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• Power Mean Inequality
  ...'' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] Inequality. ...ights <math>w_i</math> with sum <math>\sum_{i=1}^n w_i=1</math>, the power mean with exponent <math>t</math>, where <math>t\in\mathbb{R}</math>, is defined
  3 KB (606 words) - 23:59, 1 July 2022
• Power of a Point Theorem
  The '''Power of a Point Theorem''' is a relationship that holds between the lengths of t ...ugh <math>P</math> that intersects the circle. This constant is called the power of point <math>P</math>. For example, in the figure below
  5 KB (827 words) - 17:30, 21 February 2024
• Reader's Digest National Word Power Challenge
  The '''Reader's Digest National Word Power Challenge''' is the first nationwide vocabulary competition for middle scho ...dependencies thereof, who have not previously won a scholarship from Word Power, can participate.
  2 KB (293 words) - 21:13, 20 March 2008
• Power of a Point Theorem/Introductory Problem 1
  Applying the [[Power of a Point Theorem]], we get <math> 3\cdot(3+5) = x (x+10) \rightarrow x^2 ''Back to the [[Power of a Point Theorem]].''
  448 bytes (67 words) - 15:15, 23 March 2020
• Power of a Point Theorem/Introductory Problem 2
  Applying the Power of a Point Theorem gives <math> 6\cdot x = 4\cdot 1 </math>, so <math> x = ''Back to the [[Power of a Point Theorem]].''
  289 bytes (45 words) - 13:14, 16 July 2017
• Power of a Point Theorem/Introductory Problem 3
  From the Power of a Point Theorem, we have that ''Back to the [[Power of a Point Theorem]].''
  969 bytes (154 words) - 14:40, 3 July 2006
• Power of a Point Theorem/Introductory Problem 4
  ...3 </math> (or by just knowing your [[Pythagorean Triple]]s). Applying the Power of a Point Theorem gives <math> AE\cdot BE = CE\cdot DE </math>, or <math> ''Back to the [[Power of a Point Theorem]].''
  1 KB (177 words) - 02:14, 26 November 2020
• Power of a point
  #REDIRECT [[Power of a Point Theorem]]
  38 bytes (6 words) - 23:18, 30 June 2006
• Power of a Point Introductory Problem 3
  #REDIRECT [[Power of a Point Theorem/Introductory Problem 3]]
  61 bytes (8 words) - 12:07, 10 July 2006
• 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
• Power of a Point Theorem Introductory Problem 1
  #REDIRECT [[Power of a Point Theorem/Introductory Problem 1]]
  61 bytes (8 words) - 09:19, 1 July 2006
• Power of a Point Theorem Introductory Problem 2
  #REDIRECT [[Power of a Point Theorem/Introductory Problem 2]]
  61 bytes (8 words) - 09:25, 1 July 2006
• Power of a Point Theorem Introductory Problem 3
  #REDIRECT [[Power of a Point Theorem/Introductory Problem 3]]
  61 bytes (8 words) - 09:27, 1 July 2006
• Power of a Point Theorem Introductory Problem 4
  #REDIRECT [[Power of a Point Theorem/Introductory Problem 4]]
  61 bytes (8 words) - 09:27, 1 July 2006
• Power set
  The '''power set''' of a given [[set]] <math>S</math> is the set <math>\mathcal{P}(S)</m Similarly, for any [[finite]] set with <math>n</math> elements, the power set has <math>2^n</math> elements.
  4 KB (757 words) - 11:44, 8 March 2018
• Perfect power
  ...math> is a perfect <math>2</math>nd, <math>3</math>rd and <math>6</math>th power. ...h>st power" is a meaningless property: every integer is a <math>1</math>st power of itself.
  870 bytes (148 words) - 16:52, 18 August 2013
• Power's of 2 in pascal's triangle
  2 KB (341 words) - 16:57, 16 June 2019
• Power
  ...oosely defined as the speed something can do [[work]]. The [[SI]] unit for power is the [[Watt]]. ...d of [[time]]. It is also the [[derivative]] of work. If <math>P</math> is power, <math>W</math> is work, and <math>t</math> is time, then:
  467 bytes (77 words) - 23:26, 2 March 2008
• Power mean inequality
  #REDIRECT [[Power Mean Inequality]]
  35 bytes (4 words) - 21:24, 20 December 2007
• Besot Power Series
  Besot's Power Series Theorem states that
  725 bytes (145 words) - 15:51, 17 February 2016
• Power of a Point
  #REDIRECT [[Power of a Point Theorem]]
  38 bytes (6 words) - 16:53, 15 April 2019
• Root-mean power
  ...Geometric Mean: 0 (theoretical, can't be solved using radicals), Harmonic Mean: -1
  539 bytes (92 words) - 01:01, 2 November 2023
• 2023 Power
  * [https://www.stanfordmathtournament.com/pdfs/smt2023/power-problems.pdf/ Problems] * [https://www.stanfordmathtournament.com/pdfs/smt2023/power-solutions.pdf/ Solutions]
  171 bytes (18 words) - 21:17, 20 January 2024
• 2022 Power
  * [https://www.stanfordmathtournament.com/pdfs/smt2022/power-problems.pdf/ Problems] * [https://www.stanfordmathtournament.com/pdfs/smt2022/power-solutions.pdf/ Solutions]
  171 bytes (18 words) - 21:30, 20 January 2024
• 2021 Power
  * [https://www.stanfordmathtournament.com/pdfs/smt2021/power-problems.pdf/ Problems] * [https://www.stanfordmathtournament.com/pdfs/smt2021/power-solutions.pdf/ Solutions]
  171 bytes (18 words) - 21:39, 20 January 2024
• 2020 Power
  * [https://www.stanfordmathtournament.com/pdfs/smt2020/power-problems.pdf/ Problems] * [https://www.stanfordmathtournament.com/pdfs/smt2020/power-solutions.pdf/ Solutions]
  171 bytes (18 words) - 22:28, 22 January 2024
• 2019 Power
  * [https://www.stanfordmathtournament.com/pdfs/smt2019/power-problems.pdf/ Problems] * [https://www.stanfordmathtournament.com/pdfs/smt2019/power-solutions.pdf/ Solutions]
  171 bytes (18 words) - 22:34, 22 January 2024
• 2018 Power
  * [https://www.stanfordmathtournament.com/pdfs/smt2018/power-problems.pdf/ Problems] * [https://www.stanfordmathtournament.com/pdfs/smt2018/power-solutions.pdf/ Solutions]
  171 bytes (18 words) - 22:20, 29 January 2024
• 2014 Power
  * [https://www.stanfordmathtournament.com/pdfs/smt2014/power-problems.pdf/ Problems] * [https://www.stanfordmathtournament.com/pdfs/smt2014/power-solutions.pdf/ Solutions]
  171 bytes (18 words) - 22:25, 29 January 2024
• 2013 Power
  * [https://www.stanfordmathtournament.com/pdfs/smt2013/power-problems.pdf/ Problems] * [https://www.stanfordmathtournament.com/pdfs/smt2013/power-solutions.pdf/ Solutions]
  171 bytes (18 words) - 22:20, 5 February 2024
• 2012 Power
  * [https://www.stanfordmathtournament.com/pdfs/smt2012/power-problems.pdf/ Problems] * [https://www.stanfordmathtournament.com/pdfs/smt2012/power-solutions.pdf/ Solutions]
  171 bytes (18 words) - 22:25, 5 February 2024
• 2011 Power
  * [https://www.stanfordmathtournament.com/pdfs/smt2011/power-problems.pdf/ Problems] * [https://www.stanfordmathtournament.com/pdfs/smt2011/power-solutions.pdf/ Solutions]
  171 bytes (18 words) - 22:29, 5 February 2024
• Power of a point theorem
  6 KB (1,010 words) - 02:38, 7 May 2024

Page text matches

• Pythagorean Theorem
  ...> and <math>AX = AB + BC </math>. By considering the [[Power of a Point | power of point]] <math>A </math> with respect to <math>\omega </math>, we see
  5 KB (886 words) - 21:12, 22 January 2024
• Physics
  ...the study of movement. [[Kinematics]], mechanical [[force]]s, [[work]], [[power]], [[energy]], and [[matter]] are all part of mechanics. ...defined as <math>P =\int^{v_f}_{v_i} F\,dv</math> where <math>P</math> is power delivered and <math>v</math> is velocity. [[Energy]] is the other basic int
  9 KB (1,355 words) - 07:29, 29 September 2021
• Inequality
  * [[Arithmetic Mean-Geometric Mean | Arithmetic Mean-Geometric Mean Inequality]] * [[Power mean inequality]]
  12 KB (1,798 words) - 16:20, 14 March 2023
• 2015 IMO Problems
  is a power of 2. (<i>A power of 2 is an integer of the form <math>2^n</math> where <math>n</math> is a n
  4 KB (692 words) - 22:33, 15 February 2021
• American Regions Mathematics League
  *A power round, where the questions are proof-oriented and are in a theme
  2 KB (267 words) - 17:06, 7 March 2020
• Joining an ARML team
  ...ractices which are usually composed of individual tests, team tests, and a power round test. ...n Individual Exam, Individual Short Answer Section, Team Questions, a Team Power Question, and 2 sets of relays of 5 each (there are 10 members in each team
  21 KB (3,500 words) - 18:41, 23 April 2024
• Princeton University Mathematics Competition
  ...ustification (referring to previous parts is allowed). Groups turn in the Power Round solutions the day of the event.
