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
  2 KB (316 words) - 16:03, 1 January 2024
• 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>.
  10 KB (1,761 words) - 03:16, 12 May 2023
• 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?
  8 KB (1,192 words) - 17:20, 16 June 2023
• 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>\
  12 KB (1,993 words) - 23:49, 19 April 2024
• 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.
  2 KB (296 words) - 15:04, 5 August 2022
• 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
  3 KB (496 words) - 22:14, 5 January 2024
• 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
  9 KB (1,555 words) - 20:05, 2 November 2023
• 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}
  5 KB (892 words) - 21:52, 1 May 2021
• 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
  1 KB (274 words) - 19:50, 29 August 2023
• 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).
  2 KB (351 words) - 10:39, 1 October 2015
• 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
  2 KB (401 words) - 09:46, 31 January 2018
• 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
  860 bytes (142 words) - 22:51, 26 January 2021

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