Create the page "If only" on this wiki! See also the search results found.

Page title matches

• 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
• Mill's Constant
  ...we are not sure) and Mill's Constant is the smallest element in that set. If the [[Riemann Hypothesis]] is true, Mill's constant is approximately <math>
  794 bytes (105 words) - 01:59, 15 January 2022
• 2020 CIME II Problems/Problem 10
  ...ath>2c+6</math>. It follows that <math>a+b\in\{c,2c,3c,4c,5c\}</math>, for if <math>a+b\geq 6c</math> we have <math>6c\leq 2c+6\Rightarrow c\leq3/2\Right
  2 KB (332 words) - 09:37, 30 December 2021
• Cauchy-Schwarz Inequality
  ..._i</math> for all <math>1 \leq i \leq n</math>, or if one list consists of only zeroes. Along with the [[AM-GM Inequality]], Cauchy-Schwarz forms the found ...math> such that <math>\overrightarrow{v} = t \overrightarrow{w}</math>, or if one of the vectors is zero. This formulation comes in handy in linear algeb
  13 KB (2,048 words) - 15:28, 22 February 2024
• Discriminant
  ...the discriminant is negative, the equation has two [[nonreal]] roots; and if the discriminant is 0, the equation has a real [[double root]]. *If <math>D=0</math>, then at least two of the roots are equal;
  4 KB (734 words) - 19:19, 10 October 2023
• Menelaus' theorem
  If line <math>PQ</math> intersecting <math>AB</math> on <math>\triangle ABC</m The converse of Menelaus' theorem is also true. If <math>\frac{BP}{PC} \cdot \frac{CQ}{QA} \cdot \frac{AR}{RB} = 1</math> in t
  5 KB (804 words) - 03:01, 12 June 2023
• Number theory
  Some branches of number theory may only deal with a certain subset of the real numbers, such as [[integer]]s, [[pos ...p://www.amazon.com/exec/obidos/ASIN/0262680289 (details)] Out of print but if you can find it in a library or used, you might love it and learn a lot. Wr
  3 KB (399 words) - 23:08, 8 January 2024
• Fermat's Little Theorem
  ...r arithmetic]] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is If <math>{a}</math> is an [[integer]], <math>{p}</math> is a [[prime number]]
  16 KB (2,675 words) - 10:57, 7 March 2024
• Parabola
  ...p) upwards/right if <math>a</math> is positive, and will be downwards/left if <math>a</math> is negative.
  3 KB (551 words) - 16:22, 13 September 2023
• Chebyshev's Inequality
  '''Chebyshev's inequality''', named after [[Pafnuty Chebyshev]], states that if On the other hand, if <math> a_1\geq a_2\geq ... \geq a_n </math> and
  1 KB (214 words) - 20:32, 13 March 2022
• Euler's Totient Theorem
  ...h>\{1,2,3\cdots{,n}\}</math> which are relatively prime to <math>n</math>. If <math>{a}</math> is an integer and <math>m</math> is a positive integer [[r .... All elements of <math>B</math> are relatively prime to <math>m</math> so if all elements of <math>B</math> are distinct, then <math>B</math> has the sa
  3 KB (542 words) - 17:45, 21 March 2023
• Geometric inequality
  The [[Isoperimetric Inequality]] states that if a figure in the plane has [[area]] <math>A</math> and [[perimeter]] <math>P ...e the total value of the expression. To prove this, all we need to show is if <math>0<A,B<180</math>, then <math>\sin(A+B)+\sin(A-B)<2\sin A</math>. This
  7 KB (1,296 words) - 14:22, 22 October 2023
• Brahmagupta's Formula
  ...formula]] for determining the [[area]] of a [[cyclic quadrilateral]] given only the four side [[length]]s, given as follows: <cmath>[ABCD] = \sqrt{(s-a)(s- If we draw <math>AC</math>, we find that <math>[ABCD]=\frac{ab\sin B}{2}+\frac
  3 KB (465 words) - 18:31, 3 July 2023
• Ptolemy's theorem
  ...> be the regular heptagon. Consider the quadrilateral <math> ABCE </math>. If <math> a </math>, <math> b </math>, and <math> c </math> represent the leng
  7 KB (1,198 words) - 20:39, 9 March 2024
• Elementary symmetric sum
  ...th>k</math> of those numbers (<math>1 \leq k \leq n</math>). For example, if <math>n = 4</math>, and our set of numbers is <math>\{a, b, c, d\}</math>,
  2 KB (275 words) - 12:51, 26 July 2023
• Cyclic quadrilateral
  ...rty is both sufficient and necessary (Sufficient & necessary = if and only if), and is often used to show that a quadrilateral is cyclic.
  1 KB (162 words) - 20:39, 9 March 2024
• Algebra
  As if to add to the confusion, "[[algebra (structure) |algebra]]" is the name for
  3 KB (369 words) - 21:18, 18 June 2021
• Number theory/Advanced
  ...mmarize the group law by saying <math>P+Q+R=\mathcal{O}</math> if and only if <math>P,Q</math> and <math>R</math> lie on a line. ...ssociated to <math>z</math> and <math>z'</math> are isomorphic if and only if there is a relation
  5 KB (849 words) - 16:14, 18 May 2021
• 1959 IMO Problems/Problem 2
  ...<math>A=\sqrt{2}</math>, (b) <math>A=1</math>, (c) <math>A=2</math>, where only non-negative real numbers are admitted for square roots? '''Case I:''' If <math>x \le 1</math>, then <math>|x-1| = 1 - x</math>, and the equation red
  3 KB (466 words) - 12:04, 12 April 2024
• Harmonic mean
  ...to positive numbers is necessary to avoid division by zero. For instance, if we tried to take the harmonic mean of the set <math>\{-2, 3, 6\}</math> we
  1 KB (196 words) - 00:49, 6 January 2021
• Euler's totient function
  ...an or equal to <math> n </math> that are divisible by <math> p_1 </math>. If we do the same for each <math> p_i </math> and add these up, we get (d) If <math>p</math> is prime and <math>n\ge{1},</math> then <math>\phi(p^n)=p^n-
  5 KB (898 words) - 19:12, 28 January 2024
• Probability
  ...slate a very basic situation into measure-theoretic language. Furthermore, if one wanted to determine whether the coin was fair or weighted, it would be
  4 KB (588 words) - 12:47, 2 October 2022
• Contest strategies
  ...imation can narrow down the possible values. It can also be used to check if an answer is reasonable. ...ompletely solving the problem, especially when the problem stumps you, and if you can narrow the problem down to a couple of choices, you may be able to
  3 KB (538 words) - 13:13, 16 January 2021
• Divisibility
  ...— or using [[modular arithmetic]], that <math>b \equiv 0 \pmod a</math>. If <math>b</math> does ''not'' divide <math>a</math>, we write that <math>b \n
  2 KB (277 words) - 16:21, 29 April 2023
• Rearrangement Inequality
  ...zed when <math>A</math> and <math>B</math> are oppositely sorted (that is, if <math>a_k</math> is less than or equal to exactly <math>{i}</math> of the o ...number of tens, etc. Meanwhile, you would find yourself very disappointed if there were more pennies than nickels, more nickels than dimes, and so on.
