...ber of successful free throws was one more than their number of successful two-point shots. The team's total score was <math>61</math> points. How many fr
  13 KB (1,994 words) - 13:52, 3 July 2021
• 2011 AMC 12A Problems/Problem 10
  ...mes \frac{1}{6} \times \frac{1}{6} = \frac{2}{36}</math>. The sum of these two probabilities now gives the final answer: <math>\frac{1}{36} + \frac{2}{36}
  2 KB (248 words) - 14:51, 5 May 2021
• 1958 AHSME Problems
  Which of these four numbers <math> \sqrt{\pi^2},\,\sqrt[3]{.8},\,\sqrt[4]{.00016},\,\sqrt[3]{-1}\cdot \sqrt{(.09)^{-1}}< The sum of two numbers is <math> 10</math>; their product is <math> 20</math>. The sum of
  25 KB (3,872 words) - 14:21, 20 February 2020
• 2011 AMC 10A Problems
  ...math>¢ per text message sent, plus <math>10</math>¢ for each minute used over <math>30</math> hours. In January Juan sent <math>100</math> text messages ...ber of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws d
  13 KB (1,903 words) - 18:09, 19 April 2021
• 2011 AMC 10B Problems/Problem 18
  ...cmath>\tan (\theta) = \frac{3}{x}</cmath> YAY!!! We have two equations for two variables... that are relatively difficult to deal with. Well, we'll try to ...{6 - 3 \sqrt{3}} = 2 + \sqrt{3}</math>, <math>\theta = \frac{5 + 12n}{12} \pi</math>, where <math>n</math> is any integer. Converting to degrees, we have
  5 KB (782 words) - 14:29, 1 April 2024
• 2011 AMC 12B Problems
  ...went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and ...integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the c
  13 KB (1,978 words) - 16:28, 12 July 2020
• 2002 AMC 10B Problems/Problem 5
  ...d \mathrm{(C) \ } 6\pi\qquad \mathrm{(D) \ } 9\pi\qquad \mathrm{(E) \ } 12\pi </math> A line going through the centers of the two smaller circles also goes through the diameter. The length of this line wit
  2 KB (247 words) - 17:30, 5 January 2021
• 2011 AIME I Problems
  ...um of the numbers on every three consecutive vertices is a multiple of 3. Two acceptable arrangements are considered to be indistinguishable if one can b Suppose <math>x</math> is in the interval <math>[0,\pi/2]</math> and <math>\log_{24 \sin x} (24 \cos x) = \frac{3}{2}</math>. Fin
  10 KB (1,634 words) - 22:21, 28 December 2023
• 2011 AMC 12B Problems/Problem 8
  ...)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math> The curves of the track are semicircles, but since there are two of them, we can consider both of the at the same time by treating them as a
  2 KB (389 words) - 19:35, 7 August 2023
• 2011 AIME I Problems/Problem 9
  Suppose <math>x</math> is in the interval <math>[0, \pi/2]</math> and <math>\log_{24\sin x} (24\cos x)=\frac{3}{2}</math>. Find <ma There are now two ways to finish this problem.
  2 KB (330 words) - 20:47, 10 December 2023
• 2011 AIME I Problems/Problem 14
  ...7}}</math> are parallel, and its side length is the distance between these two lines. However, this is given to be the side length of the octagon, or <mat ...nt-and-a-line/ point to line distance formula], the distance between these two lines is <cmath>\frac{|c_{2}-c_{1}|}{\sqrt{a^{2}+b^{2}}}=\frac{2m\left(1+\s
  8 KB (1,344 words) - 18:39, 9 February 2023
• 2011 AIME II Problems/Problem 8
  ...the real parts of the blue dots is easily seen to be <math>8+16\cos\frac{\pi}{6}=8+8\sqrt{3}</math> and the negative of the sum of the imaginary parts o Note that <math>\sin(x) = \sin(x + \pi/2 - \pi/2) = \cos(x + \pi/2)</math>.
  5 KB (805 words) - 18:46, 27 January 2024
• 2011 AMC 10B Problems
  ...o went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and ...integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was <math>161</math>. Wh
  13 KB (2,090 words) - 18:05, 7 January 2021
• 2003 AMC 10B Problems
  ...l, and Al's pills cost a total of <math> \ </math><math>546</math> for the two weeks. How much does one green pill cost? ...math>. What is the distance between the <math>x</math>-intercepts of these two lines?