  2 KB (295 words) - 23:19, 5 January 2019
• Power Mean Inequality
  ...'' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] Inequality. ...ights <math>w_i</math> with sum <math>\sum_{i=1}^n w_i=1</math>, the power mean with exponent <math>t</math>, where <math>t\in\mathbb{R}</math>, is defined
  3 KB (606 words) - 23:59, 1 July 2022
• Polynomial
  ...mation that one can have about a polynomial of one variable is the highest power of the variable which appears in the polynomial. This number is known as t
  6 KB (1,100 words) - 01:44, 17 January 2024
• Factorial
  ...n</math>, and not divisible by any prime <math>p>n</math>. But what is the power of a prime <math>p\le n</math> ...r of <math>p</math>. Those divisible by <math>p^3</math> give yet another power of <math>p</math>. Continuing in this manner gives
  10 KB (809 words) - 16:40, 17 March 2024
• San Diego ARML
  ** Power: 33 ** Power: 17
  2 KB (378 words) - 16:34, 5 January 2010
• Fermat's Little Theorem
  .../math> as a product of cyclic groups of prime order where the set of prime power orders is unique. We can do this because if any two prime powers are not c
  16 KB (2,658 words) - 16:02, 8 May 2024
• Harmonic mean
  ...d of mean (like [[arithmetic mean]] and [[geometric mean]]). The harmonic mean of a [[set]] of <math>n</math> [[positive]] [[real number]]s <math> x_1, x_ ...to avoid division by zero. For instance, if we tried to take the harmonic mean of the set <math>\{-2, 3, 6\}</math> we would be trying to calculate <math>
  1 KB (196 words) - 00:49, 6 January 2021
• Rearrangement Inequality
  ...e to be [[positive]] or negative, unlike the [[Power mean inequality|power-mean]] family of inequalities. *[[Power Mean Inequality]]
  5 KB (804 words) - 13:54, 26 January 2023
• Divisibility rules
  .../math> if and only if the last <math>n</math> digits are divisible by that power of 5. ...zeros that should be at the end of a number for it to be divisible by that power of 10.
  8 KB (1,315 words) - 18:18, 2 March 2024
• Induction
  ...s an emphasis on proof. Induction problems can be found anywhere from the Power Round of the [[American Regions Math League | ARML]] up through the [[Unite ...number of odd binomial coefficients in any row of the Pascal triangle is a power of 2. (1956 Putnam Competition)
  5 KB (768 words) - 20:45, 1 September 2022
• Jensen's Inequality
  ...t examples of Jensen's inequality is the [[quadratic mean]] - [[arithmetic mean]] inequality. Taking <math>F(x)=x^2</math>, which is convex (because <math> Similarly, [[arithmetic mean]]-[[geometric mean]] inequality ([[AM-GM]]) can be obtained from Jensen's inequality by consid
  3 KB (623 words) - 13:10, 20 February 2024
• Generating function
  ...ath>, give the terms of a [[sequence]] which is of interest. Therefore the power series (i.e. the generating function) is <math>c_0 + c_1 x + c_2 x^2 + \cdo ...the result is uninteresting (both the generating function and the desired power series are just <math>1</math>).
  4 KB (659 words) - 12:54, 7 March 2022
• Binomial Theorem
  Since <math>e^{a+b} = e^ae^b</math>, and power series for the same function are termwise equal, the series at <math>x = a
  5 KB (935 words) - 13:11, 20 February 2024
• Prime number
  ...prime, ''n'' must not have any odd divisor larger than 1 and so must be a power of 2. Therefore all Fermat primes have the form <math>2^{2^n}+1</math>. [[
  6 KB (985 words) - 12:38, 25 February 2024
• Circle
  *Lengths of chords can be calculated by using the [[Power of a point]] theorem. * [[Power of a point]]
  9 KB (1,581 words) - 18:59, 9 May 2024
• Divisor
  ...s often useful to know that this expression grows slower than any positive power of <math>{n}</math> as <math>n\to\infty</math>.
  1 KB (274 words) - 19:50, 29 August 2023
• Base numbers
  ...r in 2746 is actually just a placeholder which shows how many of a certain power of 10 there are. The first digit to the left of the decimal place (recall
  4 KB (547 words) - 17:23, 30 December 2020
• National Online Math League
  ...ilar to that used by the [[American Regions Math League]]: a Team test, a Power question, and several Relays. However, one match used an experimental form
  3 KB (452 words) - 11:21, 25 June 2006
• Least common multiple
  ...me number]] which divides any of them, take the largest [[exponentiation | power]] with which it appears, and multiply the results together. For example, t ...; the largest power of 3 that appears is <math>3^1</math>; and the largest power of 5 that appears is <math>5^1</math>. Therefore the LCM, <math>LCM(8, 12,
  2 KB (383 words) - 10:49, 4 September 2022
• Pi
  ...> is Liouvillian, then so is any rational multiple of any positive integer power of <math>x</math> (this is a simple exercise we leave to the reader), so, i Since the largest possible power of a given [[prime number|prime]] <math>p\le n</math> that can divide one o
  8 KB (1,469 words) - 21:11, 16 September 2022
• Logarithm
  ...ogs also allow (with repetition) to turn left to right exponentiation into power towers (especially useful for tetration (exponentiation repetition with the ...to the usual logarithm by the fact that if <math>b</math> isn't an integer power of <math>a</math> then <math>\lceil \log_a(m)\rceil</math> is a lower bound
  4 KB (680 words) - 12:54, 16 October 2023
• Utah ARML
  ...d in the B division, but was mislisted in the A division. However, on the Power Question, the Utah team got the 9th best score in the nation, which would h
  565 bytes (88 words) - 12:29, 11 December 2007
• Vocabulary and spelling competitions
  * [[Reader's Digest National Word Power Challenge | Word Power Challenge]] The premier vocabulary competition, for 6-8th grades. Over 3 mi
  786 bytes (99 words) - 17:53, 22 June 2006
• Vocabulary competitions
  * [[Reader's Digest National Word Power Challenge | Word Power Challenge]] The premier vocabulary competition, for 6-8th grades. Over 3 mi
  392 bytes (50 words) - 02:36, 29 November 2018
• Mathew Crawford
  * Headed up the grading of the Power Round for the Georgia ARML site (2010)
  2 KB (360 words) - 02:20, 2 December 2010
• Fermat's Last Theorem
  ...to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. ''I have discover
  3 KB (453 words) - 11:13, 9 June 2023
• Diophantine equation
  Note that every square, and therefore every fourth power, is either <math>1</math> or <math>0\mod 4</math>. The proof of this is fai
  9 KB (1,434 words) - 13:10, 20 February 2024
• Pascal's triangle
  ...s Theorem, which states that given a prime p and integers m,n, the highest power of p dividing <math>\binom{m}{n}</math> is the number of carries in adding
  5 KB (838 words) - 17:20, 3 January 2023
• Newton's Sums
  ...cient way of finding the sums of [[root]]s of a [[polynomial]] raised to a power. They can also be used to derive several [[factoring]] [[identity|identiti
  4 KB (690 words) - 13:11, 20 February 2024
• Exponentiation
  ...ten in the form <math>a^b</math>, where <math>b</math> is the exponent (or power) and <math>a</math> is the [[base]]. ...times. The base is 3 (what is repeatedly multiplied) and the exponent (or power) is 5 (the number of times to repeat multiplication).