  5 KB (804 words) - 13:54, 26 January 2023
• Real number
  We say that <math>\alpha</math> is a '''cut''' if and only if (2) If <math>a\in\alpha</math>, <math>p\in\alpha\forall p<a</math>
  3 KB (496 words) - 23:22, 5 January 2022
• Divisibility rules
  ...her [[integer]]s. All of these rules apply for [[Base number| base-10]] ''only'' -- other bases have their own, different versions of these rules. ...r is divisible by 2 if and only if its units digit is divisible by 2, i.e. if the number ends in 0, 2, 4, 6 or 8.
  8 KB (1,315 words) - 18:18, 2 March 2024
• Induction
  ...t shown to hold for a certain value (the Base Case); it is then shown that if the desired result holds for a certain value, it then holds for another, cl ...<math>{n=k}</math> gives the desired result, so does <math>n=k+2</math>. If you wish, you can similarly induct over the powers of 2.
  5 KB (768 words) - 20:45, 1 September 2022
• Orthic triangle
  The orthic triangle does not exist if <math>\triangle ABC</math> is right. The two cases of when <math>\triangle If <math>\triangle ABC</math> is acute, then the incenter of the orthic triang
  8 KB (1,408 words) - 11:54, 8 December 2021
• Vertex
  ...y</math>-value of the graph is minimal (if <math>a > 0</math>) or maximal (if <math>a < 0</math>).
  1,000 bytes (162 words) - 13:08, 20 February 2024
• Overcounting
  ...order to count the number of handshakes. It's not a matter of a great deal if there are less than 10 persons. But assume a party from the President, ther ...ople. According to the formula we get 2 handshakes, but wait, we will have only 1 handshake between two persons. That means we have overcounted somewhere.
  4 KB (635 words) - 12:19, 2 January 2022
• Constructive counting
  ...idea, but there are multiple ways one might proceed with the construction. If we were to go like before and break the problem down by each box, we'd get ...ir position is actually predetermined by choosing the <math>A</math>s; the only place the three <math>B</math>s can go is in the three empty boxes, so we d
  12 KB (1,896 words) - 23:55, 27 December 2023
• Jensen's Inequality
  If <math>{F}</math> is a concave function, we have: We only prove the case where <math>F</math> is concave. The proof for the other cas
  3 KB (623 words) - 13:10, 20 February 2024
• Casework
  ...xists, casework may be considered [[brute force]]. This is especially true if that alternative solution uses [[complementary counting]]. ''How many words are less than four letters long and contain only the letters A, B, C, D, and E? Here, 'word' refers to any string of letters
  5 KB (709 words) - 10:28, 19 February 2024
• Euclidean algorithm
  ...<math>mb</math> and, thereby, their difference <math>r</math>. Similarly, if <math>{d}</math> divides both <math>b</math> and <math>r</math>, it should * If <math>b=0</math>, then <math>\gcd(a,b)=a</math>.
  6 KB (924 words) - 21:50, 8 May 2022
• Conic section
  * An [[ellipse]] is formed if the cutting plane makes an [[angle]] with an [[axis]] that is larger than t If the cutting plane passes through the vertex of the cone, the result is a de
  5 KB (891 words) - 01:14, 9 January 2023
• Generating function
  If we let <math>A(k)={n \choose k}</math>, then we have <math>{n \choose 0}+{n ...0</math>) or <math>x \in \left( \frac{1}{a}, -\frac{1}{a} \right)</math> (if <math>a < 0</math>), generating the powers of <math>a</math> for any real <
  4 KB (659 words) - 12:54, 7 March 2022
• Binomial Theorem
  Therefore, if the theorem holds under <math>n+1</math>, it must be valid. ...e complicated [[polynomial]]s with [[binomial coefficient]]s. For example, if a contest problem involved the polynomial <math>x^5+4x^4+6x^3+4x^2+x</math>
  5 KB (935 words) - 13:11, 20 February 2024
• Prime number
  ...' (or simply '''prime''') is a [[positive integer]] <math>p>1</math> whose only positive [[divisor | divisors]] are 1 and itself. ...s being neither prime nor [[composite number|composite]] because it is its only factor among the [[natural number|natural numbers]].
  6 KB (985 words) - 12:38, 25 February 2024
• Recursion
  ...example of a recursive definition is that of the [[Fibonacci sequence]]. If we let <math>F_n</math> be the <math>n</math>th Fibonacci number, the seque
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• Function
  ...A</math> to <math>B</math>'' (written <math>f: A \to B</math>) if and only if * if <math>(a,b)\in f</math> and <math>(a,c)\in f</math> then <math>b=c</math>.
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• Complementary counting
  More formally, if <math>B</math> is a [[subset]] of <math>A</math>, complementary counting ex ...any house a single color. In how many ways can she color the seven houses if at least one pair of consecutive houses must have the same color?
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• Combinatorial identity
  If <math>k > n</math> then <math>\binom{n}{k} = 0 = \binom{n - 1}{k - 1} + \bi Also, we can look at Pascal's Triangle to see why this is. If we were to extend Pascal's Triangle to row n, we would see the term <math>\
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• Integer
  ...), or the negatives of the natural numbers (<math>-1,-2,-3,\ldots</math>). If <math>a</math> and <math>b</math> are integers, then their sum <math>a+b</m ...egers). Integers in Python can be of arbitrary size, theoretically limited only by computer memory.
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• Prime factorization
  ...s (or powers of primes) are a factor and then move up to a different prime if said prime is not a factor. When a prime is a factor, we factor out the p
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• Circle
  ...ometry)|tangent]] of that circle. Note that any point on a circle can have only one tangent. ...nt that has endpoints on the circle is called the [[chord]] of the circle. If the chord is extended to a line, that line is called a secant of the circle
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• Ellipse
  ...on <math>\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1</math>. In particular, if the center of the ellipse is the origin this simplifies to <math>\frac{x^2}
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• Divisor
  ...al number <math>{k}</math> such that <math>n=kd</math> or, in other words, if <math>\frac nd</math> is also a natural number (i.e <math>d</math> divides See the main article on [[counting divisors]]. If <math>n=p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}}\cdot\dots\cdot p_m^{\al
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• Base numbers/Common bases
  ...11103_8</math> only marginally exceeds the time of decimal multiplication (if you are good in division by 8 with remainder, of course).
