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• 1992 AHSME Problems/Problem 17
  ...th>N=192021\cdots9192</math>. Suppose that <math>3^k</math> is the highest power of 3 that is a factor of <math>N</math>. What is <math>k</math>?
  2 KB (342 words) - 13:19, 4 January 2021
• 1993 USAMO Problems/Problem 5
  or, after taking both sides to the <math>(n^2 - 1)</math> power, simplifying, and taking the <math>n</math>-th root of both sides, to prove
  3 KB (598 words) - 17:08, 13 August 2023
• 2012 AIME I Problems/Problem 12
  ..., by the Pythagorean Theorem, <math>EQ=\frac{13}{\sqrt{3}}</math>. Now, by Power of a Point, we know that <math>(DE)(BE)=(EQ)(EC)</math>, which means that
  9 KB (1,523 words) - 12:23, 7 September 2022
• 2012 AIME II Problems/Problem 13
  ...complex plane geometry, which is similar to vector geometry only with the power of easy rotation. Place the triangle in the complex plane by letting <math
  13 KB (2,052 words) - 18:02, 5 February 2024
• 2012 AIME II Problems/Problem 15
  ...ath>, and use Stewart's Theorem to find <math>AD=\tfrac{15}{8}</math>. Use Power of Point <math>D</math> to find <math>DE=\tfrac{49}{8}</math>, and so <math ...ath>, and use Stewart's Theorem to find <math>AD=\tfrac{15}{8}</math>. Use Power of Point <math>D</math> to find <math>DE=\tfrac{49}{8}</math>, and so <math
  13 KB (2,298 words) - 12:56, 10 September 2023
• Mock AIME I 2012 Problems/Problem 7
  ...d <math>XQ=YQ=1</math>. Let <math>P_{O_1}(P), P_{O_2}(P)</math> denote the power of <math>P</math> with respect to the circles <math>O_1, O_2</math>, respec
  2 KB (380 words) - 17:38, 7 April 2012
• Mock AIME I 2012 Problems/Problem 8
  ...\frac{1+\sqrt{5}}{2},\text{ the golden ratio.} </cmath> Taking the fourth power, the desired answer is <math>\dfrac{7+\sqrt{45}}{2} \implies r+s+t=\boxed{0
  3 KB (543 words) - 18:51, 7 May 2020
• Mock AIME I 2012 Problems/Problem 12
  ...>, the minimal polynomial for which is <math>x^2 + x + 1</math>. Since any power of this base polynomial will work, <math>P(x) = (x^2 + x + 1)^5</math>, mak
  1 KB (196 words) - 01:56, 25 November 2016
• Mock AIME I 2012 Problems/Problem 13
  ...have <math>p \mid n - p</math> and so <math>n - p</math> is even but not a power of <math>2</math>. By induction, all such positions are wins for the start If <math>n</math> is even but not a power of <math>2</math>, <math>n</math> has an odd factor <math>p > 1</math>. Th
  2 KB (445 words) - 17:43, 7 April 2012
• 1998 USAMO Problems/Problem 2
  .../math>. Therefore, <math>DM = x - 1</math> and <math>MC = 4 - x</math>. By Power of a Point, <math>AE \cdot AF = AB^2 = 4</math>, so <math>AF = \frac{4}{AE}
  4 KB (656 words) - 17:26, 20 June 2019
• 2012 USAMO Problems/Problem 3
  ...}\right\}</math>, where <math>p^{m_p\left(i\right)}</math> is the greatest power of <math>p</math> that divides <math>i</math>, gives a valid sequence. Ther
  4 KB (790 words) - 06:38, 27 October 2022
• 2012 USAJMO Problems/Problem 1
  ...same circle. Since <math>AP = AQ</math>, <math>AP^2 = AQ^2</math>, so the power of A with respect to both circumcircles is the same. Thus, <math>A</math> l ..., with <math>M</math> on the circumcircle of triangle <math>PRS</math>. By Power of a Point, <math>AQ^2 = AM \cdot AS</math> and <math>AP^2 = AN \cdot AS</m
  4 KB (613 words) - 20:50, 19 December 2023
• 2006 Romanian NMO Problems/Grade 10/Problem 1
  ...osed of <math>n</math> elements and let <math>\mathcal P (M)</math> be its power set. Find all functions <math>f : \mathcal P (M) \to \{ 0,1,2,\ldots,n \}</
  4 KB (796 words) - 10:53, 8 May 2012
• 1988 USAMO Problems/Problem 5
  The <math>32</math>nd power of each of these roots is just <math>1</math>, hence the sum of the <math>3 ...id the question of what an infinite product means in the context of formal power series, we could instead view the problem statement as saying that
  8 KB (1,348 words) - 09:44, 25 June 2022
• CSS
  ...the most preferred style, although white-on-black (reverse video) can save power on the reader's computer, depending on the display technology. Don't make a
  5 KB (846 words) - 22:23, 13 July 2022
• 1954 AHSME Problems
  ...as the fifth root of <math>z</math>, then <math>x</math> varies as the nth power of <math>z</math>, where n is:
  23 KB (3,535 words) - 16:29, 24 April 2020
• 2010-2011 Mock USAJMO Problems/Solutions/Problem 1
  Coordinate bash with the origin as the midpoint of BC using Power of a Point.
  558 bytes (94 words) - 12:24, 7 January 2018
• 2020 Mock Combo AMC 10
  ...teger coefficients for all <math>0 \leq i \leq 45</math>. Find the largest power of two that evenly divides the expected value of <math>\sum_{i=0}^{44} a_i<
  15 KB (2,452 words) - 03:03, 4 July 2020
• 2012 AMC 8 Problems/Problem 12
  ...pattern, we can subtract 1 from 2012 and divide by 4. The remainder is the power of three that we are looking for, plus one. <math>2011</math> divided by <m
  2 KB (293 words) - 18:59, 15 April 2023
• 2012 AMC 12B Problems/Problem 24
  Now let's look at the other cases. Any first power of prime in a prime factorization will not contribute the unboundedness bec
  6 KB (969 words) - 10:06, 5 November 2021

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