  5 KB (803 words) - 16:25, 10 August 2020
• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  ...root-mean power]], [[arithmetic mean]], [[geometric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> th ...,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math>n</math> is the root mean power.
  5 KB (912 words) - 20:06, 14 March 2023
• Power of a Point Theorem
  The '''Power of a Point Theorem''' is a relationship that holds between the lengths of t ...ugh <math>P</math> that intersects the circle. This constant is called the power of point <math>P</math>. For example, in the figure below
  5 KB (827 words) - 17:30, 21 February 2024
• Alt front page
  ...><br>Cacti is a complete network graphing solution designed to harness the power of RRDTool's data storage and graphing functionality. Cacti provides a fast
  2 KB (329 words) - 17:25, 8 June 2008
• Rational approximation of famous numbers
  ..., that <math>x</math> is not a [[Liouvillian number]], i.e., that for some power <math>M<+\infty</math>, the [[inequality]] <math>\left|x-\frac pq\right|\ge Since the largest possible power of a given prime <math>p\le n</math> that can divide one of the numbers <ma
  8 KB (1,431 words) - 13:48, 26 January 2008
• Reader's Digest National Word Power Challenge
  The '''Reader's Digest National Word Power Challenge''' is the first nationwide vocabulary competition for middle scho ...dependencies thereof, who have not previously won a scholarship from Word Power, can participate.
  2 KB (293 words) - 21:13, 20 March 2008
• Bertrand's Postulate
  Note that the power with which a prime <math>p</math> satisfying <math>\frac{2n}3<p\le n</math>
  2 KB (309 words) - 21:43, 11 January 2010
• Perfect square
  * [[Perfect power]]
  954 bytes (155 words) - 01:14, 29 November 2023
• Uncountable
  ...th>|S|<|\mathcal{P}(S)|</math>, where <math>\mathcal{P}(S)</math> is the [[power set]] of <math>S</math>. First, we note that the [[Cantor set]] <math>\math
  2 KB (403 words) - 20:53, 13 October 2019
• Subset
  The set of all subsets of a given set <math>S</math> is called the [[power set]] of <math>S</math> and is denoted <math>\mathcal{P}(S)</math> or <math
  1 KB (217 words) - 09:32, 13 August 2011
• 2006 AIME I Problems
  ...tive integer <math> x, </math> let <math> g(x) </math> denote the greatest power of 2 that divides <math> x. </math> For example, <math> g(20)=4 </math> and
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 13
  ...ve integer]] <math> x </math>, let <math> g(x) </math> denote the greatest power of 2 that [[divisor | divides]] <math> x. </math> For example, <math> g(20) ...ath> that divides <math>n+1</math>. Thus by the above formula, the highest power of <math>2</math> that divides <math>S_n</math> is <math>2^{k+n-1}</math>.
  10 KB (1,702 words) - 00:45, 16 November 2023
• 2006 AIME I Problems/Problem 9
  The product of <math>a^2</math> and <math>r^{11}</math> is a power of 2. Since both numbers have to be integers, this means that <math>a</mat ...ly not possible. Thus the only restriction r has is that it must be an odd power of 2, so <math>2^{1}</math>, <math>2^{3}</math>, <math>2^{5}</math> .... al
  4 KB (651 words) - 18:27, 22 May 2021
• 2006 AIME I Problems/Problem 4
  A number in decimal notation ends in a zero for each power of ten which divides it. Thus, we need to count both the number of 5s and ...h> - every <math>n!</math> term for <math>n\geq25</math> has an additional power of <math>5</math> dividing it, for <math>76</math> extra; every n! for <mat
  2 KB (278 words) - 08:33, 4 November 2022
• 2006 AMC 12B Problems/Problem 20
  ...eq 0</math> is the lowest value such that <math>4x</math> becomes a higher power of 10.
  3 KB (485 words) - 14:09, 21 May 2021
• 2006 AMC 12B Problems/Problem 22
  The power of <math>10</math> for any factorial is given by the well-known algorithm ...of <math>2</math> that divides <math>n!</math> is larger or equal than the power of <math>5</math> which divides
  5 KB (881 words) - 15:52, 23 June 2021
• 2006 AMC 12A Problems/Problem 17
  ...> is an isosceles right triangle. Thus <math>DG = r\sqrt{2}</math>. By the Power of a Point Theorem,
  6 KB (958 words) - 23:29, 28 September 2023
• 2006 AMC 12A Problems/Problem 24
  ...rent expression is irreducible as each term has a different <math>x</math> power. Thus, when we write <math>a</math> and <math>b</math> back to their origin
  8 KB (1,332 words) - 17:37, 17 September 2023
• 2005 AMC 12B Problems/Problem 23
  We can then put <math>x+y</math> to the third power or <math>(x+y)^{3}=10^{3z}</math>. Basic polynomial multiplication shows us
  5 KB (786 words) - 16:49, 31 January 2023
• Lassiter Invitational
  ...consists of a multiple-choice test, ten ciphering questions, and a pair of power questions, i.e., more in-depth questions on which members of teams collabor
  1 KB (161 words) - 18:35, 25 November 2007
• Power of a Point Theorem/Introductory Problem 1
  Applying the [[Power of a Point Theorem]], we get <math> 3\cdot(3+5) = x (x+10) \rightarrow x^2 ''Back to the [[Power of a Point Theorem]].''
  448 bytes (67 words) - 15:15, 23 March 2020
• Power of a Point Theorem/Introductory Problem 2
  Applying the Power of a Point Theorem gives <math> 6\cdot x = 4\cdot 1 </math>, so <math> x = ''Back to the [[Power of a Point Theorem]].''
  289 bytes (45 words) - 13:14, 16 July 2017
• Power of a Point Theorem/Introductory Problem 3
  From the Power of a Point Theorem, we have that ''Back to the [[Power of a Point Theorem]].''
  969 bytes (154 words) - 14:40, 3 July 2006
• Power of a Point Theorem/Introductory Problem 4
  ...3 </math> (or by just knowing your [[Pythagorean Triple]]s). Applying the Power of a Point Theorem gives <math> AE\cdot BE = CE\cdot DE </math>, or <math> ''Back to the [[Power of a Point Theorem]].''
  1 KB (177 words) - 02:14, 26 November 2020
• Power of a Point Introductory Problem 3
  #REDIRECT [[Power of a Point Theorem/Introductory Problem 3]]
  61 bytes (8 words) - 12:07, 10 July 2006
• 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
• 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) - 01:55, 5 December 2022
• 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 (2,149 words) - 18:44, 5 February 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
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1995 IMO Problems/Problem 2
  By the [[Power Mean Inequality]],
  6 KB (1,122 words) - 12:23, 6 January 2022
• 1985 AIME Problems/Problem 13
  ...ath>n</math>, we can multiply the left integer, <math>100+n^2</math>, by a power of two without affecting the greatest common divisor. Since the <math>n^2</
  4 KB (671 words) - 20:04, 6 March 2024
• 1985 AIME Problems/Problem 7
  ...[perfect power | perfect fourth power]], <math>b</math> is a perfect fifth power, <math>c</math> is a [[perfect square]] and <math>d</math> is a [[perfect c
  1 KB (222 words) - 11:04, 4 November 2022
• 1986 AIME Problems/Problem 11
  We want the coefficient of the <math>y^2</math> term of each power of each binomial, which by the binomial theorem is <math>{2\choose 2} + {3\ ...c{d^2f}{dx^2} = 2\cdot 1 - 3\cdot 2x+\cdots-17\cdot 16x^{15}</math> by the power rule.