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• Henstock-Kurzweil integral
  ...h> is ''Generalized Riemann Integrable'' on <math>[a,b]</math> if and only if, <math>\forall\epsilon>0</math>, there exists a [[gauge]] <math>\delta:[a,b if <math>\mathcal{\dot{P}}</math> is a <math>\delta</math>-fine [[Partition of
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• Multiple
  ...me other integer. Thus <math>k</math> is a multiple of <math>m</math> only if <math>k</math> can be written in the form <math>mn</math>, where <math>m</m
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• Economics 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
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• Math Day at the Beach
  ...to work. The first to ring in is given a chance to say the correct answer; if that is wrong, the second person is given a limited amount of time (varying
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• Ptolemy's Inequality
  ...clic quadrilateral with diagonals <math>AC </math> and <math>BD </math> OR if <math>A, B, C, D</math> are collinear. This also holds if <math>A,B,C,D</math> are four points in space not in the same plane, but eq
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• Pi
  Note that if <math>x</math> is Liouvillian, then so is any rational multiple of any posi If <math>k=\ell</math>, we get the sum of all inverse squares starting with <m
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• Phi
  ...dical]] <math>\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}</math>. It is the only positive real number that is one more than its [[perfect square| square]] a ..._n</math> is the nth number in the [[Fibonacci sequence]]. In other words, if you divide the <math>n</math>^th term of the Fibonacci series ove
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• Fibonacci sequence
  ## How about if the colony starts with <math>a</math> rabbits and <math>b</math> offspring? #Find <math>a</math> if <math>a</math> and <math>b</math> are [[integer]]s such that <math>x^2 - x
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• Muirhead's Inequality
  '''Muirhead's Inequality''' states that if a sequence <math>p</math> [[Majorization|majorizes]] a sequence <math>q</ma ...application of AM-GM instead. Thus, it is suggested that Muirhead be used only to verify that an inequality ''can'' be proved with AM-GM before demonstrat
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• Injection
  ...on <math>f:A\to B</math> is an injection if for all <math>x,y\in A</math>, if <math>f(x)=f(y)</math> then <math>x=y</math>.
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• 1988 AJHSME Problems/Problem 5
  If <math>\angle \text{CBD}</math> is a [[right angle]], then this protractor i
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• Fundamental Theorem of Calculus
  ...)-F(x_i)</math>. But this is approximately <math>F'(x_i)\Delta x</math>. If <math>\Delta F_i</math> is the amount that <math>F</math> changes on the in If we repeat this process, using more and more subintervals, then our approxim
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• Polygon
  ...gon can be [[regular polygon| regular]] or irregular. A polygon is regular if all sides are the same length and all angles are [[congruent]].
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• Isoperimetric Inequalities
  If a figure in a plane has area <math>A</math> and perimeter <math>P</math> th
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• 2011 AMC 10B Problems/Problem 4
  ...ve Bernardo <math>50 - 45 = 5</math>. Looking at the answer choices we see only <math>\frac{50-40}{2} = 5</math> works. <math>\boxed{\textbf{(C) } \;\frac{
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• Bretschneider's formula
  Then if <math>a, b, c, d</math> represent <math>|\vec{a}|, |\vec{b}|, |\vec{c}|, |\
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• Vector
  ...vectors can be compared by only looking at their endpoints. This is why we only require <math>n</math> values for an <math>n</math> dimensional vector writ ...h>\bold{b}</math>. Note that the dot product is <math>0</math> if and only if the two vectors are perpendicular.
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• Logarithm
  left to right parenthesized exponentiation) is only 7 digits before the decimal point. Comparing the logs of the numbers to a g An only partially related value is the discrete logarithm, used in [[cryptography]]
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• Law of Cosines
  1. If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the (2) If <math>DC=2\sqrt{2}</math>, find <math>BC</math>.
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• Iff
  '''Iff''' is an abbreviation for the phrase "if and only if." * if <math>p</math> then <math>q</math>
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• Complex number
  ...t turns out that in the system that results from this addition, we are not only able to find the solutions of <math> x^2 = -1 </math> but we can now find ' ...a complex number is real: for example, <math>\Im(3 + 4i) = 4</math>. So, if <math>z\in \mathbb C</math>, we can write <math>z=\mathrm{Re}(z)+i\mathrm{I
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• Rational number
  ...er satisfies exactly one of the last two conditions. The same remark holds if "decimal" is replaced with any other [[number base|base]].
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• Polar form
  ...\cdot\mathrm{cis }(\theta)=r(\cos \theta+i\sin\theta)</math>. (See [[cis]] if you do not understand this notation.) This represents a complex number <mat
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• Roots of unity
  ...ering the magnitudes of both sides of the equation <math> x^n = 1</math>. If we let <math> x = re^{i\theta} </math>, we see that <math> r^n = 1</math>, ...> n>1 </math>, the sum of the ''n''th roots of unity is 0. More generally, if <math>\zeta</math> is a primitive ''n''th root of unity (i.e. <math>\zeta^m
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• Without loss of generality
  ...marked off. For example, you can't say "WLOG, let <math>a > b > c</math>." if <math>a</math> could equal <math>b</math> or <math>c</math>. * If three objects are each painted either red or blue, then there must be at le
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• Law of Sines
  This method only works to prove the regular (and not extended) Law of Sines. *If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the
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• Sequence
  Intuitively, a sequence '''converges''' if its terms approach a particular number. ...integers <math>n \ge k</math>, we have <math>|x_n - L| < \epsilon</math>. If <math>(x_n)</math> converges to <math>L</math>, <math>L</math> is called th
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• Science
  ...[[randomisation]], and regulation (i.e. [[experimental control|control]].) If data is obtained from some source that does not use these three benchmarks
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• Geometric sequence
  ...math>b</math>, and <math>c</math> are in geometric progression if and only if <math>b / a = c / b</math>. A common lemma is that a sequence is in geometric progression if and only if <math>a_n</math> is the [[geometric mean]] of <math>a_{n-1}</math> and <mat
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• Arithmetic sequence
  ...ath>b</math>, and <math>c</math> are in arithmetic progression if and only if <math>b - a = c - b</math>. ...r common lemma is that a sequence is in arithmetic progression if and only if <math>a_n</math> is the [[arithmetic mean]] of <math>a_{n-1}</math> and <ma
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• Fermat's Last Theorem
  ...ematicians today doubt that Fermat actually had a proof for this theorem. If he did have one, he never published it, though he did publish a proof for t
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• Homothety
  ...h>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math>. If the sides of triangle <math>ABC</math> are <math>13,</math> <math>14,</math
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• 2005 USAMO Problems/Problem 4
  (The table is ''stable'' if it can be placed so that all four of the leg ends touch the floor. Note th ...btain a set of cuts that is counted in <math>C_{i-1}</math>. For example, if <math>\{2,3,4,5\}</math> was counted in <math>C_5</math>, then <math>\{1,2,
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• 2000 AMC 12 Problems/Problem 1
  ...lowest only if, of the <math>3</math> [[factor]]s, two of them are small. If we want to make sure that this is correct, we could test with a smaller num The sum is the highest if two [[factor]]s are the lowest.
