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• 2006 iTest Problems/Problem U6
  ...f a line connecting a point on the circle to the origin. That line must be tangent to the circle. ...Identity yields <math>\tan 2\theta = \frac34</math>, so <math>\tan (90 - 2\theta) = \frac43</math>.
  4 KB (722 words) - 20:53, 27 March 2019
• 2019 AIME II Problems
  ...le of <math>3</math>, and negative otherwise. The degree measure of <math>\theta</math> is <math>\tfrac{p}{q}</math>, where <math>p</math> and <math>q</math ....</math> Circle <math>\omega_2</math> passes through <math>C</math> and is tangent to line <math>AB</math> at <math>A.</math> Let <math>K</math> be the inters
  7 KB (1,254 words) - 14:45, 21 August 2023
• 2019 AIME I Problems/Problem 11
  ...nt to <math>\omega</math>, and the other two excircles are both externally tangent to <math>\omega</math>. Find the minimum possible value of the perimeter of ...C=a</math>, <math>AB=b</math>, <math>s</math> be the semiperimeter, <math>\theta=\angle ABC</math>, and <math>r</math> be the inradius. Intuition tells us t
  21 KB (3,915 words) - 19:55, 10 October 2023
• 2006 SMT/Team Problems/Problem 13
  ...theta </math> with the positive <math> x </math> axis. Compute <math> \cos\theta </math>. ...gin has an equation of <math>y=mx</math>. If the line <math>y=mx</math> is tangent to the hyperbola than the equation <math>(mx)^2=x^2-x+1</math> will have on
  1 KB (261 words) - 18:38, 14 January 2020
• 2020 AIME I Problems/Problem 15
  ...e intersection of the altitudes of <math>\triangle ABC.</math> Suppose the tangent to the circumcircle of <math>\triangle HBC</math> at <math>H</math> interse var phi=75.5, theta=130, r=4.8;
  16 KB (2,678 words) - 22:45, 27 November 2023
• 2000 SMT/Calculus Problems
  Find the slope of the tangent at the point of inflection of <math>y = x^3 - 3x^2 + 6x + 2000</math>. An envelope of a set of lines is a curve tangent to all of them. What is the envelope of the family of lines y = <math>\frac
  3 KB (413 words) - 13:10, 21 January 2020
• 2020 AMC 12B Problems/Problem 10
  ...triangle <math>\bigtriangleup ADM</math>. Let <math>\measuredangle AMD = \theta</math>. ...ega</math>. Therefore, <math>\measuredangle POM = 180 - 2(90 - \theta) = 2\theta</math>.
  9 KB (1,328 words) - 16:14, 11 September 2023
• 2021 AIME I Problems/Problem 15
  ...<math>(-5,20)</math>, and by drawing the graph, you realize this is not a tangent point and there is in fact another intersection nearby, due to slope. There ...ection of the ordinate axis <math>2\theta,</math> its length <math>\rho_1(\theta),</math> the angle between the vector connecting the focus of the second pa
  10 KB (1,742 words) - 02:31, 13 November 2023
• 2021 Fall AMC 10A Problems/Problem 15
  ...AB = AC = 3\sqrt6</math>, and a circle with radius <math>5\sqrt2</math> is tangent to line <math>AB</math> at <math>B</math> and to line <math>AC</math> at <m ...th>\odot O_1</math> be the circle with radius <math>5\sqrt2</math> that is tangent to <math>\overleftrightarrow{AB}</math> at <math>B</math> and to <math>\ove
  7 KB (1,026 words) - 13:43, 5 May 2024
• 2021 AIME II Problems
  ...h>\ell</math> to the point where the sphere with radius <math>13</math> is tangent to plane <math>\mathcal{P}</math> is <math>\tfrac{m}{n}</math>, where <math ...acute angle formed by the diagonals of the quadrilateral. Then <math>\tan \theta</math> can be written in the form <math>\tfrac{m}{n}</math>, where <math>m<
  8 KB (1,429 words) - 14:31, 26 February 2024
• 2021 AMC 10B Problems/Problem 21
  ...nd perpendicular lines have slopes negative inverses of each other). Using tangent double angle formula, the slope of <math>\overline{EC'}</math> is <math>\fr
  8 KB (1,252 words) - 23:49, 3 March 2024
• 2021 AIME II Problems/Problem 12
  ...acute angle formed by the diagonals of the quadrilateral. Then <math>\tan \theta</math> can be written in the form <math>\tfrac{m}{n}</math>, where <math>m< ...</math>, <math>CX = c</math>, and <math>DX = d</math>. We know that <math>\theta</math> is the acute angle formed between the intersection of the diagonals
  10 KB (1,669 words) - 17:33, 12 January 2024
• 2021 AIME II Problems/Problem 14
  ...entroid <math>G</math>. Let <math>X</math> be the intersection of the line tangent to the circumcircle of <math>\Delta ABC</math> at <math>A</math> and the li ...lies \beta + \theta = 11\alpha.</cmath> However, from before, <math>\beta+\theta = 180 - 17 \alpha</math>, so <math>11 \alpha = 180 - 17 \alpha \implies 180
  10 KB (1,526 words) - 16:50, 25 December 2022
• 2022 AIME II Problems/Problem 11
  ...DAM = \alpha</math>, <math>\angle BAD = \beta</math>, <math>\angle BMA = \theta</math>, <math>\angle CMD = \phi</math>. Hence, <math>MP = MB</math> and <math>\angle AMP = \theta</math>.
  14 KB (2,254 words) - 18:26, 8 February 2024
• 2022 AIME II Problems/Problem 15
  Two externally tangent circles <math>\omega_1</math> and <math>\omega_2</math> have centers <math> Denote <math>\angle O_1 O O_2 = 2 \theta</math>.
  14 KB (2,217 words) - 00:28, 29 June 2023
• 2021 Fall AMC 12B Problems/Problem 22
  ...and passes through <math>A</math>. A circle centered at <math>P</math> is tangent to line <math>AC</math> at <math>A</math> and passes through <math>B</math> ...bel("$\theta$", A, 7*dir(162)); label("$\theta$", B, 7*dir(-20)); label("$\theta$", P, 7*dir(-110)); label("$6$", B--C, left); label("$8$", A--C, down); lab
  6 KB (943 words) - 00:41, 6 August 2023
• 2021 Fall AMC 10B Problems/Problem 17
  ...tively, that meet at an angle <math>\theta</math> is a rotation by <math>2\theta</math> around the intersection of <math>l</math> and <math>m</math>. by the tangent addition formula. Since the slope of line <math>m</math> is <math>\frac{2}{
  8 KB (1,331 words) - 22:44, 16 December 2023
• 2021 Fall AMC 12A Problems/Problem 13
  Now, we can take the tangent and apply the tangent subtraction formula: Denote by <math>\theta</math> the acute angle formed by lines <math>y = x</math> and <math>y = 3 x
  16 KB (2,526 words) - 00:53, 6 May 2023
• 2023 AIME I Problems/Problem 5
  Using the half-angle formula for tangent, Denote by <math>\theta</math> the argument of point <math>P</math> on the circle.
  19 KB (3,107 words) - 23:31, 17 January 2024
• Kimberling’s point X(24)
  Let <math>\Theta</math> be the circle with diameter <math>OQ.</math> ...\cdot QC = (QO – R) \cdot (QO + R) = QP^2</math> <cmath>\implies P \in \Theta, \Omega \perp \omega.</cmath>
  10 KB (1,751 words) - 15:34, 25 November 2022

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