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• 2013 AMC 12A Problems/Problem 17
  ...le is multiplied by some <math>\frac{n}{12}</math>. This means the highest power of 2 that divides the number of coins is continually decreasing or staying
  3 KB (502 words) - 01:36, 11 October 2020
• 2013 AMC 12A Problems/Problem 19
  ...ct <math>AC</math> at <math>D</math> and <math>E</math> as shown. We apply Power of a Point on point <math>C</math> with respect to circle <math>A.</math> T ...h>Y</math> on <math>AC</math>. Then <math>AZ = AY = 86</math>. Using the Power of a Point, we get that <math>p(p+q) = 11(183) = 11 * 3 * 61</math>. We kn
  3 KB (546 words) - 15:24, 19 September 2021
• 2013 AMC 10A Problems/Problem 23
  ==Solution 1 (Number Theoretic Power of a Point)== ...Then <math>AZ = AY = 86</math>. Using the Power of a Point (Secant-Secant Power Theorem), we get that <math>p(p+q) = 11(183) = 11 * 3 * 61</math>. We know
  5 KB (846 words) - 23:02, 21 August 2023
• 2013 AMC 12B Problems/Problem 19
  ==Solution 4 (Power of a Point)== ...point where the circle intersects <math>AC</math> be <math>G</math>. Using power of a point, we can write the following equation to solve for <math>AG</math
  6 KB (1,004 words) - 22:38, 18 June 2023
• 2013 AIME II Problems/Problem 15
  ==Video Solution by The Power Of Logic==
  8 KB (1,424 words) - 17:13, 28 June 2024
• 2013 AIME II Problems/Problem 2
  ...h> and <math>b</math> to be integers. Thus, <math>2^a</math> can only be a power of <math>2</math>. To help us see the next step, we factorize <math>1000</m
  2 KB (415 words) - 23:19, 10 January 2024
• 2013 USAMO Problems/Problem 1
  ...C</math> again at <math>D</math> and <math>E</math>, respectively. Then by Power of a Point, we have <cmath>XM \cdot XE = XZ \cdot XP \quad\text{and}\quad X
  14 KB (1,830 words) - 18:22, 10 May 2023
• 2013 USAMO Problems/Problem 6
  Now, <math>X_1 X^2 = XA \cdot XP = X_2 X^2</math> by Power of a Point, and so <math>X</math> is the midpoint of <math>X_1 X_2</math>.
  4 KB (691 words) - 18:29, 10 May 2023
• 2013 USAJMO Problems/Problem 1
  ...v 7\pmod 9</math>. Note that since <math>a^6b^6</math> is a perfect sixth power, and since neither <math>a</math> nor <math>b</math> contains a factor of < ...3 - 3</math>. If it has power <math>r_1</math> in <math>x^3 - 3</math> and power <math>r_2</math> in <math>y^3 - 3</math>, then <math>5r_1</math> - <math>r_
  9 KB (1,494 words) - 02:45, 10 May 2024
• 2013 USAJMO Problems/Problem 4
  ...h>, where <math>\textrm{pow}_{\textrm{larg}}</math> represents the largest power of <math>2</math> that is smaller than <math>n</math>. I will call this sum ...Already it looks like <math>f(n)</math> is only odd if <math>n=2^{\textrm{power}}-1</math>.
  6 KB (1,069 words) - 14:05, 28 November 2022
• 2013 IMO Problems/Problem 4
  ...th>. Furthermore, we are given that <math>NTWB</math> is cyclic. Hence, by Power of a Point, The converse of Power of a Point then proves <math>THLW</math> cyclic.
  11 KB (1,991 words) - 01:31, 19 November 2023
• Mock AIME I 2015 Problems
  (b) <math>(a+7b)c</math> is a power of <math>2</math>,
  9 KB (1,463 words) - 14:48, 12 February 2017
• 2016 AMC 10A Problems
  The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 5 ...r variables <math>a, b,c,</math> and <math>d</math>, each to some positive power. What is <math>N</math>?
  13 KB (1,979 words) - 11:34, 27 June 2024
• 1974 IMO Problems/Problem 2
  ...</math> on side <math>AB</math> such that <math>CD</math> is the geometric mean of <math>AD</math> and <math>DB</math> if and only if <math>\sin{A}\sin{B} Suppose that <math>DC</math> is the geometric mean of <math>DA,DB</math>.
  3 KB (553 words) - 15:56, 29 January 2021
• 2013 AMC 8 Problems/Problem 10
  Then, to find the LCM, we have to find the greatest power of all the numbers there are; if one number is one but not the other, use i For the GCF of 180 and 594, use the least power of all of the numbers that are in both factorizations and multiply. <math>3
  2 KB (293 words) - 22:42, 5 January 2024
• 1967 AHSME Problems/Problem 32
  ...math>AO \cdot OC = BO \cdot OD</math>, <math>ABCD</math> is cyclic through power of a point. From the given information, we see that <math>\triangle{AOB}\si
  4 KB (560 words) - 18:56, 17 December 2023
• 2014 AMC 10A Problems/Problem 25
  ...een how many consecutive powers of <math>5</math> are there <math>3</math> power of <math>2</math>s The first power of <math>2</math> is between <math>5^n</math> and <math>2 \cdot 5^n</math>.
  3 KB (399 words) - 15:51, 15 August 2023
• 2000 PMWC Problems/Problem I4
  ...teger that, when multiplied by <math>75600</math>, yields a perfect fourth power. The prime factorization of <math>75600</math> is <math>2^4 * 3^3 * 5^2 * 7
  632 bytes (93 words) - 12:03, 23 December 2019
• 2000 PMWC Problems/Problem T1
  The LCM is determined by taking the highest power of each prime that appears in the factorizations:
  3 KB (487 words) - 16:44, 24 May 2024
• 2014 AMC 12B Problems
  A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible ...k}</math>, the number <math>e^p</math> is an integer. What is the largest power of 2 that is a factor of <math>e^p</math> ?
  13 KB (2,066 words) - 14:08, 1 November 2022

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