  6 KB (872 words) - 16:51, 9 June 2023
• 1986 AIME Problems/Problem 8
  ...question asks for proper divisors, we exclude <math>2^65^6</math>, so each power is actually <math>141</math> times. The answer is thus <math>S = \log 2^{14
  3 KB (487 words) - 20:52, 16 September 2020
• 1986 AIME Problems/Problem 7
  ...math>k</math>th term after the <math>n</math>th power of 3 is equal to the power plus the <math>k</math>th term in the entire sequence. Thus, the <math>100< ...= <math>2187</math>. Writing out more terms of the sequence until the next power of 3 again (81) we can see that the (<math>2^n</math>+<math>2^{n+1}</math>)
  5 KB (866 words) - 00:00, 22 December 2022
• 1987 AIME Problems/Problem 11
  ...of the two factors will be a power of three, and the other will be twice a power of three. <math>(2n + m + 1)</math> will represent the greater factor while
  3 KB (418 words) - 18:30, 20 January 2024
• 1989 AIME Problems/Problem 9
  ...>'s units digit is <math>0, 2, 4, 6,</math> or <math>8.</math> When to the power of <math>5,</math> they each give <math>0, 2, 4, 6,</math> and <math>8</mat
  6 KB (874 words) - 15:50, 20 January 2024
• 1990 AIME Problems/Problem 13
  '''Lemma''': For all positive integers n, there's exactly one n-digit power of 9 that does not have a left-most digit 9 ...ove by contradiction that there must be at least either one or two n-digit power of 9 for all n.
  5 KB (762 words) - 01:18, 10 February 2023
• 1990 AIME Problems/Problem 5
  ...<math>5</math>, the greatest of all the factors, to be raised to the least power. Therefore, <math>n = 2^43^45^2</math> and <math>\frac{n}{75} = \frac{2^43^
  1 KB (175 words) - 03:45, 21 January 2023
• 1990 AIME Problems/Problem 2
  The <math>3/2</math> power is quite irritating to work with so we look for a way to eliminate that. No
  5 KB (765 words) - 23:00, 26 August 2023
• 2006 AMC 10B Problems/Problem 25
  ...visibility by any powers lower than these means indivisibility by a higher power of the prime (for example, indivisibility by <math>2^2=4</math> means indiv
  5 KB (878 words) - 14:39, 3 December 2023
• 1996 AIME Problems/Problem 3
  ...will contain <math>(n+1)^2</math> terms, as each term will have an unique power of <math>x</math> or <math>y</math> and so none of the terms will need to b
  3 KB (515 words) - 04:29, 27 November 2023
• 1998 AIME Problems/Problem 13
  ...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 ...(for now, including the empty subset, which we will just define to have a power sum of zero) with <math>9</math> in it is equal to the number of subsets wi
  2 KB (384 words) - 19:02, 20 October 2023
• 1998 AIME Problems/Problem 1
  ...rs an be found by writing out their factorizations and taking the greatest power for each factor. <math>[6^6,8^8] = 2^{24}3^6</math>. Therefore <math>12^{12 ...</math> wouldn't be <math>12</math>) and <math>0\le a\le 24</math> (or the power of <math>2</math> in the <math>\operatorname{lcm}</math> would be <math>a</
  2 KB (289 words) - 22:50, 23 April 2024
• 1999 AIME Problems/Problem 4
  ..., etc. are congruent by symmetry (you can prove it rigorously by using the power of a point to argue that exactly two chords of length <math>1</math> in the
  3 KB (398 words) - 13:27, 12 December 2020
• 2000 AIME I Problems/Problem 15
  To simplify matters, we want a power of <math>2</math>. Hence, we will add <math>48</math> 'fake' cards which we ...<math>31-15=16</math> cards remaining. Since <math>16</math> is a perfect power of <math>2</math>, we will not need to worry about this scenario again in t
  15 KB (2,673 words) - 19:16, 6 January 2024
• 2000 AIME I Problems/Problem 11
  ...hen divide by 1000. To do this, write the corresponding divisor under each power. e.g. 2 - 500, 4 - 250, 5 - 200, etc. Call this the "partner" of any diviso ...minator, then every power of five will be multiplied by the partner of the power of 2. Essentially, all we have to do is a large scale distributive property
  4 KB (667 words) - 13:58, 31 July 2020
• 2000 AIME I Problems/Problem 1
  ...>2</math>s and the <math>5</math>s separated, so we need to find the first power of 2 or 5 that contains a 0.
  1 KB (163 words) - 17:44, 16 December 2020
• 2001 AIME I Problems/Problem 5
  ...is a diameter of the unit circle. Then <math>XC=2-2n\sqrt{3}.</math> Using power of a point on X,
  6 KB (1,043 words) - 10:09, 15 January 2024
• 2002 AIME I Problems/Problem 13
  By the [[Power of a Point Theorem]] on <math>E</math>, we get <math>EF = \frac{12^2}{27} = Using Power of a Point, we have
  6 KB (974 words) - 13:01, 29 September 2023
• 2024 AMC 8 Problems/Problem 8
  ==Video Solution 1 (easy to digest) by Power Solve==
  2 KB (384 words) - 22:57, 17 February 2024
• 2003 AIME II Problems/Problem 13
  ====4.1.4: Using the JCF to calculate the transition matrix to the power of any n, large or small==== ...then we must find the sum of the coefficients that share a variable with a power divisible by <math>3</math>.
  15 KB (2,406 words) - 23:56, 23 November 2023
• 2002 AIME II Problems/Problem 14
  ...\triangle MPA</math> are similar. Also note that <math>AM = BM</math> by [[power of a point]]. Using the fact that the ratio of corresponding sides in simil
  4 KB (658 words) - 19:15, 19 December 2021
• 2001 AIME II Problems/Problem 8
  ...proach is to consider the graph of <math>f(x)</math>, which iterates every power of <math>3</math>, and resembles the section from <math>1 \le x \le 3</math
  3 KB (545 words) - 23:41, 14 June 2023
• 2000 AIME II Problems/Problem 13
  ...(or in this case, nearly symmetric) polynomials is to divide through by a power of <math>x</math> with half of the polynomial's degree (in this case, divid ...can see that the number of zeros in a term more or less correlates to the power of <math>x^2</math>. Thus, we let <math>y=10x^2</math>. The equation then b
  6 KB (1,060 words) - 17:36, 26 April 2024
• 2000 AIME II Problems/Problem 4
  ...her is (<math>6 = 2 \cdot 3</math>) a prime raised to the <math>5</math>th power, or two primes, one of which is squared. The smallest example of the former ...e now divide all of the odd factors from <math>n</math>; then we require a power of <math>2</math> with <math>\frac{18}{6} = 3</math> factors, namely <math>
  2 KB (397 words) - 15:55, 11 May 2022
• 2006 USAMO Problems/Problem 5
  ...> 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\ge 2</math> i ...made from a number that is divisible by <math>2^M</math> (and by no higher power of 2). Thus we must have <math>M < i_0</math>, since otherwise a number div
  7 KB (1,280 words) - 17:23, 26 March 2016
• Holomorphic function
  <math>z</math> if <math>f</math> has a convergent [[power series]] expansion on some its power series diverges when <math>\lvert x \rvert > 1</math>. But in the
  9 KB (1,537 words) - 21:04, 26 July 2017
• Minnesota ARML
  ...f Wayzata HS (1991-2006), founder and long-time problem writer of the ARML Power Contest, and Mike Reiners of Christ's Household of Faith School (2007-14),
  4 KB (680 words) - 16:45, 10 June 2015
• New York State Math League
  ...f [[ARML]], with 15-member teams competing in Individual, Team, Relay, and Power rounds, although the scoring is slightly different. Teams compete in A and
  1 KB (201 words) - 14:39, 25 June 2023
• 1977 Canadian MO Problems/Problem 3
  ...est positive integer <math>b</math> for which <math>N</math> is the fourth power of an integer.
  713 bytes (114 words) - 01:45, 19 August 2012
• Mock AIME 1 2006-2007 Problems/Problem 7
  ...h> is cyclic, since <math>\angle AEB=\angle AFB</math>. We then have, from Power of a Point, that <math>CE\cdot CA=CF\cdot CB</math>. In other words, <math> As above, use Power of a Point to compute <math>CF=2</math> and <math>FB=1</math>. Since triang
  3 KB (518 words) - 16:54, 25 November 2015
• Mock AIME 2 2006-2007 Problems/Problem 12
  Also, from the Power of a Point Theorem, <math>DO \cdot BO=AO\cdot CO\Rightarrow CO=\frac{3a}{2}
  2 KB (311 words) - 10:53, 4 April 2012
• Algebra/Olympiad
  ** [[Arithmetic Mean-Geometric Mean | Arithmetic Mean-Geometric Mean Inequality]] ** [[Power Mean Inequality]]
  1 KB (100 words) - 16:22, 18 May 2021
• 1977 Canadian MO Problems
  ...est positive integer <math>b</math> for which <math>N</math> is the fourth power of an integer.
