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• Unique factorization domain
  ...arithmetic]] holds. More precisely an integral domain <math>R</math> is a unique factorization domain if for any nonzero element <math>r\in R</math> which i * This representation is unique up to units and reordering, that is if <math>r = p_1p_2\cdots p_n = q_1q_2\
  6 KB (1,217 words) - 23:05, 23 August 2009

Page text matches

• Art of Problem Solving
  The accomplishments of every student are unique, and there is no way to measure that success. However, we try to record an
  5 KB (667 words) - 17:09, 3 July 2023
• Wilson's Theorem
  ...math>a</math> or <math>b-c</math>&mdash;a contradiction.) This inverse is unique, and each number is the inverse of its inverse. If one integer <math>a</ma
  4 KB (639 words) - 01:53, 2 February 2023
• Polynomial
  ...n the coefficients of the polynomial. We can solve this system and find a unique solution when we have as many equations as we do coefficients. Thus, given
  6 KB (1,100 words) - 01:44, 17 January 2024
• Fermat's Little Theorem
  ...uct of cyclic groups of prime order where the set of prime power orders is unique. We can do this because if any two prime powers are not coprime then <math
  16 KB (2,658 words) - 16:02, 8 May 2024
• Orthic triangle
  Orthic triangles are not unique to their mother triangle; one acute and one to three obtuse triangles are g
  8 KB (1,408 words) - 11:54, 8 December 2021
• Prime number
  ...a product of primes ([[permutation|permutations]] not withstanding). This unique [[prime factorization]] plays an important role in solving many kinds of [[
  6 KB (985 words) - 12:38, 25 February 2024
• Prime factorization
  ...tal Theorem of Arithmetic]] tells us that every [[positive integer]] has a unique prime factorization, up to changing the order of the terms.
  3 KB (496 words) - 22:14, 5 January 2024
• National Online Math League
  ...League''' ('''NOML'''), formerly known as the '''Cody Bowl''', is a highly unique and challenging [[mathematics competition]] for high school and college stu
  3 KB (452 words) - 11:21, 25 June 2006
• Chain Rule
  ...at <math>{x}</math>.") If such a matrix <math>M</math> exists, then it is unique, and it is called <math>F'(x)</math>. Intuitively, the fact that <math>\fr
  12 KB (2,377 words) - 11:48, 22 July 2009
• Fermat's Last Theorem
  ...e this video link for detailed explanation of the proof and the concept of unique factorization: https://youtu.be/jfDbnz-Bp_g
  3 KB (453 words) - 11:13, 9 June 2023
• Improper fractional base
  ...ation in an improper fractional base. (Note that this means there is not a unique representation for each number in an improper fractional base.)
  787 bytes (118 words) - 19:20, 23 October 2010
• Incenter
  ...is the center of the [[incircle]]. Every [[nondegenerate]] triangle has a unique incenter.
  2 KB (381 words) - 19:38, 24 November 2011
• Mathematicial notation
  <math>\mathbb{Z}</math>: the [[integer]]s (a [[unique factorization domain]]).
  8 KB (1,401 words) - 13:11, 17 June 2008
• Riemann zeta function
  of a unique combination of [[prime number]]s, the zeta function can be
  9 KB (1,547 words) - 03:04, 13 January 2021
• Perfect square
  ...adratic equation that have <math>3</math> terms and contain <math>1</math> unique root.
  954 bytes (155 words) - 01:14, 29 November 2023
• Limit
  ...e unique. Fortunately, it is always the case that if a limit exists, it is unique. a contradiction. Therefore limits are unique, as we wanted.
  7 KB (1,325 words) - 13:51, 1 June 2015
• 2002 AMC 12A Problems
  The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The l
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2006 AMC 12A Problems/Problem 24
  ...f(x)</math> where the coefficient of <math>x^k</math> equals the number of unique terms in <math>(x+y+z)^k + (x-y-z)^k</math>. ...ome constant. Therefore, the generating function for the MAXIMUM number of unique terms possible in the simplified expression of <math>(x+y+z)^k + (x-y-z)^k<
  8 KB (1,332 words) - 17:37, 17 September 2023
• OpenCourseWare
  '''OpenCourseWare''' is a unique resource offered by [[MIT]]. There are course decriptions, sample syllabi,
  289 bytes (40 words) - 12:20, 3 July 2006
• Empty set
  ...erally denoted <math>\emptyset</math> or <math>\varnothing</math>) is the (unique) [[set]] containing no elements. It is therefore a [[subset]] of every set.
