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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

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