  3 KB (560 words) - 19:23, 10 March 2015
• Mean
  ...] of the set (also known as the [[average]]). For example, the arithmetic mean of the members of the set {3, 5, 10} is ==Mean, Median, Mode==
  1 KB (148 words) - 15:28, 12 November 2023
• 1971 Canadian MO Problems/Problem 1
  ...the circle at <math>P.</math> Let the radius be <math>r.</math> Applying [[power of a point]],
  680 bytes (114 words) - 21:38, 9 July 2019
• 2006 Romanian NMO Problems
  ...n</math> elements and let <math>\displaystyle \mathcal P (M)</math> be its power set. Find all functions <math>\displaystyle f : \mathcal P (M) \to \{ 0,1,2
  11 KB (1,779 words) - 14:57, 7 May 2012
• Geometry/Introduction
  * [[Power of a point]]
  1 KB (122 words) - 16:25, 18 May 2021
• Geometry/Olympiad
  * [[Power of a point theorem]]
  2 KB (242 words) - 10:16, 18 June 2023
• Divisor function
  ...ath>\sigma_k(n)</math> and is defined as the sum of the <math>k</math>th [[power]]s of the [[divisor]]s of <math>n</math>. Thus <math>\sigma_k(n) = \sum_{d ...lities. Likewise, since each divisor can have a power of 3, and since this power can be 0, 1, or 2, we have 3 possibilities. By an elementary [[counting]] p
  3 KB (511 words) - 06:58, 21 May 2009
• Leonhard Euler
  ...of [[mathematics]] called [[analysis]]. His achievements often involved [[power series]]. He is also credited with discovering [[Euler's constant]], denote He also discovered the power series for the [[tangent function|arctangent]], which is
  3 KB (500 words) - 21:28, 15 September 2008
• 1973 IMO Shortlist Problems/Bulgaria 1
  ...</math> is invariant, we may multiply both sides of the condition by that power to obtain
  2 KB (346 words) - 14:59, 30 July 2006
• Navier-Stokes Equation
  ...in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of the effects of pollution, etc. Coupled with [[Max
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• Binary relation
  ...to each other. We could also define a relation <math>\leq</math> on the [[power set]] of a set <math>S</math>, so that <math>(A,B) \in \leq</math>, or <mat
  4 KB (655 words) - 22:33, 18 May 2008
• Perfect cube
  ...raised to the third power. We refer to raising a [[number]] to the third power as ''cubing'' the number. * [[Perfect power]]
  704 bytes (91 words) - 14:12, 24 August 2023
• System of equations
  A system of [[linear]] equations is where all of the variables are to the power 1. There are three elementary ways to solve a system of linear equations.
  5 KB (784 words) - 23:27, 30 July 2020
• Graph of a function
  ...> and <math>x=1</math>. Since the coefficient of the term with the highest power (in this case <math>x^2</math>) is <math>2>0</math>, the graph is above the
  3 KB (497 words) - 05:11, 8 June 2023
• Stanford Mathematics Tournament
  The tournament consists of an individual round, a team round, a power round, and a guts round. '''Power Round'''
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• Divisibility rules/Rule for 5 and powers of 5 proof
  ...y <math>5^n</math> if the last <math>n</math> digits are divisible by that power of 5.
  2 KB (292 words) - 09:58, 17 August 2006
• Power set
  The '''power set''' of a given [[set]] <math>S</math> is the set <math>\mathcal{P}(S)</m Similarly, for any [[finite]] set with <math>n</math> elements, the power set has <math>2^n</math> elements.
  4 KB (757 words) - 11:44, 8 March 2018
• North Fulton Varsity Invitational Math Tournament
  ...rty-question written test, ten ciphering questions, and a proof-based team power round. Calculutors will not be permitted for any part of the competition.
  785 bytes (102 words) - 19:40, 27 August 2006
• Partially ordered set
  ...= (\mathcal{P}(S), \leq)</math> where <math>\mathcal{P}(S)</math> is the [[power set]] (that is, the set of [[subset]]s) of <math>S</math> and for <math>x,
  4 KB (717 words) - 20:01, 25 April 2009
• Rockdale Mathematics Competition
  ...h problem after the first relies on answers to the previous problems. The power round, worth sixty points, is a multi-part question for which the team has
  2 KB (267 words) - 21:50, 6 March 2016
• Boolean lattice
  ...dered set]] whose elements are those of <math>\mathcal{P}(S)</math>, the [[power set]] of <math>S</math>, ordered by inclusion (<math>\subset</math>).
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• 1981 IMO Problems/Problem 4
  ...equal to 3 must be divisible by a number that is greater than two and is a power of a prime. ...note that for any integer not equal to 6 and greater than 4 must have some power-of-a-prime factor greater than 3.
  3 KB (516 words) - 09:43, 28 March 2012
• 1981 IMO Problems/Problem 6
  ...^a 2</math> represents tetration or 2 to the power 2 to the power 2 to the power 2 ... where <math>a</math> amount of 2s. So therefore the answer is <math>f
  2 KB (306 words) - 18:15, 12 April 2024
• Duke Math Meet
  ...t of [[ARML]]. Teams are of six students each. The test commences with a power round, a multi-part proof round that lasts for one hour. This is followed
  1 KB (174 words) - 09:48, 10 November 2012
• Euler line
  The power of point <math>O</math> with respect to <math>\omega_1, \omega_2,</math> an The power of point <math>H</math> with respect to <math>\omega_1</math> is <math>AH \
  59 KB (10,203 words) - 04:47, 30 August 2023
• Division
  When dividing by decimals, multiply both sides by a power of 10 so the divisor is an integer.
  2 KB (259 words) - 09:52, 23 January 2020
• 2003 AMC 10A Problems/Problem 16
  ...on 2000, and starts once again on 2001. As our expression is raised to the power of 2003, we know that the units digit of our expression must end with the t
  1 KB (218 words) - 15:52, 19 August 2023
• Chinese Remainder Theorem
  ...iquity derives from the fact that many results can be easily proven mod (a power of a prime), and can then be generalized to mod <math>m</math> using the Ch
  6 KB (1,022 words) - 14:57, 6 May 2023
• 1985 IMO Problems
  ...a subset of <math>4</math> elements whose product is the <math>4</math>th power of an integer.
  3 KB (465 words) - 03:00, 29 March 2021
• 1985 IMO Problems/Problem 3
  Let <math>k</math> be the largest power of 2 that is less than or equal to <math>i_n</math>. We proceed by inducti
  2 KB (354 words) - 04:56, 11 March 2023
• Perfect power
  ...math> is a perfect <math>2</math>nd, <math>3</math>rd and <math>6</math>th power. ...h>st power" is a meaningless property: every integer is a <math>1</math>st power of itself.
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• 1987 IMO Problems/Problem 3
  We first note that by the [[Power Mean Inequality]], <math> \sum_{i=1}^{n} x_i \le \sqrt{n} </math>. Therefore al
  2 KB (349 words) - 04:36, 28 May 2023
• Cherokee County Open Math Tournament
  .... Calculators are not permitted on any part of the competition except the power round.
