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• 2022 AMC 10A Problems/Problem 21
  //Comment two lines below to remove red edges ...tersection by symmetry. Therefore, the triangle that is wedged between the two hexagons has the same angle as the square at the bottom wedged between the
  13 KB (2,080 words) - 11:27, 25 October 2023
• 2022 AMC 12A Problems/Problem 13
  ...o semicircles will be formed with a radius of 1 for a total area of <math>\pi \approx 3</math>. Therefore, the total area is <math>5(2) + \pi \approx 10 + 3 = \boxed{\textbf{(A) } 13}</math>.
  2 KB (344 words) - 15:41, 25 October 2023
• 2022 AMC 10B Problems/Problem 22
  ...\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad</math> There are four cases with two circles belonging to each:
  7 KB (1,202 words) - 01:15, 10 June 2023
• 2022 AMC 12B Problems/Problem 11
  -1 + i\sqrt{3} &= 2e^{\frac{2\pi i}{3}}, \\ -1 - i\sqrt{3} &= 2e^{-\frac{2\pi i}{3}} = 2e^{\frac{4\pi i}{3}}.
  5 KB (866 words) - 22:17, 27 October 2023
• 2021 EGMO Problems
  (ii) no two points in <math>P</math> lie on a line through the origin. ...real number <math>x</math>. Thus <math>\lfloor\sqrt{2}\rfloor = 1, \lfloor\pi\rfloor =\lfloor 22/7 \rfloor = 3, \lfloor 42\rfloor = 42,</math> and <math>
  3 KB (492 words) - 14:07, 24 December 2022
• 2023 AMC 12B Problems
  ...) of a <math>3 \times 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 ...ath>-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?
  13 KB (2,107 words) - 22:19, 20 April 2024
• 2023 AMC 8 Problems/Problem 12
  ...he total area of the radius <math>3</math> circle is simply just <math>9* \pi</math> when using our area of a circle formula. ...\frac{1}{4} \pi + 4 \pi -\pi - \pi = \frac{3}{4} \pi + 2 \pi = \frac{11 * \pi}{4}</math>.
  3 KB (453 words) - 17:31, 18 May 2024
• Random Problem
  ...maximum possible volume of this cylinder in the form of <math>\frac{a}{b}\pi</math>? <math>V=\pi{r^2}h</math>
  4 KB (697 words) - 17:07, 24 March 2023
• 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

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