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• 1960 IMO Problems/Problem 3
  <cmath>\tan {\angle PAR} = \tan (\angle RAD - \angle PAD) = \frac{\frac{PR}{h}}{1 + \frac{DP \cdot DR}{h^2}} = \frac{PR \cdot h}{h^2
  3 KB (501 words) - 23:14, 16 May 2015
• 2003 USAMO Problems/Problem 2
  ...2} AB \cdot AP \sin BAP}{\frac{1}{2} AP \cdot AD \sin PAD} = \frac{[BAP]}{[PAD]} = \frac{BP}{PD} </math> is rational. Define <math>r</math> to be equal t
  2 KB (335 words) - 19:50, 29 April 2014
• 2008 AMC 12B Problems/Problem 25
  Also, <cmath>\sin(\angle PAD)=\sin(\frac12\angle XDA)=\sqrt{\frac{1-\cos(\angle XDA)}{2}}=\sqrt{\frac{3}
  13 KB (2,244 words) - 21:31, 13 October 2024
• 1998 AHSME Problems/Problem 28
  ...ath> bisects <math>\angle CAD</math>. Thus, angles <math>CAP</math>, <math>PAD</math>, and <math>DAB</math> are congruent. Applying the angle bisector the
  4 KB (662 words) - 23:51, 2 October 2023
• 2004 USAMO Problems/Problem 2
  ...tainable if the sequence is of a length which is a power of 2. If not, we "pad" the sequence with many copies of an existing element of the sequence until
  3 KB (508 words) - 13:23, 17 July 2014
• 2011 USAMO Problems/Problem 5
  By the terms of the problem, <math>S=\frac{\sin \angle PBC}{\sin \angle PAD}\cdot\frac{\sin \angle PDA}{\sin \angle PCB}\cdot\frac{\sin \angle PCD}{\si ...c{\sin \angle PBC}{\sin \angle PCB}\cdot\frac{\sin \angle PDA}{\sin \angle PAD}\cdot\frac{\sin \angle PCD}{\sin \angle PDC}\cdot\frac{\sin \angle PAB}{\si
  4 KB (660 words) - 00:04, 15 February 2024
• 1996 AHSME Problems/Problem 9
  Since the two planes are perpendicular, it follows that <math>\triangle PAD</math> is a [[right triangle]]. Thus, <math>PD = \sqrt{PA^2 + AD^2} = \sqrt
  3 KB (425 words) - 13:07, 5 July 2013
• ASIA TEAM Problems
  ...umps to lily pad <math>\tfrac x2</math> if <math>x</math> is even and lily pad <math>x+1</math> if <math>x</math> is odd. ...mps. In particular, <math>o(1)=0</math> because Kelvin is already at lily pad <math>1</math>.
  10 KB (1,710 words) - 22:23, 10 January 2020
• 2014 AMC 12B Problems
  ...pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that th
  13 KB (2,066 words) - 13:08, 1 November 2022
• 2014 AMC 10B Problems
  ...</math> it will be eaten by a patiently waiting snake. If the frog reaches pad <math>10</math> it will exit the pond, never to return. What is the probabi
  13 KB (2,011 words) - 20:54, 8 November 2022
• 2014 AMC 12B Problems/Problem 22
  ...pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that th ...ath>P(N)</math> to be the probability that the frog survives starting from pad N.
  7 KB (1,082 words) - 11:12, 2 August 2024
• 2019 AMC 12B Problems
  ...eaches pad <math>10</math> without landing on either pad <math>3</math> or pad <math>6</math>?
  16 KB (2,480 words) - 17:31, 16 June 2024
• 2019 AMC 12B Problems/Problem 16
  ...eaches pad <math>10</math> without landing on either pad <math>3</math> or pad <math>6</math>? ...t if Fiona jumps over the predator on pad <math>3</math>, she must land on pad <math>4</math>. Similarly, she must land on <math>7</math> if she makes it
  8 KB (1,274 words) - 08:32, 15 September 2024
• 2019 AIME II Problems
  ...th> and independently of other jumps. The probability that the frog visits pad <math>7</math> is <math>\tfrac{p}{q}</math>, where <math>p</math> and <math
  7 KB (1,254 words) - 13:45, 21 August 2023
• 2019 AIME II Problems/Problem 2
  ...th> and independently of other jumps. The probability that the frog visits pad <math>7</math> is <math>\tfrac{p}{q}</math>, where <math>p</math> and <math .../math> be the probability the frog visits pad <math>7</math> starting from pad <math>n</math>. Then <math>P_7 = 1</math>, <math>P_6 = \frac12</math>, and
  3 KB (526 words) - 20:27, 24 October 2023
• 2022 AIME I Problems/Problem 3
  ...ng \angle APP'</math> by interior angles and <math>\angle PAB \cong \angle PAD</math> by the problem statement. Thus, <math>\triangle P'AP</math> is isosc ...th>\angle DAB + \angle ADC = 180^{\circ}</math>, <math>\angle ADP + \angle PAD = 90^{\circ}</math>. Therefore, triangles <math>APD</math>, <math>APZ</math
  24 KB (3,832 words) - 19:59, 2 March 2024
• 2020 IMO Problems/Problem 1
  <cmath>\angle PAD : \angle PBA : \angle DPA = 1 : 2 : 3 = \angle CBP : \angle BAP : \angle BP Let <math>O</math> be the circumcenter of <math>\triangle ABP, \angle PAD = \alpha, OE</math> is the perpendicular bisector of <math>AP,</math> and p
  3 KB (481 words) - 23:20, 13 November 2024
• 2020 IMO Problems
  <cmath>\angle PAD : \angle PBA : \angle DPA = 1 : 2 : 3 = \angle CBP : \angle BAP : \angle BP
  4 KB (607 words) - 11:15, 21 May 2021
• 2023 AIME I Problems/Problem 5
  ...5.</math> Then, <math>\angle PAB = 135-x</math>, <math>\angle PCD = \angle PAD = (135-x)-90 = 45-x</math>, and <math>\angle PDC = 90+x.</math> Letting <ma
  22 KB (3,480 words) - 18:33, 1 August 2024
• 2023 AMC 8 Problems
  ...left. What is the fewest number of jumps Greta must make to reach the lily pad located <math>2023</math> pads to the right of her starting point?
  29 KB (4,048 words) - 20:26, 1 October 2024

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