  15 KB (2,166 words) - 21:17, 16 February 2021
• 2007 AMC 10B Problems
  ...nd <math>1.5</math> points for each problem left unanswered. After looking over the <math>25</math> problems, Sarah has decided to attempt the first <math> Two points <math>B</math> and <math>C</math> are in a plane. Let <math>S</math>
  15 KB (2,297 words) - 12:57, 19 February 2020
• 2007 AMC 10B Problems/Problem 13
  ...(0,2).</math> What is the area of the intersection of the interiors of the two circles? ...\frac{\pi \sqrt{3}}{3} \qquad\textbf{(D) } 2(\pi -2) \qquad\textbf{(E) } \pi</math>
  898 bytes (142 words) - 20:42, 15 February 2024
• 1998 AHSME Problems/Problem 16
  The figure shown is the union of a circle and two semicircles of diameters <math>a</math> and <math>b</math>, all of whose ce The area of the whole circle is <math>A_{big} = \pi\cdot (a + b)^2</math>
  2 KB (307 words) - 18:58, 11 January 2014
• Area of an equilateral triangle
  ''Method 1:'' Dropping the altitude of our triangle splits it into two triangles. By HL congruence, these are congruent, so the "short side" is <m ...ac{ab\sin{C}}{2}</math>. Plugging in <math>a=b=s</math> and <math>C=\frac{\pi}{3}</math> (the angle at each vertex, in radians), we get the area to be <m
  1 KB (189 words) - 04:06, 18 June 2018
• Trigonometric Equations
  Solve for <math>x</math> for all answers in the domain <math>[0, 2\pi]</math>. We have a problem! The domain for values of <math>x</math> is <math>[0, 2\pi]</math>. However, no real value of <math>x</math> can become imaginary when
  8 KB (1,351 words) - 20:30, 10 July 2016
• 2012 AMC 12B Problems/Problem 23
  ...with <math>|z_0|=1.</math> What is the sum of all values <math>P(1)</math> over all the polynomials with these properties? ...these cases, <math>P(-1)=4-4+t-t+0=0</math>. The sum of <math>P(1)</math> over these cases is <math>\sum_{t=0}^{4} (4+4+t+t) = 40+20=60</math>.
  11 KB (1,979 words) - 17:25, 6 September 2021
• 1967 AHSME Problems
  The difference in the areas of two similar triangles is <math>18</math> square feet, and the ratio of the larg <math>\textbf{(A)}\ \frac{\pi m^2}{2}\qquad
  20 KB (3,108 words) - 14:14, 20 February 2020
• Interior angle
  The '''interior angle''' is the [[angle]] between two line segments, having two endpoints connected via a path, facing the path connecting them. ...erior angles of an <math>n</math> sided regular polygon,are <math>180(1-{2\over n})</math> degrees.
  673 bytes (105 words) - 22:28, 27 February 2020
• 2003 AMC 10B Problems/Problem 19
  ...qquad\textbf{(D) } \frac{\pi +\sqrt{3}}{2} \qquad\textbf{(E) } \frac{7}{6}\pi - \frac{\sqrt{3}}{2}</math> ...n into <math>\frac{5}{6}</math> of a circle with radius <math>1</math> and two equilateral triangles with side length <math>1</math>.
  4 KB (606 words) - 13:19, 9 July 2021
• 1984 AHSME Problems/Problem 15
  This looks a lot like the formula relating the slopes of two [[perpendicular]] [[Line|lines]], which is <math> m_1\times m_2=-1 </math>, ...> 2x+3x=5x </math>, so <math> 5x=\frac{\pi}{2} </math> and <math> x=\frac{\pi}{10} </math>, or <math> 18^\circ, \boxed{\text{A}} </math>.