  1 KB (178 words) - 21:59, 21 December 2006
• 2002 IMO Shortlist Problems/N3
  ...<math> 2^{p_1} +1 </math> is clearly greater than 3 and cannot be a larger power of three, since this would require <math> p_1 \equiv 3 \pmod{6} </math>. T ...is not the square of the latter (i.e., that the former is either a higher power of the latter, or some other prime divides the latter, either of which impl
  10 KB (1,739 words) - 06:38, 12 November 2019
• Mock AIME 4 2006-2007 Problems/Problem 15
  ...R</math> be the length of the [[radius]] of <math>\omega</math>. By the [[Power of a Point Theorem]], <math>MD \cdot (2R - MD) = AM \cdot MC = 24^2</math>
  3 KB (532 words) - 20:29, 31 August 2020
• Degree of a polynomial
  ...ree''' of a [[polynomial]] in one [[variable]] is the largest [[exponent | power]] with which the variable appears with non-zero [[coefficient]]. Thus, for
  1 KB (203 words) - 19:38, 30 January 2007
• 2005 AMC 10A Problems/Problem 23
  ...ince <math>\overline{AB}</math> is the diameter, <math>CD=CG</math>. Using power of points, 2. Use the geometric mean theorem,
  6 KB (1,045 words) - 09:46, 4 April 2023
• Mock AIME 3 Pre 2005 Problems/Problem 4
  Let <math>s_n = \zeta_1^n + \zeta_2^n + \zeta_3^n</math> (the [[power sums]]). Then from <math>(1)</math>, we have the [[recursion]] <math>s_{n+3
  2 KB (221 words) - 02:49, 19 March 2015
• Mock AIME 3 Pre 2005 Problems/Problem 11
  ...>. Remark that from the perimeter condition <math>AB=45-b</math>. Now from Power of a Point we have the system of two equations <cmath>\begin{cases}7\cdot 1
  2 KB (325 words) - 15:32, 22 March 2015
• National Vocabulary Championship
  ...pps Howard National Spelling Bee]] and the [[Reader's Digest National Word Power Challenge]]. Based on the best of both contests, students can compete in on
  2 KB (259 words) - 23:12, 28 February 2007
• 2007 AIME I Problems/Problem 7
  ...the <math>\log 2</math> term to cancel out, <math>k</math> is a [[exponent|power]] of <math>2</math>. Thus, <math>N</math> is equal to the sum of all the nu
  2 KB (242 words) - 20:26, 20 April 2023
• 2007 AIME II Problems/Problem 12
  ...a value greater than 2 times <math>x_7</math> (amount needed to raise the power of 3 by 1). This confirms that <math>3^{n+7m} = 3^{56}</math>. (2)
  5 KB (829 words) - 12:22, 8 January 2024
• 2005 IMO Shortlist Problems/G1
  ...E} = \frac{r\cdot \frac{1}{2}b}{b} = \frac{r}{2} </math>. Considering the power of the point <math> \displaystyle T </math> to the circumcircle of <math> \ ...eorem, <math> AQ = \frac{c}{3}; CQ = \frac{a}{3} </math>. Considering the power of the point <math> \displaystyle Q </math> with respect to the circumcircl
  7 KB (1,088 words) - 16:57, 30 May 2007
• 2004 IMO Shortlist Problems/G2
  By considering the power of the point <math> \displaystyle D </math> with respect to <math> \display
  10 KB (1,539 words) - 23:37, 6 June 2007
• 2004 IMO Shortlist Problems/N2
  ...nnot have any odd divisors greater than 1, i.e., <math>a </math> must be a power of 2. ...n <math>\psi(x) = ax </math> is unique if and only if <math>a </math> is a power of 2. Q.E.D.
  6 KB (1,007 words) - 09:10, 29 August 2011
• LaTeX:Layout
  far above our poor power to add or detract. The world will little note nor long dead who struggled here have hallowed it far above our poor power to add or
  30 KB (5,171 words) - 10:16, 4 April 2021
• Root-mean-square
  ...qrt{\frac{x^2_1+x^2_2+\dots+x^2_n}{n}}</math>. This is the second [[power mean]] of the <math>x_i</math>. It is so-named because it is the square root of the mean of the squares of the <math>x_i</math>.
  543 bytes (86 words) - 13:41, 8 May 2013
• Zermelo-Fraenkel Axioms
  ..., there are models of <math>\sf{ZF}</math> where <math>\in</math> does not mean set membership, but due to the [[Mostowski Collapse lemma]] this is often o === The Power Set Axiom ===
  4 KB (732 words) - 20:49, 13 October 2019
• Proportion
  A proportion in which one number is equal to a constant raised to the power of the other, or the [[logarithm]] of the other, is called an exponential p
  2 KB (325 words) - 16:34, 1 June 2022
• 2005 AMC 12A Problems/Problem 15
  By [[Power of a Point Theorem]], <math>CD^2= AC \cdot BC = 2\cdot AC^2</math>, and thu
  14 KB (1,970 words) - 17:02, 18 August 2023
• Linear equation
  ...of the first [[degree]] - i.e. each term does not have any variables to a power other than one.
  1 KB (257 words) - 13:39, 14 July 2021
• 2006 AIME II Problems/Problem 3
  Any power of 3 above 81 doesn't fit into our sequence.
  4 KB (562 words) - 18:37, 30 October 2020
• Order relation
  ...r set]] of another set <math>A</math>, and define <math>a \leq b</math> to mean "<math>a</math> is a [[subset]] of <math>b</math>." In this case, <math>a<
  2 KB (384 words) - 12:50, 25 November 2007
• Omega
  ...he interesting property that whenever one item in the ring is taken to any power, another item in the ring is the result.
  613 bytes (96 words) - 15:54, 5 June 2020
• Omega ring
  ...he interesting property that whenever one item in the ring is taken to any power, another item in the ring is the result.
  265 bytes (44 words) - 18:57, 9 September 2008
• Watt
  ...t''' (abbreviated W) is the [[Système international|metric]] measure of [[power]], named for James Watt.
  398 bytes (58 words) - 23:34, 23 December 2008
• *[[Watt]] - [[power]]
  2 KB (293 words) - 01:06, 20 June 2011
• Force
  *[[Power]]
  1,006 bytes (169 words) - 22:17, 18 October 2017
• Work
  *[[Power]]
  1 KB (192 words) - 00:46, 16 July 2018
• Power
  ...oosely defined as the speed something can do [[work]]. The [[SI]] unit for power is the [[Watt]]. ...d of [[time]]. It is also the [[derivative]] of work. If <math>P</math> is power, <math>W</math> is work, and <math>t</math> is time, then:
  467 bytes (77 words) - 23:26, 2 March 2008
• South Carolina ARML
  ...RML practices which usually compose of individual tests, team tests, and a power round test.
  2 KB (383 words) - 13:19, 20 June 2008
• De Longchamps point
  The de Longchamps point of a triangle is the radical center of the power circles of the triangle. Prove that De Longchamps point lies on Euler line. We call A-power circle of a <math>\triangle ABC</math> the circle centered at the midpoint
  10 KB (1,780 words) - 09:23, 17 November 2022
• Proofs of AM-GM
  ...x_{n-1}}{n-1} \right)}.</cmath> Raising both sides to the <math>n</math>th power yields <cmath>\left( \frac{x_1 + x_2 + \cdots + x_{n-1}}{n-1}\right)^n \geq
  12 KB (2,171 words) - 07:55, 11 May 2023
• 2000 AMC 12 Problems/Problem 23
  ...To see this, for each number subtract the power of <math>5</math> from the power of <math>2</math>. This yields <math>0,1,2,-1,3,0,4,1,-2,5,2</math>, and in
  2 KB (348 words) - 22:26, 24 October 2021
• 2007 AMC 10A Problems/Problem 20
  ...mial Theorem, we raise both sides of <math>a+a^{-1}=4</math> to the fourth power: <li>To find the fourth power of a sum/difference, we can first square that sum/difference, then square t
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• 2007 AMC 10A Problems/Problem 17
  <math>3 \cdot 5^2m</math> must be a perfect cube, so each power of a prime in the factorization for <math>3 \cdot 5^2m</math> must be divis
  1 KB (201 words) - 08:04, 11 February 2023
• 1995 AHSME Problems/Problem 28
  ...secting the circle at <math>G</math> and let <math>FG = x</math>. By the [[Power of a Point Theorem]],
  3 KB (509 words) - 22:56, 5 December 2023
• Three Greek problems of antiquity
  ...raic number]] and that the degree of it's [[minimal polynomial]] must be a power of <math>2</math>.
  2 KB (359 words) - 17:06, 11 June 2022
• 2004 AMC 12B Problems/Problem 25
  Given <math>n</math> digits, there must be exactly one power of <math>2</math> with <math>n</math> digits such that the first digit is <
  3 KB (383 words) - 20:12, 15 October 2016
• 1990 USAMO Problems/Problem 5
  Now we use the Power of a Point theorem with respect to point <math>H</math>. From the circle wi
  3 KB (604 words) - 20:52, 24 October 2018
• 2007 AMC 10B Problems/Problem 11
  We also know that, from the [[Power of a Point Theorem]], <cmath>AD\cdot DE=BD\cdot DC.</cmath>
  5 KB (851 words) - 22:02, 26 July 2021
• 2008 AMC 12A Problems/Problem 19
  ...I choose <math>x^2</math> from <math>A</math> , then there is exactly one power of <math>x</math> in <math>B</math> that I can choose; in this case, it wou
  5 KB (895 words) - 22:54, 9 January 2021
• 1992 USAMO Problems/Problem 3
  Note that the first 8 numbers are power of <math>2</math> from <math>0</math> to <math>7</math>, and realize that a
  5 KB (858 words) - 07:52, 19 July 2016
• 2005 Alabama ARML TST Problems/Problem 14
  ...lutions we want to the equation. Raising the first equation to the fourth power gives us
  3 KB (478 words) - 23:41, 5 January 2014
• 2008 AIME I Problems
  ...< 1000</math>, and <math>m</math> is not divisible by the <math>n</math>th power of any prime. Find <math>m + n</math>.
