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• If I have four boxes arranged in a $2 \times 2$ grid, in how many distinct ways can I place the digits $1$, $2$, and $3$ in the boxes, using each digit exactly once, such that each box contains at most one digit? (I only have one of each digit, so one box
  328 bytes (60 words) - 17:19, 12 June 2022
• Points $A$, $B$, $C$, and $D$ lie on a line, in that order. If $AB=2$ units, $BC=5$ units and $AD=14$ units, what is the ratio of $AC$ to $BD$? Express your answer as a common fraction.
  119 bytes (19 words) - 11:01, 20 March 2019
• If no one shares an office, in how many ways can 3 people be assigned to 5 different offices? (Each person gets exactly one office).
  If we number the three people 1, 2, and 3, there are 5 offices that person 1 c
  254 bytes (42 words) - 16:52, 28 February 2021
• Suppose that a and b are positive integers for which a has 3 factors and b has a factors. If b is divisible by a, then what is the least possible value of b?
  ...has <math>3</math> factors and <math>b</math> has <math>a</math> factors. If <math>b</math> is divisible by <math>a</math>, then what is the least possi
  262 bytes (49 words) - 13:14, 3 August 2021
• The table shows the percent of families in Mathville that have $0, 1, 2, 3$ and $4$ or more children. If there are a total of $10,250$ families, how many are there with no children?
  ...lle that have <math>0, 1, 2, 3</math> and <math>4</math> or more children. If there are a total of <math>10,250</math> families, how many are there with
  214 bytes (35 words) - 09:55, 1 August 2022

Page text matches

• Pythagorean Theorem
  Note that (-1,0,1) and (3,4,5) are the only pythagoren triplets that consist of consecutive integers. Also, if (a,b,c) are a pythagorean triplet it follows that (ka,kb,kc) will also form
  5 KB (886 words) - 21:12, 22 January 2024
• Main Page
  During AMC 10/12 testing week, the AoPS Wiki is in read-only mode. No edits can be made. If you don't know much about how to edit a wiki, you might want to read the [[
  2 KB (258 words) - 14:05, 15 November 2023
• 2016 AMC 10A Problems/Problem 3
  If Ben paid <math>\$ 12.50</math> more than David, then he paid <math>\frac{12 If for every dollar Ben spent on bagels David spent <math>25</math> cents less
  1 KB (176 words) - 10:58, 16 June 2023
• 2016 AMC 10A Problems/Problem 5
  ...math>x</math>. Then the volume is <math>x \cdot 3x \cdot 4x =12x^3</math>. If <math>x=2</math>, then <math>12x^3 = 96 \implies \boxed{\textbf{(D) } 96.}< ...math>(E) 12</math> and the final answer has to equal <math>x^3</math>. The only answer choice that works is <math>(D)</math>.
  1 KB (184 words) - 13:58, 22 August 2023
• 2016 AMC 10A Problems/Problem 7
  ...ed and the mean of the remaining numbers will be the same. This means that if using the mean in your reasoning, you can just take the mean of the other 6
  2 KB (268 words) - 18:19, 27 September 2023
• 2016 AMC 10A Problems/Problem 8
  If you started backwards you would get: <cmath>0\Rightarrow (+40)=40 , \Righta If you have <math>x</math> as the amount of money Foolish Fox started with we
  1 KB (169 words) - 14:59, 8 August 2021
• 2016 AMC 10A Problems/Problem 13
  ...and 3. This means that either Edie or Dee was originally in an edge seat. If Edie and Dee were in seats 1 and 2, then Bea must have been in seat 3, whic
  2 KB (402 words) - 14:54, 25 June 2023
• List of mathematics competitions
  This list is intended to be global. If other international or contests from other nations or regions are documente
  4 KB (406 words) - 11:20, 26 February 2021
• MathILy (serious mathematics infused with levity)
  ...mitted, placement in a given program is determined by academic suitability only.
  5 KB (706 words) - 23:49, 29 January 2024
• MATHCOUNTS
  ...nts may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target R ...ter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be
  10 KB (1,497 words) - 11:42, 10 March 2024
• ICTM Math Contest
  ...submitted. Correct answers are worth 2 points each. Bonus points are given if all four answers are correct and the answer sheet is submitted at the 3 min ...e. This trial program will be reviewed at the end of the year to determine if it will continue in future years.