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• Diophantine equation
  ...always be an infinite number of solutions when <math>\gcd(a,b)=1</math>. If <math>\gcd(a,b)\nmid c</math>, then there are no solutions to the equation. ...viously in the form <math>(bk,-ak)</math> for all integers <math>k</math>. If they are not, we simply divide them by their greatest common divisor.
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• Area
  ...ctangle, we can easily find the area of a [[triangle]] by just noting that if our triangle has base <math>b</math> and height <math>h</math>, then the re ...ith an [[angle]] of <math>\frac{180}{n}^{\circ}</math> at the top vertex. If the polygon has side length <math>s</math>, the height of the triangle can
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• Fraction
  ...fractions should be converted have the same denominator. This is because only quantities with the same units can be combined (we can't add 1 banana and 1
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• Functional equation
  Now, if we multiply the first equation by 3 and the second equation by 4, and add t
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• Exponential form
  ...he sum of a [[real]] and an [[imaginary]] component, <math>z=a+bi</math>. If you consider complex numbers to be [[coordinate]]s in the [[complex plane]]
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• Floor function
  * If <math>x</math> is a positive real number, and <math>n</math> is a positive ...f <math>x</math>, i.e., the greatest integer not exceeding <math>x</math>. If <math>n</math> is a positive integer, express as a simple function of <math
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• Pascal's triangle
  if((i==2 && n<6)||(i==3 && n==6)){ if(n==6){label(string(chew(n,i)),(11+n/2-i,-n),p=red+2.5);}
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• Exponentiation
  In the [[order of operations]], it is the second operation performed if a equation has [[parentheses]] or the first one performed when there is no ...+ 3 +3 + 3 +3 + 3 +3 + 3 +3 + 3</math>, but this gets burdensome quickly: if we wanted to capture the idea of "the amount equal to two hundred 3s." Thu
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• Rational approximation
  ...oximation than <math>\frac 1q</math>. Surprisingly enough, it is possible, if not for all <math>q</math>, then, at least for some of them. ...}{n},1\right),</math>. Since we have <math>n+1</math> fractional parts and only <math>n</math> subintervals, the [[Pigeonhole Principle|pigeonhole principl
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• Trigonometry
  ...<math>90^{\circ}</math> for the triangle to stay right, these definitions only work for acute angles. ...ngle ABC</math> is right, which means that the trigonometric functions are only defined when angle <math>A</math> is acute.
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• Euler's number
  ...that annoying limit on the right turn out to be equal to <math>1</math>? If this were possible, then for that special value of <math>b</math>, it would ...ould happen to the expression <math>\frac{b^{\Delta x}-1}{\Delta x}</math> if <math>b</math> were equal to <math>(1+\Delta x)^{\frac{1}{\Delta x} }</math
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• Exponential function
  :If <math>b > 1</math>, then the function will show growth. :If <math>0 < b < 1</math>, then the function will show decay.
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• Complex analysis
  ...> be an [[entire]] function (i.e. holomorphic on the whole complex plane). If <math>\lvert f(z)\rvert \le A</math> for all <math>z</math> for some real This is a powerful generalization of Liouville's Theorem. If <math>f</math> is an entire function so that there exist two complex number
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• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  The geometric mean is the theoretical existence if the root mean power equals 0, which we couldn't calculate using radicals be with equality if and only if <math>x_1=x_2=\cdots=x_n</math>. This inequality can be expanded to the [[
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• Goldbach Conjecture
  ...math> are prime are statistically independent of each other. Nevertheless, if one pursues this heuristic, one might expect the total number of ways to wr ...ial relation because (besides 2) only odd numbers can be prime. Similarly, if <math>n</math> is divisible by 3, and <math>m</math> was already a prime di
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• Twin Prime Conjecture
  ...twin primes is an idea adopted from the proof of [[Dirichlet's Theorem]]. If one can show that the sum ...o only if one of variables is 0 will the factorization be trivial (contain only 1 and itself).
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• Riemann Hypothesis
  ...he Riemann Hypothesis would hold. The Riemann Hypothesis would also follow if <math>M(n)\le C\sqrt{n}</math> for any constant <math>C</math>.
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• Quadratic residues
  ...that <math>a</math> is a '''quadratic residue''' [[modulo]] <math>m</math> if there is some integer <math>n</math> so that <math>n^2-a</math> is [[divisi ...d }\ a\ \mathrm{\ is\ a\ quadratic\ residue\ modulo\ }\ p, \\ -1 & \mathrm{if }\ p\nmid a\ \mathrm{ and }\ a\ \mathrm{\ is\ a\ quadratic\ nonresidue\ mod
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• Power of a Point Theorem
  Notice how this definition still works if <math>A_k</math> and <math>B_k</math> coincide (as is the case with <math>X ...The intersection of <math>P</math> and <math>S</math> must be a circle. If we consider the lines and <math>P</math> with respect simply to that circle
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• Collatz Problem
  ...eneralizing <math>n</math> to an integral, integer coefficient polynomial. If <math>n=2^zx+b</math> for example, it can be shown by parity argument, that ...well as: <cmath>{6m+4\over 2}=3m+2</cmath> we can then observe that; only if <math>m</math> is even will another division by 2 be possible.
