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• Power of a Point Introductory Problem 4
  #REDIRECT [[Power of a Point Theorem/Introductory Problem 4]]
  61 bytes (8 words) - 18:49, 10 July 2006
• Set
  ==Power Sets== {{main|power set}}
  11 KB (2,021 words) - 00:00, 17 July 2011
• 2006 USAMO Problems
  ...> or to <math>n+2^{m_n+1}</math> where <math>2^{m_n}</math> is the largest power of 2 that is a factor of <math>n</math>. Show that if <math>k\ge2</math> is
  3 KB (520 words) - 09:24, 14 May 2021
• 2022 USAJMO Problems/Problem 1
  ...other. Indeed, if m is square-free, then each prime dividing m only has a power of 1 in the prime factorization, so given (4) that m|<math>x\cdot (r-1)^2</
  5 KB (883 words) - 01:05, 2 June 2024
• 2005 AIME II Problems/Problem 4
  If <math>n</math> is the power of a single prime, then there are 11 possibilities (<math>2^1</math> to <ma
  3 KB (377 words) - 18:36, 1 January 2024
• 2005 AIME II Problems/Problem 5
  ...> and <math>3^6</math>. However, the ordered pairs where b is to the sixth power are distinct, so they are not redundant. (For example, the pairs (4, 64) an
  3 KB (547 words) - 19:15, 4 April 2024
• 2005 AIME II Problems/Problem 8
  ...until it reaches the circle on both sides; call them <math>P,Q</math>. By Power of a Point,
  4 KB (693 words) - 13:03, 28 December 2021
• Problem solving
  ...es, students’ formula sheets were the source of knowledge, the source of power that fueled the top students and the top schools. They were studied, memori
  6 KB (1,039 words) - 17:43, 30 July 2018
• Euler's identity
  *[[Power series]]
  3 KB (452 words) - 23:17, 4 January 2021
• 2005 AIME I Problems/Problem 6
  ...1)^4</math>. Thus, we add 1 to each side in order to complete the fourth power and get If we don't see the fourth power, we can always factor the LHS to try to create a quadratic substitution. Ch
  4 KB (686 words) - 12:52, 13 June 2024
• 2005 AIME I Problems/Problem 15
  ...ength of the median be <math>3m</math>. Then by two applications of the [[Power of a Point Theorem]], <math>DE^2 = 2m \cdot m = AF^2</math>, so <math>DE = Our earlier result from Power of a Point was that <math>2m^2 = (10 - c)^2</math>, so we combine these two
  5 KB (906 words) - 23:15, 6 January 2024
• 2004 AIME I Problems/Problem 14
  Note that by Power of a Point, the point the unicorn is at has power <math>4 \cdot 20 = 80</math> which implies that the tangent from that point
  4 KB (729 words) - 01:00, 27 November 2022
• 2004 AIME II Problems/Problem 10
  ...so if we raise <math>2^{3}</math>, which we know already works, to an odd power, we will also satisfy the congruence. Thus, <math>2^{3}, 2^{9}, 2^{15},</ma
  8 KB (1,283 words) - 19:19, 8 May 2024
• 2004 AIME II Problems/Problem 8
  For a single power of 2004, we have three choices (2, 3, and 167) to give a power of 2003 to. ...501, there are three choices to give a power of 500 to and the rest get a power of 1.
  2 KB (353 words) - 18:08, 25 November 2023
• 1998 AIME Problems
  ...re <math>i^2 = - 1.</math> Let <math>S_n</math> be the sum of the complex power sums of all nonempty [[subset]]s of <math>\{1,2,\ldots,n\}.</math> Given t
  7 KB (1,084 words) - 02:01, 28 November 2023
• 1983 AIME Problems/Problem 1
  ...w</math>, and <math>(xyz)^{12}=w</math>. If we now convert everything to a power of <math>120</math>, it will be easy to isolate <math>z</math> and <math>w< ...ng both sides of <math>y^{40}=w</math> to the <math>\frac{12}{40}</math>th power gives <math>y^{12}=w^{\frac{3}{10}}</math>.
  4 KB (642 words) - 03:14, 17 August 2022
• 1983 AIME Problems/Problem 9
  The derivative of <math>f(y)</math>, using the Power Rule, is
  4 KB (722 words) - 20:25, 14 January 2023
• 1983 AIME Problems/Problem 12
  Since H is the midpoint of <math>CD</math>, by [[Power of a Point]], <math>CH^2=(AH)(BH)</math>. Because <math>AH=r-OH</math> and
  2 KB (412 words) - 18:23, 1 January 2024
• 1983 AIME Problems/Problem 14
  Thus by Power of a Point in the circle passing through <math>Q</math>, <math>R</math>, an
  13 KB (2,151 words) - 17:48, 27 May 2024
• 1983 AIME Problems/Problem 15
  Suppose that <math>\Delta > 0</math>, which would mean that there could be two real roots of <math>f(x)</math>, one lying in the i ...which does not make sense in the original problem statement. (For it would mean that the point <math>A</math> lies in the half-plane above the line <math>3
  20 KB (3,497 words) - 15:37, 27 May 2024

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