  9 KB (1,536 words) - 00:46, 26 August 2023
• 2008 AIME I Problems/Problem 6
  ...ath>r</math> by <math>2^{r-1}</math> (we can do this because dividing by a power of 2 doesn't affect divisibility by <math>67</math>). The second row will b
  3 KB (509 words) - 17:21, 22 March 2018
• 2008 AIME I Problems/Problem 15
  ...m<1000</math>, and <math>m</math> is not divisible by the <math>n</math>th power of any prime. Find <math>m+n</math>.
  6 KB (1,041 words) - 00:54, 1 February 2024
• 2006 IMO Shortlist Problems/A5
  By the [[Power Mean Inequality]],
  3 KB (529 words) - 08:03, 29 March 2008
• 2006 IMO Shortlist Problems/C5
  ...yers in any minimal subtournament of an <math>(n,k)</math>-tournament is a power of 2. ...ple of <math>2^{t+1}</math> players, since each must have a size that is a power of 2, and no two players meet more than once. It follows that <math>2^{t+1
  4 KB (641 words) - 11:09, 30 March 2008
• Mock AIME 1 2007-2008 Problems/Problem 1
  ...e sum of the degrees of the power of each of the two terms must sum to the power being raised to, here <math>17</math>. However, <math>3 + 13 = 16</math>, c
  552 bytes (84 words) - 17:16, 2 April 2008
• Mock AIME 1 2007-2008 Problems/Problem 9
  ...th> upon division by <math>3</math>). The third condition implies that the power of each prime factor of <math>n</math> must be divisible by <math>5</math> ...th>n</math>, we want to just use the prime factors <math>2,3,5</math>. The power of <math>2</math> must be divisible by <math>3,5</math>, and <math>2^{15}</
  2 KB (323 words) - 00:32, 31 August 2015
• 2008 AIME II Problems/Problem 7
  ...> and <math>(t+r)</math> are roots of this polynomial, we know that (using power reduction)
  7 KB (1,251 words) - 19:18, 2 January 2024
• 2005 USAMO Problems/Problem 2
  We now apply the lifting the exponent lemma to examine the power of 3 that divides each side of the equation when <math>m > 0</math> to obta
  7 KB (1,053 words) - 10:38, 12 August 2015
• 2008 USAMO Problems/Problem 1
  Now, for <math>1 \le i \le n</math>, take <math>k_i</math> to be the greatest power of <math>p_i</math> that divides <math>a^2 + a +1</math>, and let <math>k_0
  11 KB (1,964 words) - 03:38, 17 August 2019
• 2008 USAMO Problems/Problem 4
  ...stinct powers of <math>2</math>, where <math>1= 2^0</math> is considered a power of <math>2</math>.
  8 KB (1,318 words) - 12:37, 20 April 2022
• 2008 USAMO Problems/Problem 6
  ...er of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form <math>2^k</math> for some positive integer <ma ...Therefore, the number of good configurations of the mathematicians was a power of two.
  13 KB (2,414 words) - 14:37, 11 July 2016
• 2008 USAMO Problems
  ...er of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form <math>2^k</math> for some positive integer <ma
  4 KB (674 words) - 21:48, 12 August 2014
• Mathleague.org
  ..., which is a proof-based problem set of around 10 problems for a team. The Power Round is scored out of 100, with possible partial credit on most problems. ...hool). The total points possible for a team is 400. At contests where the Power Round is taken, that score is also included in the total sum, making it out
  5 KB (801 words) - 12:47, 23 September 2023
• Mock ARML
  ...rom March to May of the year (some times including the full set with Team, Power, Individual, Relay, Super Relay, and Tiebreaker, other times with only Indi
  1 KB (153 words) - 11:44, 16 June 2019
• Nilpotent group
  ...rder subgroup generated by the <math>P_p</math> must be divisible by every power of a prime that divides <math>G</math>, but it must also divide <math>G</ma
  9 KB (1,768 words) - 17:55, 5 June 2008
• P-group
  ...up''' is a [[finite]] [[group]] whose [[order (group theory) |order]] is a power of a [[prime]] <math>p</math>.
  4 KB (814 words) - 22:50, 3 November 2023
• Sylow Theorems
  ...G</math> modulo <math>H</math>. Since the order of <math>K</math> is some power of <math>p</math>, say <math>p^s</math>, the size of each orbit must divide
  11 KB (2,071 words) - 12:25, 9 April 2019
• Poncelet
  ...wer]] named after [[Jean-Victor Poncelet]]. One poncelet is defined as the power required to raise a hundred-[[kilogram]] [[mass]] (quintal) at a [[velocity
  284 bytes (40 words) - 11:36, 6 June 2008
• 2003 AMC 12B Problems/Problem 18
  ...p + 1}</math> to the fifth root. Similarly, we want to look for the lowest power <math>n</math> of <math>7</math> such that <math>13n - 1 \equiv 0 \pmod{5}<
  6 KB (914 words) - 11:07, 7 September 2023
• Topological space
  ...</math> is a [[set]] and <math>\mathcal{T}</math> is a [[subset]] of the [[power set]] of <math>X</math> satisfying the following relations: ...ts <math>\{ \varnothing, X\}</math>, and <math>\mathfrak{P}(X)</math> (the power set of <math>X</math>) are both topologies. They are the least and greates
  6 KB (1,142 words) - 15:38, 21 June 2008
• Mechanics
  Mechanics is the study of movement. Kinematics, mechanical forces, work, power, energy, and matter are all part of mechanics. ...states respectively. Similarly, mechanical power is defined as where is power delivered and is velocity. Energy is the other basic intrinsic property of
  2 KB (309 words) - 15:41, 1 December 2015
• 1995 AHSME Problems/Problem 27
  ...few rows: <math>0, 2, 6, 14, 30, 62</math>. They are each two less than a power of <math>2</math>, so we try to prove it: ...2S_n-2(n-1)+2(n-1)+2=2S_n+2</math>. If <math>S_n</math> is two less than a power of 2, then it is in the form <math>2^x-2</math>. <math>S_{n+1}=2^{x+1}-4+2=
  5 KB (682 words) - 09:45, 18 February 2022
• 1995 AHSME Problems/Problem 26
  By [[Power of a point]], <math>AE \cdot EB = DE \cdot EF \Rightarrow (r+x)(r-x)=r^2-x^
  2 KB (359 words) - 20:01, 23 January 2017
• 1988 IMO Problems/Problem 1
  Since <math>PB\cdot PC</math> is the power of the point <math>P</math>, it stays constant as <math>A</math> varies. T
  3 KB (545 words) - 11:32, 30 January 2021
• 2003 USAMO Problems/Problem 4
  ...D = \angle MBC</math>. This exact condition can also be visualized through power of M with respect to the circumcircle of triangle <math>BDC</math> and gett
  5 KB (820 words) - 02:39, 10 January 2023
• 1985 IMO Problems/Problem 4
  ...a subset of <math>4</math> elements whose product is the <math>4</math>th power of an integer.
  1 KB (261 words) - 23:56, 29 January 2021
• 1995 IMO Problems/Problem 1
  Think about Radical Axis, Power of a Point and Radical Center. ...ath>, we have <math>PN\cdot PB=PM\cdot PC</math> and so by the converse of Power of a Point, the quadrilateral <math>MNBC</math> is cyclic. Thus, <math>90-
  5 KB (847 words) - 19:03, 12 October 2021
• 2008 iTest Problems
  ...e people will have reasonably inexpensive options for switching to cleaner power sources. Even now most families could switch to biomass for between <math>\ element which is the [[arithmetic mean]] of all the elements in that subset?
