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• Mock AIME 1 2006-2007 Problems/Problem 14
  <math>\tan{\angle{OAB}} = \dfrac{OT}{AT} = \dfrac{r}{m}</math>
  3 KB (541 words) - 17:32, 22 November 2023
• 2008 AIME I Problems/Problem 14
  ...y that <math>\triangle APC \sim \triangle OTC</math>. Thus <math>\frac{AP}{OT} = \frac{CA}{CO} \Longrightarrow AP = \frac{9(x-9)}{x}</math>. By the [[Law where <math>\cos \angle BAP = \cos (180 - \angle TOA) = - \frac{OT}{OC} = - \frac{9}{x}</math>, so:
  8 KB (1,339 words) - 13:29, 3 September 2024
• 2008 AIME II Problems/Problem 11
  .../math> we have <math>\frac{\overline{PX}}{\overline{XC}} = \frac{\overline{OT}}{\overline{TC}} \rightarrow \frac{16}{\overline{XC}} = \frac{21}{28} \righ
  6 KB (1,092 words) - 22:22, 18 August 2024
• 1996 AJHSME Problems/Problem 17
  If <math>PT = 4</math>, and <math>PO = 2</math>, then <math>OT = 2</math>, and <math>T</math> must be <math>2</math> units to the left of
  2 KB (257 words) - 11:20, 22 March 2015
• 2015 AIME I Problems/Problem 15
  ...egrees and right triangle calculations, we find <math>MT</math> = 3, <math>OT</math> = 4, <math>OM</math> = 5). We will be integrating with respect to th
  11 KB (1,830 words) - 07:00, 10 October 2024
• 2015 USAMO Problems/Problem 2
  As <math>OP = OT</math>, <math>OP^2-OM^2 = MT^2</math> from right triangle <math>OMT</math>.
  5 KB (902 words) - 09:58, 20 August 2021
• 2015 USAJMO Problems/Problem 3
  As <math>OP = OT</math>, <math>OP^2-OM^2 = MT^2</math> from right triangle <math>OMT</math>.
  4 KB (760 words) - 16:45, 29 April 2020
• 2016 USAMO Problems/Problem 3
  <math>V</math> is the foot ot perpendicular from <math>O</math> to <math>AI_A.</math>
  6 KB (998 words) - 21:36, 17 October 2022
• 2020 AIME II Problems/Problem 15
  ...math>O</math> be the circumcenter of <math>\triangle ABC</math>; say <math>OT</math> intersects <math>BC</math> at <math>M</math>; draw segments <math>XM
  7 KB (1,221 words) - 16:46, 29 January 2023
• 2020 AIME II Problems/Problem 7
  ...ph paper, with the scale of <math>\sqrt{2}:1</math>. So, we can find <math>OT</math> then divide by <math>\sqrt{2}</math> to convert to our desired units
  8 KB (1,255 words) - 13:43, 24 June 2024
• 2021 AIME I Problems/Problem 13
  ...<math>PQ</math> and <math>O_{1}O_{2}</math> are perpendicular, lines <math>OT</math> and <math>O_{1}O_{2}</math> meet at a <math>60^{\circ}</math> angle. ...\triangle O_{2}OT&:(r+R_{2})^{2}=u^{2}+(r+x)^{2}-u(r+x) \\ \triangle O_{1}OT&:(r+R_{1})^{2}=v^{2}+(r+x)^{2}+v(r+x)\end{align*}</cmath>
  17 KB (2,852 words) - 03:59, 7 February 2024
• 2023 AMC 12A Problems/Problem 18
  Then <math>OT = BO + BT = BO + AT - \frac{1}{2} = \frac{1}{4} + 1 - \frac{1}{2} = \frac{3
  4 KB (547 words) - 01:11, 25 September 2024
• Symmedians, Lemoine point
  Denote <math>T</math> the midpoint <math>FE \implies OT \perp FE.</math>
  30 KB (5,152 words) - 17:38, 6 August 2024
• 1959 AHSME Problems/Problem 32
  // Segments OT, OA ...tangent to a circle, we know that <math>OB=OT=r</math> and <math>\overline{OT} \perp \overline{TA}</math>. By the [[Pythagorean Theorem]], <math>OA=\sqrt
  2 KB (311 words) - 21:15, 20 July 2024