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

• 2023 AIME I Problems/Problem 8
  ...h>XY</math> respectively. Through this, we know that the distance from the two pairs of opposite lines of rhombus <math>XYZW</math> is <math>25</math> and ...ively, and <math>BC \parallel AD</math>, the distance between each pair of two parallel sides of <math>ABCD</math> is <math>16 + 9 = 25</math>.
  17 KB (2,612 words) - 14:54, 3 July 2023
• 2023 AIME II Problems/Problem 8
  Let <math>\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},</math> where <math>i = \sqrt{-1}.</math> Find the value of the product ...>z_n = \left(\textrm{cis }\frac{2n\pi}{7}\right)^3 + \textrm{cis }\frac{2n\pi}{7} + 1</math>.
  9 KB (1,284 words) - 23:37, 31 January 2024
• 2023 AMC 10A Problems/Problem 24
  ...ame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of // Function to calculate the center of a side given two vertices
  13 KB (1,886 words) - 17:36, 27 May 2024
• 2023 AMC 10A Problems
  ...a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of th ...number of circles needed to make the total shaded area at least <math>2023\pi</math>?
  16 KB (2,411 words) - 03:51, 22 May 2024
• 2022 SSMO Relay Round 2 Problems
  ...e diameter of the circle can be expressed as <math>\frac{a\pi+b\sqrt{c}}{d\pi},</math> where <math>a,b,c,d</math> are integers such that <math>a</math> a ...<math>f(x)</math> is tangent to the function <math>g(x)=|x+2|-T</math> at two points, when graphed on the coordinate plane. Then <math>|f(1)|</math> can
  2 KB (291 words) - 21:56, 31 May 2023
• 2022 SSMO Relay Round 5 Problems
  ...o outside the chessboard. How many squares can the knight reach in exactly two moves? ...}+a_{2k+2})|}</cmath> can be expressed in the form of <math>a+b\cos(\frac{\pi}{c})</math>, where <math>\text{cis}(x) = \cos(x) + i\sin(x)</math>. Find <m
  1 KB (217 words) - 21:59, 31 May 2023
• 2024 AIME I Problems
  ...bers are randomly chosen from <math>S.</math> She wins a prize if at least two of her numbers were <math>2</math> of the randomly chosen numbers, and wins ...of <cmath>y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).</cmath>
  8 KB (1,307 words) - 20:00, 6 February 2024
• 2024 AIME I Answer Key
  ...and zoom in the region near <math>\left( 1, 1 \right)</math>, you can see two distinct solutions. ...and <math>\cos \theta \approx 1 - \frac{\theta^2}{2}</math>. This reduces two complicated equations to one linear and one quadratic equation. I can then
  3 KB (515 words) - 03:57, 4 February 2024
• 2019 Mock AMC 10B Problems/Problem 12
  ...0</math> degrees, then <math>\angle BIO = 30</math> degrees (radii are the two legs of the triangle <math>\rightarrow</math> isosceles triangle). That mea ...area <math>= \frac{75\sqrt{3}}{4} + \frac{25}{2} \pi = \frac{75\sqrt{3}+50\pi}{4}</math>. Our answer <math>= 75 + 3+ 50 + 4 = \boxed{132}</math>.
  2 KB (276 words) - 20:22, 1 July 2023
• Know the Facts: Gmaas
  ...s uncover a portrait of Gottfried Leibniz holding a cat, sparking a debate over whether the cat is Gmaas's demigod son or Gmaas's long-lost brother. - 1977: Historians settle the century long debate over Leibniz' cat, revealed to be Gmaas' demigod son. Turns out Schrodinger's ca
  88 KB (14,928 words) - 13:54, 29 April 2024
• 1999 IMO Problems/Problem 1
  For any two distinct points <math>A</math> and <math>B</math> in <math>S</math>, the pe ...h>P_{i}=\left\langle Rcos\left( \frac{2\pi}{n}i \right),Rsin\left( \frac{2\pi}{n}i \right) \right\rangle</math> for <math>i=0,1,2,...,(n-1)</math>.
  5 KB (955 words) - 02:21, 21 November 2023
• 2023 AMC 10B Problems
  ...h>\times</math> <math>3</math> grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is cover ...{2}</math>. What is the length of the longest interior diagonal connecting two vertices of <math>P</math>?
  13 KB (1,959 words) - 00:37, 27 May 2024
• 2023 AMC 12B Problems/Problem 3
  ...s to get <math>25\pi</math> and <math>169\pi</math>. Cancel out the <math>\pi</math>, and lastly, divide, to get your answer <math>=\boxed{\textbf{(D) }\ ...meter of a circle is <math>90</math> degrees, the hypotenuse of each these two triangles is respectively the diameter of circles <math>A</math> and <math>
  3 KB (473 words) - 21:13, 29 January 2024
• 2023 AMC 12B Problems/Problem 25
  ...math> by AA similarity. Therefore, if we can write the areas of the latter two triangles as a ratio of the first triangle, we can express the whole equati Now the ratio between the area of two similar triangles is the square of the ratio of their analogous side length
  19 KB (2,971 words) - 00:31, 4 June 2024
• 2023 AMC 10B Problems/Problem 20
  shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is <math>\pi\sqrt{n}</math>. What is <math>n</math>?
  5 KB (829 words) - 11:53, 29 February 2024
• Polya’s method for extremums
  Let ABC be a section of a triangular prism, two faces of which AB and AC are fixed, and the third (BC) is a piston whose wi ...length dividing a given equilateral triangle with side <math>a</math> into two equal parts.
  12 KB (2,104 words) - 14:11, 24 February 2024
• 2023 AMC 12B Problems/Problem 11
  ...on is equal to zero when <math>x=\pm\sqrt{2}</math>, so <math>A</math> has two critical points at <math>\pm\sqrt{2}</math>. But given the bounds of the pr .../cmath> Hence, we maximize <math>3\sin t\cos t</math> for <math>0<t<\frac{\pi}{2}</math>.
  5 KB (886 words) - 13:55, 29 November 2023
• 2023 AMC 12B Problems/Problem 20
  ...c{\arctan \frac{1}{2}}{\pi}\qquad\textbf{(E)}~\frac{2\arcsin \frac{1}{4}}{\pi}</math> ...<math>1</math> unit away from <math>S</math> is <math>\frac{4 \alpha }{ 2 \pi}</math>.
  4 KB (587 words) - 18:15, 2 January 2024
• 2023 SSMO Relay Round 3 Problems
  ...th>, respectively. Each of the three circles are externally tangent to the two other circles. If the radius of a circle <math>\omega</math> such that <mat ...he blue-painted region and the area of the red-painted region is <math>a+b\pi.</math> Find <math>a+b.</math> (Note: if a spray painter paints an area wit
  3 KB (562 words) - 21:35, 15 December 2023
• 2024 AIME I Problems/Problem 12
  ...of <cmath>y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).</cmath> ...</math> and <math>1</math>). Thus by precariously drawing the graph of the two functions in the square bounded by <math>(0,0)</math>, <math>(0,1)</math>,
  8 KB (1,315 words) - 19:41, 18 February 2024

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