  489 bytes (84 words) - 21:33, 27 February 2020
• 2005 AIME I Problems
  ...> A(0,12), B(10,9), C(8,0), </math> and <math> D(-4,7). </math> There is a unique square <math> S </math> such that each of the four points is on a different
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 14
  ...h> A(0,12), B(10,9), C(8,0),</math> and <math> D(-4,7). </math> There is a unique [[square]] <math> S </math> such that each of the four points is on a diffe
  3 KB (561 words) - 14:11, 18 February 2018
• 2004 AIME II Problems/Problem 4
  ...digit, there are <math>9</math> possibilities. When <math>n</math> has two unique digits there are two cases. Case 1: two digits are the same with each other
  3 KB (508 words) - 01:16, 19 January 2024
• 1983 AIME Problems
  For <math>\{1, 2, 3, \ldots, n\}</math> and each of its nonempty subsets a unique '''alternating sum''' is defined as follows. Arrange the numbers in the sub
  7 KB (1,104 words) - 12:53, 6 July 2022
• 1987 AIME Problems
  What is the largest positive integer <math>n</math> for which there is a unique integer <math>k</math> such that <math>\frac{8}{15} < \frac{n}{n + k} < \fr
  6 KB (869 words) - 15:34, 22 August 2023
• 1988 AIME Problems
  ..., <math>w_4 = 1 + 27i</math>, and <math>w_5 = -14 + 43i</math>, there is a unique mean line with y-intercept <math>3</math>. Find the slope of this mean line
  6 KB (902 words) - 08:57, 19 June 2021
• 1989 AIME Problems
  ...h between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let <math>D^{}_{}</math> be the difference betw is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_
  7 KB (1,045 words) - 20:47, 14 December 2023
• 1991 AIME Problems
  ...\ldots,a_n^{}</math> are positive real numbers whose sum is 17. There is a unique positive integer <math>n^{}_{}</math> for which <math>S_n^{}</math> is also
  7 KB (1,106 words) - 22:05, 7 June 2021
• 1997 AIME Problems
  ...f(f(x))=x</math> for all values except <math>\frac{-d}{c}</math>. Find the unique number that is not in the range of <math>f</math>.
  7 KB (1,098 words) - 17:08, 25 June 2020
• 2000 AIME II Problems
  Every positive integer <math>k</math> has a unique factorial base expansion <math>(f_1,f_2,f_3,\ldots,f_m)</math>, meaning tha
  6 KB (947 words) - 21:11, 19 February 2019
• 1983 AIME Problems/Problem 13
  ...or <math>\{1, 2, 3, \ldots, n\}</math> and each of its non-empty subsets a unique '''alternating sum''' is defined as follows. Arrange the numbers in the sub
  5 KB (894 words) - 22:02, 5 April 2024
• 1983 AIME Problems/Problem 15
  ...assume that the answer is unique, but would need to prove that this is the unique solution. This can be proven as follows.
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1984 AIME Problems/Problem 11
  ...remove <math>4</math> trees that aren't birch. What you are left with is a unique arrangement of <math>5</math> birch trees and <math>3</math> other trees th ...ve <math>4</math> trees. Adding a tree between each pair of people gives a unique arrangement of <math>5</math> nonadjacent birch trees.