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• Algebraic number
  A [[complex number]] is said to be '''algebraic''' if it is a [[root]] of a [[polynomial]] with [[rational]] [[coefficient]]s. Ex Although it seems that the number of algebraic numbers is large, there are only [[countable|countably]] many of them. That is, the algebraic numbers have
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• Nonnegative number
  A [[real number]] is called '''nonnegative''' if it is greater than or equal to [[zero (constant)|zero]]. Basically, a nonne
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• Triangle Inequality
  ...ts, a_n</math> can only be the sides of a nondegenerate <math>n</math>-gon if <math>a_i < a_1 + \ldots + a_{i -1} + a_{i + 1} + \ldots + a_n = \left(\sum
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• Majorization
  ...arily nonincreasing, then we still write <math>\{a_i\}\succ\{b_i\} </math> if this is true after the sequences have been sorted in nonincreasing order. ...th> a_1, \ldots, a_n </math>, and write <math>\{b_i\}\prec\{a_i\} </math>, if <math>\{a_i\}\succ\{b_i\} </math>.
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• Range
  Thus, if we have <math>f: \mathbb{R} \to \mathbb{R}</math> given by <math>f(x) = x^2
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• Prime Number Theorem
  If follows that <math>\limsup \vartheta(x)/x = \liminf \vartheta(x)/x =1</math
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• If <math>a_1, a_2, \dotsc, a_n, b_1, b_2, \dotsc, b_n, \dotsc, z_1, z_2, \dots If <math>p,q>1</math>, <math>1/p+1/q=1</math>, <math>f\in L^p, g\in L^q</math>
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• Infinite
  ...'''infinite''' if there is a [[surjection]] <math>f:S\to\mathbb{Z}</math>. If this is not the case, <math>S</math> is said to be [[finite]]. In simplified language, a set is infinite if it doesn't end, i.e. you can always find another element that you haven't e
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• Rhombus
  * If all of a rhombus' [[angle]]s are [[right angle]]s, then the rhombus is a [[ If two lines are cut by a transversal and same-side interior angles add up to
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• Kite
  ...ram]] (if the members of each pair are opposite each other), and the kite (if the members of each pair are adjacent to each other). The properties of ki
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• Trapezoid
  ...we say it is an [[isosceles trapezoid]]. A trapezoid is cyclic if and only if it is isosceles. If <math>A</math> denotes the area of a trapezoid, <math>b_1,b_2</math> are th
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• Mock AMC
  Anyone can write a Mock AMC and administer it. If you are interested in writing one, here are some tips: * Look at famous theorems and formulas and see if there's any way you can make a good problem out of them.
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• Commutative property
  ...]) is said to have the '''commutative property''' or to ''be commutative'' if the order of its arguments does not affect the value. Formally, an operation <math>G: S \to S</math> is commutative if and only if <math>\forall a, b \in S, G(a, b) = G(b, a)</math>.
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• Algebraic topology
  ...]] <math>\sim</math> on <math>L</math> by saying that <math>p\sim q</math> if there is a continuous function <math>g:[0,1]\times[0,1]\to X</math> with <m More generally, if <math>X</math> is an [[h-space]], then <math>\pi_1(X)</math> is abelian,
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• Modular arithmetic/Introduction
  '''[[Modular arithmetic]]''' is a special type of arithmetic that involves only [[integers]]. This goal of this article is to explain the basics of modula ...rform tedious computations to discover solutions to interesting problems. If all we need to know about are remainders when integers are divided by <math
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• Perfect square
  An [[integer]] <math>n</math> is said to be a '''perfect square''' if there is an integer <math>m</math> so that <math>m^2=n</math>. The first fe
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• Convex function
  ...math> is [[convex set | convex]]. Equivalently, <math>f </math> is convex if for every <math> \lambda \in [0,1] </math> and every <math> x,y \in I</math We say that <math>f </math> is '''strictly convex''' if equality occurs only when <math>x=y </math> or <math> \lambda \in \{ 0,1 \} </math>.
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• Uncountable
  A [[set]] <math>S</math> is said to be '''uncountable''' if there is no [[injection]] <math>f:S\to\mathbb{N}</math>. Assuming the [[Axi
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• AIME Problems and Solutions
  ...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
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• Modular arithmetic/Intermediate
  ...'modulo'' <math>n</math>, or <math>a \equiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. ...divide the set of integers into sets of equivalent elements. For example, if <math>n = 3</math>, then the integers are divided into the following sets:
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• Right triangle
  If the lengths of the legs and hypotenuse are integral, they are said to form
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• Subset
  ...y a [[set]] <math>A</math> is a '''subset''' of another set <math>B</math> if every [[element]] of <math>A</math> is also an element of <math>B</math>, a
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• Limit
  ...s follows: <cmath>f(x) = \begin{cases} 0 & \text{if } x\neq 0,\\ 1 & \text{if } x=0. \end{cases}</cmath> Although the value of the function <math>f(x)</m ...(x)</math> as <math>x</math> goes to <math>c</math> equals <math>L</math>) if for every positive real <math>\epsilon</math> there exists a positive real
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• 2006 AIME I Problems/Problem 15
  This lower bound can be achieved if we use <math>b_1 = -1</math>, <math>b_2 = 0</math>, <math>b_3 = -1</math>, ...the negative number will have a magnitude three greater, so this is true. If the negative number comes first, then the positive number will have magnitu
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• 2006 AIME I Problems/Problem 13
  ...<math>S_n</math> to be a perfect square, <math>k+n-1</math> must be even. If <math>k</math> is odd, then <math>n+1</math> is even, hence <math>k+n-1</ma If <math>k=0</math>, then <math>n+1</math> is odd, so <math>k+n-1</math> is od
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• LaTeX:Symbols
  You might notice that if you use any of these to typeset an expression that is vertically large, lik If we put \left and \right before the relevant parentheses, we get a prettier
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• 2006 AIME I Problems/Problem 11
