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• 2017 AMC 12B Problems
  ...(x)</math> and <math>\cos(x)</math> are periodic with least period <math>2\pi</math>. What is the least period of the function <math>\cos(\sin(x))</math> ...i}{2}\qquad\textbf{(B)}\ \pi\qquad\textbf{(C)}\ 2\pi \qquad\textbf{(D)}\ 4\pi \qquad\textbf{(E)}</math> It's not periodic.
  15 KB (2,343 words) - 18:26, 25 December 2020
• 2017 AMC 12B Problems/Problem 12
  ...f{(D)}\ 2\sqrt{2}+\sqrt{6} \qquad \textbf{(E)}\ (1+\sqrt{3}) + (1+\sqrt{3})i</math> ...12} = 64</math>, it is easy to see <math>\pm\sqrt{2}</math> and <math>\pm {i} \sqrt{2}</math> as roots. Graphing these in the complex plane, we have fou
  3 KB (449 words) - 01:54, 11 February 2019
• 2017 AIME I Problems/Problem 8
  Q = (10*cos(pi/3), 10*sin(pi/3)); R = (10*cos(5*pi/6), 10*sin(5*pi/6));
  9 KB (1,539 words) - 15:47, 17 February 2024
• 2017 AIME I Problems/Problem 10
  ...th>z_1=18+83i,~z_2=18+39i,</math> and <math>z_3=78+99i,</math> where <math>i=\sqrt{-1}.</math> Let <math>z</math> be the unique complex number with the ...f thinking of complex numbers as purely a real plus a constant times <math>i</math>, let’s graph them and hope that the geometric visualization adds i
  13 KB (2,252 words) - 15:46, 6 January 2024
• 2017 AIME I Problems/Problem 15
  ...lie on the hypotenuse <math>\frac{x}{5} + \frac{y}{2\sqrt{3}} = 1</math>, i.e. <math>a,b</math> must satisfy ...+bi)\cdot\cos(-\frac{\pi}{3})=(-\frac{a+\sqrt{3}b}{2}+\frac{\sqrt{3}a+b}{2}i)</math>. We know that the slope of <math>AC</math> is <math>-\frac{2\sqrt{3
  22 KB (3,622 words) - 17:11, 6 January 2024
• 2017 USAMO Problems/Problem 5
  ...distance between any two points labeled <math>i</math> is at least <math>c^i</math>. ...For <math>c\le \sqrt[4]{2},</math> we can make a "checkerboard" labeling, i.e. label <math>(x, y)</math> with <math>1</math> if <math>x+y</math> is eve
  8 KB (1,495 words) - 12:19, 17 July 2023
• Reaper Archives
  | 86 || I-Can-Do-Math || 6 || 3059.612 || 509.935 | 76 || math-pi || 79 || 45682.709 || 578.262
  187 KB (10,824 words) - 18:27, 3 February 2022
• 2020 AMC 12A Problems
  for (int i = 0; i < 3; ++i) { pair A = (j,i);
  14 KB (2,073 words) - 15:15, 21 October 2021
• 2020 AMC 12B Problems
  ...f{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7</math> ...textbf {(D) } 3\sqrt{3} - \pi \qquad \textbf {(E) } \frac{9\sqrt{3}}{2} - \pi </math>
  16 KB (2,417 words) - 01:03, 28 April 2022
• MIE 2016
  ...th>. With <math>\beta</math> being a real number such that <math>0< \beta<\pi/8</math> and <math>x\neq0</math>, the value of <math>\beta</math> is: (a) <math>\frac{\pi}{9}</math>
  8 KB (1,278 words) - 09:46, 11 January 2018
• MIE 2015
  ...k=1}^{15}</math> Img<math>\left(\right.</math>cis<math>\left.^{2k-1}\frac{\pi}{36}\right)</math> (a) <math>\frac{2+\sqrt3}{4\sin\frac{\pi}{36}}</math>
  7 KB (1,127 words) - 18:23, 11 January 2018
• 2018 AMC 12A Problems/Problem 22
  ...ath>z^2=4+4\sqrt{15}i</math> and <math>z^2=2+2\sqrt 3i,</math> where <math>i=\sqrt{-1},</math> form the vertices of a parallelogram in the complex plane <li><math>z^2=4+4\sqrt{15}i</math><p>
  10 KB (1,662 words) - 12:45, 13 September 2021
• 2018 AMC 12A Problems
  ...st subset of values of <math>y</math> within the closed interval <math>[0,\pi]</math> for which ...+\sin(y)</cmath>for every <math>x</math> between <math>0</math> and <math>\pi</math>, inclusive?
  15 KB (2,380 words) - 18:52, 7 April 2022
• 2018 AMC 12B Problems
  \textbf{(A) }25\pi \qquad \textbf{(B) }50\pi \qquad
  14 KB (2,118 words) - 15:36, 28 October 2021
• 2018 AMC 12B Problems/Problem 16
  ...eta</math> is the argument of <math>z</math> such that <math>0\leq\theta<2\pi.</math> ...{4},\frac{\pi}{2},\frac{3\pi}{4},\pi,\frac{5\pi}{4},\frac{3\pi}{2},\frac{7\pi}{4}.</math></li><p>
  11 KB (1,708 words) - 12:01, 18 March 2023
• 2018 AMC 12B Problems/Problem 25
  ...<math>P_iP_{i+1}</math> is tangent to <math>\omega_i</math> for each <math>i=1,2,3</math>, where <math>P_4 = P_1</math>. See the figure below. The are Let <math>O_i</math> be the center of circle <math>\omega_i</math> for <math>i=1,2,3</math>, and let <math>K</math> be the intersection of lines <math>O_1
  13 KB (2,080 words) - 19:09, 21 October 2023
• 1960 AHSME Problems/Problem 7
  ...er of, and is tangent to, circle <math>II</math>. The area of circle <math>I</math> is <math>4</math> square inches. \textbf{(C) }8\sqrt{\pi}\qquad
  2 KB (306 words) - 18:57, 17 May 2018
• 2018 AIME I Problems/Problem 6
  ...heta = \sin (\pi - \theta)</math> and <math>\sin \theta = \sin (\theta + 2\pi)</math>. We can use these facts to create two types of solutions: <cmath>\sin \theta = \sin ((2m + 1)\pi - \theta)</cmath>
  7 KB (1,211 words) - 00:23, 20 January 2024
• 2018 AIME I Problems/Problem 12
  <math>W_S = \sum_{i=1}^{|D|} \tbinom{6}{x_i}\tbinom{6}{y_i}</math> if and only if there exists ...>S</math> is divisible by 3. Therefore, by the fact that <math>W_S = \sum_{i=1}^{|D|} \tbinom{6}{x_i}\tbinom{6}{y_i}</math>, we have that;
  26 KB (4,044 words) - 13:58, 24 January 2024
• 2018 AIME I Problems/Problem 10
  label("$I$", (2+r) * dir(162), dir(162)); ...ounter-Clockwise}</math>, and <math>\text{Switching}</math>. Let an <math>I</math> signal going clockwise (because it has to be in the ''inner'' circle
  11 KB (1,934 words) - 12:18, 29 March 2024

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