  8 KB (1,182 words) - 14:26, 3 April 2024
• Massachusetts mathematics competitions
  ...ual and team rounds, where teams are usually organized in school. However, if a school has not signed up a team, students can compete individually. This
  5 KB (733 words) - 20:58, 27 April 2024
• Art of Problem Solving
  ...n be found [http://www.artofproblemsolving.org here]. Make sure it is .org if you want to see the foundation website, otherwise, you'll see the homepage
  5 KB (667 words) - 17:09, 3 July 2023
• William Lowell Putnam Mathematical Competition
  ...ach problem is graded on a scale of 0 to 10. The top five scorers (or more if there are ties) on the exam are named Putnam Fellows. ..., which regularly lead to schools selecting the "wrong team." For example, only two of the six 2007 Putnam fellows were members of their school teams. In s
  4 KB (623 words) - 13:11, 20 February 2024
• Physics books
  ...Lewis Carroll Epstein. This book contains hundreds of conceptual problems. Only some of the problems focus on mechanics. ...I3xEclkgIG_PNylPEHv6dL21Tyfk/view solution manual] to some of these exams. If your goal is to advance to USAPhO, you should solve all of the problems on
  10 KB (1,410 words) - 13:07, 20 February 2024
• Physics
  ...y amount of matter is [[Zero]], then the object's velocity will not change if viewing from a constant reference point. #If an object has [[mass]] <math>m</math> and [[acceleration]] <math>a</math>,
  9 KB (1,355 words) - 07:29, 29 September 2021
• 2005 Alabama ARML TST Problems/Problem 7
  If you didn't understand the third case / summation, all they did was subtract
  1 KB (193 words) - 21:13, 18 May 2021
• Mathematics
  As if to add to the confusion, "[[algebra (structure) |algebra]]" is the name for
  6 KB (902 words) - 12:53, 3 September 2019
• Inequality
  *<math>a>b</math> if <math>a</math> is greater than <math>b</math>, that is, <math>a-b</math> is *<math>a<b</math> if <math>a</math> is smaller than <math>b</math>, that is, <math>a-b</math> is
  12 KB (1,798 words) - 16:20, 14 March 2023
• Academic scholarships
  If you know of scholarships that do not match any of the following categories As a general guideline for scholarships: if you're paying to apply then it is probably not a legitimate organization of
  3 KB (337 words) - 03:35, 7 September 2020
• 1999 JBMO Problems/Problem 1
  .../math>, <math> b^3 + bx + y = 0</math> and <math> c^3 + cx + y = 0</math>. If <math> a,b,c</math> are all distinct numbers prove that their sum is zero.
  1 KB (205 words) - 14:05, 30 March 2019
• AMC 10
  '''These should be taken if a student is having trouble or wants a more clear and confirmed understandi
  4 KB (574 words) - 15:28, 22 February 2024
• American Invitational Mathematics Examination
  ...]], [[AMC 12]], and [[USAMTS]] competitions. Qualification through USAMTS only is rare, however. Each answer is an integer from 000 to 999, inclusive, mak ...section which are not in the AoPSWiki, please consider adding them. Also, if you notice that a problem in the Wiki differs from the original wording, fe
  8 KB (1,057 words) - 12:02, 25 February 2024
• 1954 AHSME Problems/Problem 46
  In the diagram, if points <math>A, B</math> and <math>C</math> are points of tangency, then <m
  3 KB (415 words) - 18:01, 24 May 2020
• 2015 IMO Problems
  if, for any two different points <math>A</math>, <math>B</math> in <math>\math <math>\mathcal{S}</math> is <i>centre-free</i> if for any three points <math>A</math>, <math>B</math>, <math>C</math> in
  4 KB (692 words) - 22:33, 15 February 2021
• American Regions Mathematics League
  Find out if your area has a team already. Contact ARML for details. If your area does not already have a team, ask ARML how to start one.
  2 KB (267 words) - 17:06, 7 March 2020
• Joining an ARML team
  ...o-Xue at xinkeguoxue@gmail.com, or message Xinke's AoPS account "hurdler", if you are interested in trying out for the Alabama team. A student who wishes to attend practice should try to take the SDMO. If a student wishes to speak with one of the coaches for the team, they may do
  21 KB (3,500 words) - 18:41, 23 April 2024
• Mathematics Olympiads
  ...] and do well on it to qualify for it). Another in USA is [[USAMTS]] which if you do well can qualify for [[AIME]]. Both Olympiads are from USA & are pro
  891 bytes (116 words) - 16:42, 2 January 2021
• Biology scholarships
  ...tingy about letting students in on how they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, pl
  1 KB (184 words) - 21:01, 12 December 2018
• Chemistry scholarships
  ...tingy about letting students in on how they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, pl
  1 KB (182 words) - 22:00, 4 February 2017
• Physics scholarships
  ...tingy about letting students in on how they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, pl
  2 KB (229 words) - 12:37, 8 July 2019
• Technology scholarships
  ...tingy about letting students in on how they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, pl
  3 KB (345 words) - 17:54, 16 June 2023
• Computer science and technology scholarships