  ...ocks of size <math>1, 2, \ldots, m, m + 1</math> as there are towers using only <math>1, 2, \ldots, m</math>. There are 2 towers which use blocks <math>1, ...only worked when <math>m \geq 2</math>: when <math>m = 1</math>, there are only 2 places to insert a block of size <math>m + 1 = 2</math>, at the beginning
  3 KB (436 words) - 05:40, 4 November 2022
• 2006 AIME I Problems/Problem 10
  Assume that if unit [[square]]s are drawn circumscribing the circles, then the line will d
  4 KB (731 words) - 17:59, 4 January 2022
• 2006 AIME I Problems/Problem 5
  If it was required to solve for each variable, dividing the product of the thr
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• 2006 AIME I Problems/Problem 3
  ...visible by <math>7.</math> But <math>a_n</math> is a nonzero digit, so the only possibility is <math>a_n = 7.</math> This gives <cmath>7 \cdot 10^n = 28N_0 Note that if we let the last digit be <math>c</math> we must have <math>9c \equiv c \pmo
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• 2021 JMC 10
  A day in the format <math>mm/dd</math> is called ''binary'' if all of the digits are either <math>0</math>s or <math>1</math>s with leadin A positive integer is ''pretentious'' if it has both even and odd digits. For example, <math>74</math> and <math>83<
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2006 AMC 12B Problems
  ...hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to ca ...ne unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many differe
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• 2006 AMC 12A Problems
  ...uppose <math>x>0</math>, and let <math>S=(1,x,x^2,\ldots ,x^{100})</math>. If <math>A^{100}(S)=(1/2^{50})</math>, then what is <math>x</math>? <math>(2)</math> If <math>S</math> contains <math>k</math> elements, then <math>S</math> contai
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• 2005 AMC 12A Problems
  ...needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately? ...th>a</math> for which the equation <math>4x^2 + ax + 8x + 9 = 0</math> has only one solution for <math>x</math>. What is the sum of these values of <math>a
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2004 AMC 12A Problems
  ...points, and each problem left unanswered is worth <math>2.5</math> points. If Charlyn leaves <math>8</math> of the <math>25</math> problems unanswered, h ...s. Marketing research suggests that using wider jars would increase sales. If the diameter of the jars is increased by <math>25\%</math> without altering
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2003 AMC 12A Problems
  ...shirt for home games and another pair of socks and a shirt for away games. If the total cost is &#36;2366, how many members are in the League? ...to walk uphill <math>1</math> km from her home to school, but it takes her only <math>10</math> minutes to walk from school to her home along the same rout
  13 KB (1,955 words) - 21:06, 19 August 2023
• 2002 AMC 12A Problems
  If the order in which the exponentiations are performed is changed, how many o ...es at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 m
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2000 AMC 12 Problems
  If <math>|x - 2| = p,</math> where <math>x < 2,</math> then <math>x - p =</mat ...>13</math>, and <math>25</math> nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would ther
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2001 AMC 12 Problems
  ...}y</math> for all positive real numbers <math>x</math> and <math>y</math>. If <math>f(500) =3</math>, what is the value of <math>f(600)</math>? ...of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2002 AMC 12B Problems
  If <math>a,b,c,d</math> are positive real numbers such that <math>a,b,c,d</mat ...ts half as fast as Beth’s mower and one third as fast as Andy’s mower. If they all start to mow their lawns at the same time, who will finish first?
  10 KB (1,547 words) - 04:20, 9 October 2022
• 2003 AMC 12B Problems
  ...cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted If <math>\log (xy^3) = 1</math> and <math>\log (x^2y) = 1</math>, what is <mat
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2004 AMC 12B Problems
  If <math>x</math> and <math>y</math> are positive integers for which <math>2^x ...nteger <math>x</math>, with <math>10\leq x\leq 99</math>, is to be chosen. If all choices are equally likely, what is the probability that at least one d
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  ...She earned an A on <math>22</math> of the first <math>30</math> quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes ca
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12B Problems/Problem 1
  <math>(-1)^n=1</math> if n is even and <math>-1</math> if n is odd. So we have
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• 2006 AMC 12B Problems/Problem 3
  If the Cougars won by a margin of 14 points, then the Panthers' score would be
  910 bytes (136 words) - 13:39, 13 February 2016
• 2006 AMC 12B Problems/Problem 4
  If we simplify the equation to "x", we get <math>x=5</math>
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• 2006 AMC 12B Problems/Problem 5
  ...hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to ca
  654 bytes (115 words) - 21:47, 1 August 2020
• 2006 AMC 12B Problems/Problem 7
  There are only two possible occupants for the driver's seat. After the driver is chosen, a If there was no restriction, there would be 4!=24 ways to sit. However, only 2/4 of the people can sit in the driver's seat, so our answer is <math>\fra
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• 2006 AMC 12B Problems/Problem 9
  ...b<c</math>, none of the digits can be zero and <math>c</math> cannot be 2. If <math>c=4</math>, then <math>a</math> and <math>b</math> must each be chose ...e anything from 2-7 (If it was 1 we would have the hundreds digit to be 0, if it was more than 7, the ones digit cannot be even).
  3 KB (409 words) - 17:10, 30 April 2024
• 2006 AMC 12B Problems/Problem 10
  ...since we want the greatest perimeter, we want the greatest integer x, and if <math>x<7.5</math> then <math>x=7</math>. Then, the first side has length <
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• 2006 AMC 12B Problems/Problem 14
  ...ath>4B+5J=23</math>. Then, it can be guessed and checked quite simply that if <math>B=2</math> and <math>J=3</math>, then <math>4B+5J=4(2)+5(3)=23</math>
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• 2006 AMC 12B Problems/Problem 18
  ...ne unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many differe ...rity]] of exactly one coordinate. Hence after <math>10</math> steps we can only be in locations <math>(x,y)</math> where <math>x+y</math> is even. It can e
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• 2006 AMC 12B Problems/Problem 19
  If <math>b=1</math>, the number is not divisible by <math>2</math> (unless it' If <math>b=2</math>, then the only possible number is <math>7272</math>. <math>7272</math> is divisible by <ma
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• 2006 AMC 12B Problems/Problem 20