  7 KB (1,115 words) - 00:52, 7 September 2023
• 1987 AIME Problems/Problem 8
  What is the largest positive integer <math>n</math> for which there is a unique integer <math>k</math> such that <math>\frac{8}{15} < \frac{n}{n + k} < \fr ...\frac{7n}{8}</math>. Thus, <math>48n < 56k < 49n</math>. <math>k</math> is unique if it is within a maximum [[range]] of <math>112</math>, so <math>n = 112</
  2 KB (393 words) - 16:59, 16 December 2020
• 1988 AIME Problems/Problem 11
  ...<math>w_4 = 1 + 27i</math>, and <math>w_5 = - 14 + 43i</math>, there is a unique mean line with <math>y</math>-intercept 3. Find the [[slope]] of this mean
  2 KB (422 words) - 00:22, 6 September 2020
• 1989 AIME Problems/Problem 14
  is true for a unique choice of non-negative integer <math>m</math> and digits <math>a_0,a_1,\ldo
  2 KB (408 words) - 17:28, 16 September 2023
• 1989 AIME Problems/Problem 11
  ...h between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique [[mode]] (most frequent value). Let <math>D</math> be the difference betwee
  5 KB (851 words) - 18:01, 28 December 2022
• 1991 AIME Problems/Problem 15
  ...\ldots,a_n^{}</math> are positive real numbers whose sum is 17. There is a unique positive integer <math>n^{}_{}</math> for which <math>S_n^{}</math> is also
  4 KB (658 words) - 16:58, 10 November 2023
• 1992 AIME Problems/Problem 10
  ...he area outside the two circles but inside the square, we want to find the unique area of the two circles. We can do this by adding the area of the two circl
  2 KB (323 words) - 12:05, 16 July 2019
• 1994 AIME Problems/Problem 7
  ...of <math>\frac{12}{2}=6</math> lines. Finally, we add the <math>12</math> unique tangent lines to the circle at each of the lattice points.
  3 KB (442 words) - 19:51, 8 January 2024
• 1996 AIME Problems/Problem 3
  ...product will contain <math>(n+1)^2</math> terms, as each term will have an unique power of <math>x</math> or <math>y</math> and so none of the terms will nee ...that the coefficients in the problem statement have no effect on how many unique terms there will be in the expansion. Therefore this problem is synonymous
  3 KB (515 words) - 04:29, 27 November 2023
• 1997 AIME Problems/Problem 12
  ...f(f(x))=x</math> for all values except <math>\frac{-d}{c}</math>. Find the unique number that is not in the range of <math>f</math>. ...function, and since this is a linear function over a linear function, the unique number not in the range of <math>f</math> will be <math>e</math>. <math>\fr
  11 KB (2,063 words) - 22:59, 21 October 2023
• 2000 AIME I Problems/Problem 6
  There, a unique value of <math>x, y</math> is formed for every value of <math>k</math>. How
  6 KB (966 words) - 21:48, 29 January 2024
• 2001 AIME I Problems/Problem 15
  ...ath>7!.</math> This is because each Hamiltonian cycle corresponds to eight unique ways to label the faces. Label the vertices <math>AR,BR,CR,DR,AX,BX,CX,DX</
  11 KB (1,837 words) - 18:53, 22 January 2024
• 2003 AIME I Problems/Problem 12
  ...we know that <math>\angle{ADB}</math> is not a right angle, and there is a unique other triangle with the matching side-side-angle. ...e{ADB}</math> is not right. Therefore <math>\triangle{C'DB}</math> is the unique triangle mentioned above, so <math>\triangle{CDB}</math> is congruent, in s
  3 KB (487 words) - 22:14, 24 November 2019
• 2003 AIME II Problems/Problem 13
  ...ding to <math>\lambda_1</math>. We do the same computations for our second unique eigenvalue, but I will save the computation to you. There are actually 2 ei
  15 KB (2,406 words) - 23:56, 23 November 2023
• 2000 AIME II Problems/Problem 14
  Every positive [[integer]] <math>k</math> has a unique factorial base expansion <math>(f_1,f_2,f_3,\ldots,f_m)</math>, meaning tha
  7 KB (1,131 words) - 14:49, 6 April 2023
• 2006 USAMO Problems/Problem 3
  Such a factorization is unique. Let <math>d(g_i)</math> denote the degree of
  9 KB (1,699 words) - 13:48, 11 April 2020
• Incircle
  ...tangent]] to each side. Every [[triangle]] and [[regular polygon]] has a unique incircle, but in general polygons with 4 or more sides (such as non-[[squar
  2 KB (384 words) - 18:38, 9 March 2023
• Equation
  A unique aspect to equations is the ability to modify an original equation by perfor
  5 KB (932 words) - 12:57, 26 July 2023

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