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

• 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) - 20:58, 17 June 2024
• 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

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