  71 KB (11,749 words) - 01:31, 2 November 2023
• Gauss's Lemma (polynomial)
  ...ily commutative) [[ring]]; let <math>A[[X]]</math> be the ring of [[formal power series]] over <math>A</math>. For <math>P \in A[[X]]</math>, let <math>I(P ...t, with <math>A = \mathbb{Z}</math>, and <math>P</math> and <math>Q</math> power series with finitely many nonzero coefficients.
  3 KB (483 words) - 12:23, 30 May 2019
• POP
  #REDIRECT[[Power of a Point Theorem]]
  37 bytes (6 words) - 22:48, 3 November 2008
• Intermediate Math Open
  ...Club participants for 2 and 1/4 hours. Then, lunch is held. Next comes the Power Round, where teams work together on a series of related problems for 45 min
  1 KB (243 words) - 17:53, 1 November 2014
• Binary
  ! Power of two !! Number in base 10 !! Binary representation ...which can be odd, as in any other place the digit will be multiplied by a power of 2 that doesn't equal 1.)
  2 KB (294 words) - 12:55, 3 September 2019
• Float
  *It is an important metric in measuring computational power (in [[FLOPS]]).
  953 bytes (136 words) - 16:08, 10 March 2011
• FLOPS
  It is a commonly used metric for measuring computational power.
  187 bytes (23 words) - 17:51, 18 December 2008
• Decibel
  ...are a logarithmic units that are used to measure intensity of things like power or sound.
  140 bytes (22 words) - 17:16, 31 August 2009
• Horsepower
  '''Horsepower''' ('''hp''' or '''Hp'''), is a measurement of [[power]]. It was developed by James Watt, who wanted to market his steam engines i
  255 bytes (41 words) - 16:14, 24 December 2008
• 1992 USAMO Problems/Problem 4
  <math>m\angle APB= m\angle A'PB'</math> because they are vertical angles. By power of point, <math>(AP)(A'P)=(BP)(B'P)\rightarrow\frac{AP}{B'P}=\frac{BP}{A'P}
  5 KB (807 words) - 18:37, 25 June 2021
• 1972 USAMO Problems/Problem 1
  ...h>\alpha + \beta + \alpha) = -\beta</math>. Since every prime has the same power in both expressions, the expressions are equal. <math>\blacksquare</math>
  5 KB (1,018 words) - 11:14, 6 October 2023
• 1974 USAMO Problems/Problem 2
  This means that this inequality is homogeneous since both sides have the same power of <math>k</math> as a factor. Since the inequality is homogeneous, we can Raising <math>2</math> to the power of each side, we get
  9 KB (1,726 words) - 14:01, 1 February 2024
• 1962 IMO Problems/Problem 6
  ...e at <math>P,Q</math>, we see that <math>AI\cdot IL = PI\cdot QI</math> by Power of a Point. Therefore, <math>2r\rho = PI \cdot QI = (PO + OI)(QO - OI) = (r
  2 KB (308 words) - 06:29, 16 December 2023
• 2021 GMC 10B
  Let <math>f(n)</math> be the largest possible power of <math>2</math> that divides <math>n</math>. Find <math>f((3^2-3)(4^2-4)(
  11 KB (1,695 words) - 14:33, 7 March 2022
• 1998 AHSME Problems/Problem 30
  ...r of <math>5</math> that divides <math>a_n</math> is at least equal to the power of <math>2</math> that divides <math>a_n</math>. ...le by <math>5</math>, and at most one of them can be divisible by a higher power of <math>5</math>. As we need <math>y_n\geq x_n\geq 8</math>, one of the in
  3 KB (486 words) - 10:37, 11 August 2020
• Computer simulation
  ...ithmic Botany | plant growth]]. Currently, however, we lack the processing power and understanding of nature to do large-scale cell-level computations, so o
  1 KB (197 words) - 13:19, 26 August 2019
• 2009 AMC 12A Problems/Problem 24
  ...nt a number <math>x</math>, such that when <math>2</math> is raised to the power of <math>x</math>, we get the expression in the <math>\log</math>. So, when ...<math>124986068</math>, what happens when you raise <math>2</math> to the power of that number? We would end up getting another one of these:
  10 KB (1,595 words) - 22:13, 20 September 2021
• 2009 AMC 12A Problems/Problem 18
  ...h>, and therefore <math>5^6 + 1 \equiv 2 \pmod 4</math>. Hence the largest power of <math>2</math> that divides <math>5^6+1</math> is <math>2^1</math>, and
  6 KB (1,012 words) - 19:16, 14 September 2022
• 2009 AMC 10A Problems/Problem 10
  == Solution 3 (Power of a point)== ...ight angle triangle, AC is the diameter of the circumcircle. By applying [[Power of a Point Theorem]], we can have <math>BD=DE</math> and <math>AD\cdot CD=B
  5 KB (879 words) - 18:57, 30 April 2024
• 2002 AMC 12A Problems/Problem 24
  ...s. We know this because we are taking magnitude to the <math>2003</math>rd power, and if the magnitude of <math>a+bi</math> is larger than <math>1</math>, i
  4 KB (743 words) - 19:54, 14 March 2024
• Two Tangent Theorem
  From a simple application of the [[Power of a Point Theorem]], the result follows.
  566 bytes (93 words) - 22:47, 5 December 2023
• MaCh 8
  ...1-hour proof test on a set of related mathematical concepts. Although the Power test does not expect rigorous proofs, a clear and complete explanation is r The top 5 teams receive awards based on their performance in the individual, Power, Team, and Mach rounds. The rounds are weighted appropriately to promote a
  2 KB (350 words) - 10:09, 1 April 2009
• 2009 AIME I Problems/Problem 7
  ...(3n + 2)}</math> an integer. This means that <math>3n + 2</math> must be a power of <math>5</math>. We test <math>25</math>: <cmath>3n + 2 = 25</cmath> <cma
  2 KB (340 words) - 00:26, 9 January 2023
• Polynomial ring
  ...h the terms are written, and indeed often list them in descending order of power. So we would write: .../math>, of which the ring of polynomials is a subring. In general, formal power series are not associated with mappings of <math>R</math> into itself, as i
  12 KB (2,010 words) - 00:10, 3 August 2020
• 2009 AIME II Problems/Problem 2
  ...rocess for the other ones(you have to turn the square root to a fractional power for c), we get <math>343+121+5 = \boxed{469}</math>
  3 KB (396 words) - 15:20, 10 May 2024
• Hilbert's Basis Theorem
  <!-- I am think it doesn't work for rings of formal power
  4 KB (617 words) - 19:59, 23 April 2023
• 2009 AIME II Problems/Problem 7
  To determine <math>a</math>, we need to determine the largest power of <math>2</math> that divides <math>c</math>. ...{2009}</math> and <math>\dbinom{4016}{2008}</math> and you get the minimum power of <math>2</math> in either expression is <math>8</math> so the answer is <
  8 KB (1,312 words) - 16:23, 30 March 2024
• Determinant
  is an element of a (strictly [[power associative]]) <math>n</math>-dimensional
  8 KB (1,345 words) - 00:31, 9 May 2020
• 2004 USAMO Problems/Problem 2
  ...an existing element of the sequence until it does have a length which is a power of 2 - it is apparent that this will not change <math>S</math>.
  3 KB (508 words) - 14:23, 17 July 2014
• 2009 USAMO Problems/Problem 1
  ...essed using circle <math>\omega_2</math> and <math>\omega_3</math> and the power of <math>O_2</math> can be expressed using circle <math>\omega_1</math> and
  2 KB (290 words) - 13:16, 17 April 2021
• Radical axis
  The '''power''' of point <math>P</math> with respect to circle <math>\omega</math> (with Note that the power of a point is negative if the point is inside the circle.
  10 KB (1,797 words) - 02:05, 24 October 2023
• Differentiation Rules
  '''Power Rule:'''
  2 KB (288 words) - 00:53, 26 March 2018
• Constructible polygon
  ...<math>n</math>-gon is constructible iff <math>n</math> is the product of a power of <math>2</math> and distinct [[fermat prime]]s. For instance, a <math>17= ...)</math> (where <math>\phi(n)</math> is [[Euler's totient function]]) is a power of <math>2</math>.
  5 KB (926 words) - 18:47, 4 March 2022
• Fermat prime
  == Primes one more than a power of 2 == ...ve no odd [[divisor | factors]] other than <math>1</math> and so must be a power of 2.
  2 KB (246 words) - 18:07, 25 August 2009

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