  3 KB (493 words) - 18:16, 4 June 2021
• 2009 AMC 8 Problems
  ...mbered sectors. What is the probability that the sum of the numbers in the two sectors is prime? draw((0,0)--(cos(pi/6),sin(pi/6)));
  18 KB (2,551 words) - 18:46, 27 February 2024
• 2000 AMC 8 Problems/Problem 19
  ...\ 10+5\pi\qquad\text{(C)}\ 50\qquad\text{(D)}\ 50+5\pi\qquad\text{(E)}\ 25\pi </math> Draw two squares: one that has opposing corners at <math>A</math> and <math>B</math
  2 KB (383 words) - 16:58, 12 January 2024
• 1995 AJHSME Problems/Problem 9
  ...angle. But <math>24</math> is too low, as it is less than the area of the two circles. Thus, the only reasonable answer is <math>\boxed{C}</math>, which
  2 KB (409 words) - 00:15, 5 July 2013
• 2008 AMC 8 Problems/Problem 18
  Two circles that share the same center have radii <math>10</math> meters and <m ...extbf{(C)}\ 10\pi+40\qquad\textbf{(D)}\ 20\pi+20\qquad \\ \textbf{(E)}\ 20\pi+40</math>
  2 KB (210 words) - 13:37, 19 October 2020
• 2010 AMC 8 Problems/Problem 23
  ...the center <math>O</math>. What is the ratio of the combined areas of the two semicircles to the area of circle <math>O</math>? ...ac{\sqrt 2}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{2}{\pi}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{\sqrt 2}{2} </math
  2 KB (298 words) - 00:29, 18 December 2023
• 2009 AMC 8 Problems/Problem 12
  ...mbered sectors. What is the probability that the sum of the numbers in the two sectors is prime? draw((0,0)--(cos(pi/6),sin(pi/6)));
  1 KB (179 words) - 19:36, 15 April 2023
• 1979 USAMO Problems/Problem 4
  There are two ways to solve this problem. The first is more subtle, and the second is jus ...ngle POR,</math> and <math>b = \angle POQ.</math> Then <math>\angle ORP = \pi - x - a</math> and <math>\angle OQP = x - b.</math> Using the Law of Sines
  3 KB (568 words) - 12:24, 11 March 2018
• 1996 AHSME Problems
  ...-\pi\qquad\text{(C)}\ \frac{\pi} 2\qquad\text{(D)}\ \pi\qquad\text{(E)}\ 2\pi </math> Two opposite sides of a rectangle are each divided into <math>n</math> congruen
  15 KB (2,343 words) - 13:39, 19 February 2020
• 1992 AJHSME Problems/Problem 24
  ...e, which is <math>9\pi</math>. The area of the shaded region is <math>36-9\pi</math> which less than <math>36-9(3)=9</math>, closest to <math>\boxed{\tex
  1 KB (191 words) - 00:10, 5 July 2013
• 1950 AHSME Problems
  If the digit <math>1</math> is placed after a two digit number whose tens' digit is <math>t</math>, and units' digit is <math Two boys <math>A</math> and <math>B</math> start at the same time to ride from
  22 KB (3,306 words) - 19:50, 3 May 2023
• 1985 AHSME Problems
  ...qquad \mathrm{(D) \ } \frac{5}{6}\pi+2 \qquad \mathrm{(E) \ }\frac{5}{3}\pi+2 </math> ...h> unit apart both horizontally and vertically. A rubber band is stretched over <math>4</math> pegs as shown in the figure, forming a quadrilateral. Its ar
  17 KB (2,488 words) - 03:26, 20 March 2024
• 1985 AHSME Problems/Problem 2
  ...qquad \mathrm{(D) \ } \frac{5}{6}\pi+2 \qquad \mathrm{(E) \ }\frac{5}{3}\pi+2 </math> .... Hence the total perimeter in cm is <math>\boxed{\text{(E)} \ \frac{5}{3}\pi+2}</math>.