  ...tingy about letting students in on how they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, pl ...in exchange for service at a federal agency. At participating institutions only. [https://www.sfs.opm.gov/ website]
  3 KB (427 words) - 16:18, 25 February 2021
• Engineering scholarships
  ...tingy about letting students in on how they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, pl
  4 KB (511 words) - 14:57, 16 July 2020
• Community Service Scholarships
  ...tingy about letting students in on how they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, pl
  3 KB (350 words) - 01:18, 19 June 2016
• Mu Alpha Theta
  You may only take one type of test a year for every competition, depending on how many c ...ped if it becomes evident that one of the contestants cannot possibly win. If Big Bird has answered his questions correctly 75% of the time and Barney ha
  4 KB (632 words) - 17:09, 11 October 2020
• Power Mean Inequality
  \prod_{i=1}^n a_i^{w_i} &\text{if } t=0 \\ ...math> and <math>k_2</math>, and equal weights (i.e. <math>w_i=1/n</math>), if <math>k_1>k_2</math>, then
  3 KB (606 words) - 23:59, 1 July 2022
• Wilson's Theorem
  ..., then <math>(p-1)! + 1</math> is divisible by <math>p</math> if and only if <math>p</math> is prime. It was stated by John Wilson. The French mathemati ...) This inverse is unique, and each number is the inverse of its inverse. If one integer <math>a</math> is its own inverse, then
  4 KB (639 words) - 01:53, 2 February 2023
• 1953 AHSME Problems/Problem 13
  ...angle and a trapezoid are equal in area. They also have the same altitude. If the base of the triangle is 18 inches, the median of the trapezoid is:
  878 bytes (143 words) - 20:56, 1 April 2017
• Trivial Inequality
  ...the closure of the set of positive numbers under multiplication. Finally, if <math>x<0</math>, then <math>x^2 = (-x)(-x) > 0,</math> again by the closur ...<math>a</math> is positive and <math>ax^2+bx+c\le c-\frac{b^2}{4a}</math> if <math>a</math> is negative.
  3 KB (560 words) - 22:51, 13 January 2024
• Arithmetic mean
  ...he numbers, divided by the number of numbers - the [[average]] of the set. If we let <math>{AM}</math> denote Arithmetic Mean, For example, if I wanted to find the average of the numbers 3, 1, 4, 1, and 5, I would comp
  699 bytes (110 words) - 12:44, 20 September 2015
• Completing the square
  ...equation <math>x^2-6x+2=0</math>. Look at the <math>x^2-6x</math> part. If <math>9</math> was added to this, then we would have a [[perfect square]], What is the value of <math>x</math> if <math>x=1+\dfrac{1}{x}</math>
  2 KB (422 words) - 16:20, 5 March 2023
• Polynomial
  ...or of <math>a_n</math>. This is convenient because it means we must check only a small number of cases to find all rational roots of many polynomials. It
  6 KB (1,100 words) - 01:44, 17 January 2024
• Simon's Favorite Factoring Trick
  ...<math>a</math> are integer constants, and the coefficient of xy must be 1(If it is not 1, then divide the coefficient off of the equation.). According t If this is confusing or you would like to know the thought process behind SFFT
  7 KB (1,107 words) - 07:35, 26 March 2024
• Factoring
  If <math>n=2</math>, this creates the difference of squares factorization, <cm In addition, if <math>n</math> is [[odd integer | odd]]:
  3 KB (532 words) - 22:00, 13 January 2024
• Proof by contradiction
  ...irrationality of <math>\sqrt{2}</math> and the infinitude of primes. <!-- If you know how, please make the following proofs look better --> This techniq
  2 KB (374 words) - 14:01, 21 August 2022
• Pigeonhole Principle
  In [[combinatorics]], the '''pigeonhole principle''' states that if <math>n+1</math> or more pigeons are placed into <math>n</math> holes, one ...le'''. A common phrasing of the principle uses balls and boxes and is that if <math>n</math> balls are to be placed in <math>k</math> boxes and <math>n>k
  11 KB (1,985 words) - 21:03, 5 August 2023
• Relatively prime
  ...ntegers <math>m</math> and <math>n</math> are relatively prime if and only if <math>\frac{m}{n}</math> is in lowest terms. Consecutive positive integers are always relatively prime, since, if a prime <math>p</math> divides both <math>n</math> and <math>n+1</math>, th
  2 KB (245 words) - 15:51, 25 February 2020
• Geometric mean
  ...is the <math>n</math>th [[root]] of the product of the numbers. Note that if <math>n</math> is even, we take the positive <math>n</math>th root. It is The geometric mean arises in [[geometry]] in the following situation: if <math>AB</math> is a [[chord]] of [[circle]] <math>O</math> with [[midpoint
  2 KB (282 words) - 22:04, 11 July 2008
• Principle of Inclusion-Exclusion
  ...t and undercount, in the end making sure every element is counted once and only once. In particular, memorizing a formula for PIE is a bad idea for problem For <math>|A|</math>, we just choose <math>3</math> people and there is only one way to put them in order, then <math>3!</math> ways to order the other
  9 KB (1,703 words) - 07:25, 24 March 2024

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