  ...math>. Combining these, we get that both hold at the same time if and only if <math>10^k \leq x < \frac{10^{k+1}}4</math>. The largest value for <math>x</math> is <math>10^{0}</math>. If <math>x > 10^{-1}</math>, then <math>\lfloor\log_{10}4x\rfloor</math> doesn
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• 2006 AMC 12B Problems/Problem 22
  ...t\lfloor \frac n{25}\right\rfloor + \cdots</math>. (For any <math>n</math> only finitely many terms in the sum If we now show how to pick <math>a,b,c</math> so that we'll get the result <ma
  5 KB (881 words) - 15:52, 23 June 2021
• 2006 AMC 12B Problems/Problem 23
  ...ath> Solving this quadratic gives <math>s^2=85\pm 42\sqrt{2}.</math> Now, if this were the actual test, we stop here, noting that the question tells us If <math>s^2=85-42\sqrt{2}</math>, using <math>1.4</math> as an approximation
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• 2006 AMC 12B Problems/Problem 25
  ...d by the rule <math>a_{n+2}=|a_{n+1}-a_n|</math> for <math>n\geq 1</math>. If <math>a_1=999</math>, <math>a_2<999</math> and <math>a_{2006}=1</math>, how We say the sequence <math>(a_n)</math> completes at <math>i</math> if <math>i</math> is the minimal positive integer such that <math>a_i = a_{i +
  5 KB (924 words) - 12:02, 15 June 2022
• 2006 AMC 12A Problems/Problem 14
  It becomes evident that if you exchange <math>5</math> pigs for <math>7</math> goats, we get the small
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• 2006 AMC 12A Problems/Problem 17
  ...f of <math>s</math>, thus eliminating answer choice (A). <math>r</math> is only a little bit bigger than half of <math>s</math>, so we can reasonably assum
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• 2006 AMC 12A Problems/Problem 19
  ...h>(14,9)</math>, and let <math>L_2</math> be the line <math>y=mx+b</math>. If we let <math>\theta</math> be the measure of the acute angle formed by <mat
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• 2006 AMC 12A Problems/Problem 20
  Therefore, there is only <math>1</math> way out of <math>81</math> to get to <math>D</math> last. The bug then has only <math>1</math> way to go to a point <math>E</math> on the opposite face, th
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• 2006 AMC 12A Problems/Problem 22
  ...re six arcs. The probability of choosing a point is <math>1 / 2</math>, or if the total arc degree measures add up to <math>\frac{1}{2} \cdot 360^{\circ} If we draw the altitude of <math>APO</math> from <math>P</math>, then we get a
  2 KB (343 words) - 15:39, 14 June 2023
• 2006 AMC 12A Problems/Problem 23
  ...uppose <math>x>0</math>, and let <math>S=(1,x,x^2,\ldots ,x^{100})</math>. If <math>A^{100}(S)=(1/2^{50})</math>, then what is <math>x</math>? ...ly, <math>A^{100}(S)=\frac{(x+1)^{100}}{2^{100}}</math> To find x, we need only solve the equation <math>\frac{(x+1)^{100}}{2^{100}}=\frac{1}{2^{50}}</math
  3 KB (466 words) - 22:40, 29 September 2023
• 2006 AMC 12A Problems/Problem 24
  if the exponent of <math>y</math> is <math>1</math>, the exponent of <math>z</ if the exponent of <math>y</math> is <math>3</math>, the exponent of <math>z</
  8 KB (1,332 words) - 17:37, 17 September 2023
• 2006 AMC 12A Problems/Problem 25
  <math>(2)</math> If <math>S</math> contains <math>k</math> [[element]]s, then <math>S</math> co Note that if <math>n < 2k - 1</math>, the answer will be 0. Otherwise, the <math>k</mat
  8 KB (1,405 words) - 11:52, 27 September 2022
• 2005 AMC 12A Problems/Problem 20
  \begin{array}{clr} f(x) & = 2x, & \text { if } 0 \leq x \leq \frac {1}{2}; \\ f(x) & = 2 - 2x, & \text { if } \frac {1}{2} < x \leq 1. \end{array}
  3 KB (437 words) - 23:49, 28 September 2022
• 2005 AMC 12B Problems/Problem 4
  ...n <math>A</math> on <math>22</math> of the first <math>30</math> quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes ca Here, only the <math>A</math>'s matter... No complicated stuff!
  1 KB (197 words) - 14:16, 14 December 2021
• 2005 AMC 12B Problems/Problem 7
  ...e values, we are left with the following 4 equations (using the logic that if <math>|a|=b</math>, then <math>a</math> is either <math>b</math> or <math>-
  2 KB (357 words) - 20:15, 27 December 2020
• 2005 AMC 12B Problems/Problem 8
  ...ues (if any) such that <math>a=a^2</math>. Solving for <math>a</math>, the only values that satisfy this are <math>0</math> and <math>1</math>, so the answ
  829 bytes (135 words) - 18:23, 9 September 2020
• 2005 AMC 12B Problems/Problem 11
  ...&#36;<math>20</math> or more is if you pick a twenty and another bill, or if you pick both tens. There are a total of <math>\dbinom{8}{2}=\dfrac{8\times ...d be said for drawing anything, and then drawing a twenty. However, we can only draw something that isn't a twenty first (since we've already accounted for
  4 KB (607 words) - 21:01, 20 May 2023
• 2005 AMC 12B Problems/Problem 12
  If the roots of <math>x^2 + mx + n = 0</math> are <math>2a</math> and <math>2b
  2 KB (317 words) - 12:27, 16 December 2021
• 2005 AMC 12B Problems/Problem 15
  The only distinct positive integers that can add up to <math>11</math> is <math>(1+2 ...e can try a different approach by looking at <math>(a+b+c+d)</math> first. If <math>(e+f+g+h)=21</math>, <math>10(a+b+c+d)=221-21=200</math>, or <math>(a
  2 KB (411 words) - 21:02, 21 December 2020
• 2005 AMC 12B Problems/Problem 16
  ...ne. To find the circle enclosing these <math>4</math> circles, notice that if you connect the <math>4</math> centers as a square, the diameter of the lar
  2 KB (364 words) - 04:54, 16 January 2023
• 2005 AMC 12B Problems/Problem 19
  ..., <math>2a=15</math>, <math>a=15/2</math>, which is not a digit. Hence the only possible value for <math>a-b</math> is <math>1</math>. Now we have <math>(a
  2 KB (283 words) - 20:02, 24 December 2020
• 2005 AMC 12B Problems/Problem 20
  ...th>13</math>, this sum could only be even if it had two or four odd terms. If it had all four odd terms then it would be <math>13-7-5-3=-2</math>, and wi We start off with trying to get a <math>2(4^2) = 32</math> solution, so we only need to find one set with a sum of 4 (the other will automatically have a s
  3 KB (463 words) - 19:28, 6 November 2022
• 2005 AMC 12B Problems/Problem 25
  ...one diagonal (from top left to bottom right). EDIT: This part is wrong as if you choose the 4 vertices that have a cross section as a square, there exis Suppose an ant moved across this diagonal; then the ant at the other end can only move across the diagonal (which creates 2-cycle, bad) or it can move to ano
  10 KB (1,840 words) - 21:35, 7 September 2023
• Root (polynomial)
  <ul> <li> <math>0</math> is a root of <math>P(x)</math> if and only if the constant term <math>c_0</math> is <math>0</math>. <li> <math>1</math> is a root of <math>P(x)</math> if and only if the sum of the coefficients is <math>0</math>.
  8 KB (1,427 words) - 21:37, 13 March 2022
• Vertical line test
  The vertical line test states that a [[relation]] is a [[function]] if no vertical [[line]] intersects the graph in more than one point. In other words, for every x value, there should only be one y value.