  1 KB (181 words) - 18:35, 19 March 2024
• 2010 AMC 8 Problems
  ...ath>16</math> feet higher than the top of another tree. The heights of the two trees are in the ratio <math>3:4</math>. In feet, how tall is the taller tr ...\pi} \qquad\textbf{(C)}\ \pi \qquad\textbf{(D)}\ 2\pi \qquad\textbf{(E)}\ \pi^{2}</math>
  18 KB (2,768 words) - 21:05, 9 January 2024
• 2011 AMC 8 Problems/Problem 25
  ...s closest to the ratio of the circle's shaded area to the area between the two squares? ...area of the smaller square subtracted from the area of the circle: <math>\pi - 2.</math>
  3 KB (511 words) - 07:47, 22 May 2023
• Mathtime Version 1 Issue 2
  ...r with a tight rope. The rope is tight so that there is no extra rope left over when bound around the four circles. If the radius of a circle is <math>5</m Two triangles are combined together with overlap so that the Star of David is f
  15 KB (2,444 words) - 21:46, 1 January 2012
• 1950 AHSME Problems/Problem 34
  ...qquad \textbf{(D)}\ \dfrac{5}{2\pi}\text{ in} \qquad \textbf{(E)}\ \dfrac{\pi}{5}\text{ in}</math> ...(or else the ratio of the circumference to the diameter wouldn't be <math>\pi</math> anymore)
  2 KB (313 words) - 22:24, 24 December 2023
• 1971 Canadian MO Problems/Problem 9
  Two flag poles of height <math>h</math> and <math>k</math> are situated <math>2 ...l Angles. In effect, we can construct our desired point P by reflecting H' over IJ and finding the intersection of H'K and IJ. Therefore, there is only one
  1 KB (204 words) - 18:09, 19 September 2012
• 1989 AHSME Problems
  A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three r ...<math>100</math> does <math> x^{2}+x-n </math> factor into the product of two linear factors with integer coefficients?
  15 KB (2,247 words) - 13:44, 19 February 2020
• 2012 AMC 10A Problems
  A square with side length <math>8</math> is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles? ...>4</math> times the reciprocal of the other number. What is the sum of the two numbers?
  13 KB (1,994 words) - 01:31, 22 December 2023
• 2012 AMC 12A Problems
  ...way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of ...as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?
  14 KB (2,197 words) - 13:34, 12 August 2020
• 2012 AMC 12A Problems/Problem 16
  ...math> has its center <math>O</math> lying on circle <math>C_2</math>. The two circles meet at <math>X</math> and <math>Y</math>. Point <math>Z</math> in ...angle{OZY} \cong \angle{OZX}</math>, this is a system of two equations and two variables. Solving for <math>r</math> gives <math>r = \sqrt{30}</math>. <ma
  9 KB (1,496 words) - 02:40, 2 October 2022
• 1989 AHSME Problems/Problem 28
  ...\tan^2x-9\tan x+1=0</math> that are between <math>x=0</math> and <math>x=2\pi</math> radians. ...mathrm{(C) \frac{3\pi}{2} } \qquad \mathrm{(D) 3\pi } \qquad \mathrm{(E) 4\pi } </math>
  2 KB (282 words) - 21:14, 2 March 2019
• 1989 AHSME Problems/Problem 19
  ...uad \mathrm{(D) \frac{9}{\pi^2}(\sqrt{3}-1) } \qquad \mathrm{(E) \frac{9}{\pi^2}(\sqrt{3}+3) } </math> ...ac{6}{\pi}</math> for <math>r</math> and evaluating yields <math>\frac{9}{\pi^2}(\sqrt{3}+3)\implies{\boxed{E}}</math>.
  4 KB (649 words) - 10:04, 20 May 2021
• 2012 AMC 10B Problems
  ...y bags as possible, with 6 marbles per bag. How many marbles will be left over? ...e integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is
  18 KB (2,350 words) - 18:48, 9 July 2023
• 2012 AMC 12B Problems/Problem 15
  ...along 2 radii to form 2 sectors, one with a central angle of 120. He makes two circular cones using each sector to form the lateral surface of each cone. ...h>, and the larger is <math>16\pi</math>. Those two arc lengths become the two circumferences of the new cones; so the radius of the smaller cone is <math
  2 KB (259 words) - 13:42, 2 September 2020
• 2010 AMC 8 Problems/Problem 19
  ...erline{AD}</math> has length <math>16</math>. What is the area between the two circles? ...quad\textbf{(C)}\ 64 \pi\qquad\textbf{(D)}\ 81 \pi\qquad\textbf{(E)}\ 100 \pi </math>
  2 KB (279 words) - 09:04, 10 March 2023
• 2012 AMC 10B Problems/Problem 21
  ...)}\ 2\qquad\textbf{(C)}\ \sqrt{5}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \pi</math> ...gle inequality: <math>b<2a</math>. The only answer choice that holds these two inequalities is: <math>\sqrt{3}</math>.
  3 KB (432 words) - 20:18, 23 October 2021

View (previous 250 | next 250) (20 | 50 | 100 | 250 | 500)