  659 bytes (114 words) - 10:41, 27 April 2024
• Algebraically closed
  In [[abstract algebra]], a [[field]] is said to be '''algebraically closed''' if any nonconstant [[polynomial]] with [[coefficient]]s in the field also has
  343 bytes (52 words) - 17:17, 12 March 2014
• 2006 AMC 10A Problems
  ...mber matches the first and loses otherwise. How much should the player win if the game is fair? (In a fair game the probability of winning times the amou
  13 KB (2,028 words) - 16:32, 22 March 2022
• 2006 AMC 10A Problems/Problem 13
  ...mber matches the first and loses otherwise. How much should the player win if the game is fair? (In a fair game the probability of winning times the amou ...ity of winning is <math>\frac{1}{2}\cdot \frac{1}{6} =\frac{1}{12}</math>. If the game is to be fair, the amount paid, <math>5</math> dollars, must be <m
  1 KB (207 words) - 09:39, 25 July 2023
• 2006 AMC 10A Problems/Problem 17
  We see that if we draw a line to <math>BZ</math> it is half the width of the rectangle so
  6 KB (1,066 words) - 00:21, 2 February 2023
• 2006 AMC 10A Problems/Problem 21
  ...math> is occupying the first digit, we have <math>512</math> arrangements. If the <math>2</math> or <math>3</math> is not occupying the first digit, ther .../math> is occupying the first digit, we have <math>64</math> arrangements. If the <math>2</math> or <math>3</math> is not occupying the first digit, ther
  3 KB (525 words) - 20:25, 30 April 2024
• Set
  ...for every object <math>x</math>, we have <math>x\in S'</math> if and only if <math>x\in S''</math>. ...y many (maybe [[infinite]]ly many) elements? You cannot list them all even if you spend your entire life writing!
  11 KB (2,021 words) - 00:00, 17 July 2011
• 2006 USAMO Problems
  if and only if <math>s</math> is not a divisor of <math>p-1</math>. ...h> is the largest power of 2 that is a factor of <math>n</math>. Show that if <math>k\ge2</math> is a positive integer and <math>i</math> is a nonnegativ
  3 KB (520 words) - 09:24, 14 May 2021
• 1991 AJHSME Problems
  If <math>991+993+995+997+999=5000-N</math>, then <math>N=</math> If <math>\frac{2+3+4}{3}=\frac{1990+1991+1992}{N}</math>, then <math>N=</math>
  17 KB (2,246 words) - 13:37, 19 February 2020
• 2022 USAJMO Problems/Problem 1
  ...ath>m | gr^k - 2gr^{k+1} + gr^{k+2} = gr^k(r-1)^2 \; (6)</math>. Note that if <math>m|g,r</math>, then <math>m|gr^{n+1}-gr^n</math>, which, plugged into so if <math>r \equiv \pm 1 \pmod m</math> or <math>gr \equiv 0 \pmod m</math>, th
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• Prime ideal
  ...at an [[ideal]] <math>P</math> of a [[ring]] <math>R</math> is '''prime''' if <math>P\ne R</math> and for any ideals <math>I,J\subseteq R</math> with <ma ...</math> is prime iff <math>P\ne R</math> and for any <math>a,b\in R</math> if <math>ab\in P</math> then either <math>a\in P</math> or <math>b\in P</math>
  967 bytes (176 words) - 18:08, 7 April 2012
• 2005 AIME II Problems/Problem 4
  ...e factors which divide all three of our numbers. Luckily, we see that the only such factor is 1, so we must add 1 to our previous sum to get an answer of ...1</math> to <math>5^{10}</math>) for <math>p_1=5</math>, and 1 possibility if <math>n=1</math>. From this case, there are <math>11+22+10+1=44</math> poss
  3 KB (377 words) - 18:36, 1 January 2024
• 2005 AIME II Problems/Problem 13
  ...th>10</math>. We find that <math>x=19</math> and <math>x=22</math> are the only values that allow <math>(x-17)(x-24)</math> to be a factor of <math>10</mat .../math>. Testing every potential product of the roots, we find out that the only product that can have divisors that sum up to <math>41</math> is <math>\box
  4 KB (642 words) - 14:55, 12 August 2019
• 2005 AIME II Problems/Problem 5
  ...h>a</math>, we get <math>\log_a(b) + \frac{6 \log_a(a)}{\log_a(b)}</math>. If we substitute <math>x</math> for <math>\log_a(b)</math>, we get <math>x + \
  3 KB (547 words) - 19:15, 4 April 2024
• 2005 AIME II Problems/Problem 6
  ...nd so on, until piles <math> A </math> and <math> B </math> are exhausted. If, after the restacking process, at least one card from each pile occupies th ...over. The same has happened to pile B except replace odd with even. Thus, if 131 is still at index 131, an odd number, then 131 must be from pile A. The
  2 KB (384 words) - 00:31, 26 July 2018
• 2005 AIME II Problems
  ...nd so on, until piles <math> A </math> and <math> B </math> are exhausted. If, after the restacking process, at least one card from each pile occupies th
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 7
  It now becomes apparent that if we multiply the [[numerator]] and [[denominator]] of <math>\frac{4}{ (\sqrt
  2 KB (279 words) - 12:33, 27 October 2019
• 2005 AIME II Problems/Problem 9
  ...(\frac{\pi}2 - u\right) = \cos u</math> hold for all real <math>u</math>. If our original equation holds for all <math>t</math>, it must certainly hold ...sin nu</math>. We know that two [[complex number]]s are equal if and only if both their [[real part]] and [[imaginary part]] are equal. Thus, we need t
  6 KB (1,154 words) - 03:30, 11 January 2024
• Problem solving
  ...ing blocks, but as yet another place whose coordinates had to be memorized if ever they were to visit again. Almost to a student, the difference between ...problem solving because it is the most fundamental logical discipline. Not only is it the foundation upon which sciences are built, it is the clearest way
  6 KB (1,039 words) - 17:43, 30 July 2018
• Square root
  ...> before the root. Thus <math>y</math> is a square root of <math>x</math> if <math>x</math> is the square of <math>y</math>. ...\sqrt x</math>. For instance, <math>\sqrt 4 = 2</math>. When we consider only [[positive number|positive]] [[real number|reals]], the square root [[funct
  2 KB (275 words) - 20:58, 8 January 2024
• 2005 AIME I Problems
  ...quare formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there a # From any lattice point <math> (a,b), </math> the particle may only move to <math> (a+1,b), (a,b+1), </math> or <math>(a+1,b+1). </math>
  6 KB (983 words) - 05:06, 20 February 2019

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