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

Page title matches

• 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
• Field
  There is also a unique name for <math>(k\backslash\{0\},\cdot)</math>, which most accept as the ''
  2 KB (362 words) - 23:24, 31 December 2021
• Identity
  Identities in this sense are [[unique]]. Imagine we had two identities, <math>e</math> and <math>e'</math>, for s
  1 KB (238 words) - 13:38, 14 July 2021
• Quadratic reciprocity
  ...define the function <math>a \mapsto \genfrac{(}{)}{}{}{a}{p}</math> as the unique nontrivial multiplicative [[homomorphism]] of <math>\mathbb{F}_p^\times</ma
  7 KB (1,182 words) - 16:46, 28 April 2016
• Dedekind domain
  * Dedekind domains have unique prime factorizations of [[ideal]]s (but not necessarily of elements).
  9 KB (1,648 words) - 16:36, 14 October 2017
• Operator inverse
  If the operation <math>G</math> is associative, inverses are unique.
  1 KB (275 words) - 11:40, 23 November 2007
• Mock AIME 1 2006-2007 Problems/Problem 6
  ...h>P_2</math>, we have that <math>a(y^2 + \frac{45}4) + by = c</math> has a unique [[root]] so <math>b^2 - 4\cdot a \cdot (\frac{45}4a - c) = 0</math> or equi
  3 KB (460 words) - 15:52, 3 April 2012
• Mock AIME 1 2006-2007 Problems/Problem 13
  This is a set of three linear equations. In our case, it has a unique solution <math>(p,q,r)=(-2,-1,1)</math>, hence <math>d_7 = -2d_6 - d_5 + d_
  3 KB (568 words) - 15:50, 3 April 2012
• Mock AIME 2 2006-2007 Problems/Problem 8
  ...x_3 + x_2) = 2</math>. It is clear that <math>x_3 = x_2 = 1</math> is the unique solution to this equation in positive integers. Then <math>x_1 = 8 - x_2 =
  3 KB (470 words) - 00:33, 10 August 2019
• 2006 Romanian NMO Problems
  b) For which matrix <math>A</math> is the pseudo-inverse unique?
  11 KB (1,779 words) - 14:57, 7 May 2012
• Divisor function
  ...ermore, all of those products are unique since each positive integer has a unique prime factorization.
  3 KB (511 words) - 06:58, 21 May 2009
• Stanford Mathematics Tournament
  ...milarly proof-based team round where teams prove foundational results in a unique topic, tied together by a common theme. Topics in the past have included ga
  1 KB (214 words) - 22:37, 10 November 2023
• Free group
  ...group <math>G</math> and a function <math>\theta:I\to G</math>, there is a unique group homomorphism <math>\psi:F\to G</math> so that <math>\theta=\psi\phi</
  2 KB (454 words) - 17:54, 16 March 2012
• Odd integer
  ...Every odd integer can be written in the form <math>2k + 1</math> for some unique integer <math>k</math>.
  736 bytes (127 words) - 13:08, 20 February 2024
• Even integer
  ...en. Every even integer can be written in the form <math>2k</math> for some unique integer <math>k</math>.
  796 bytes (134 words) - 16:45, 17 January 2023
• Monoid
  Often one speaks of groups acting on sets. Since elements groups must have unique inverses, for every <math>a</math> in a group <math>G</math> acting on a se
  3 KB (670 words) - 22:45, 21 May 2008
• 2004 AMC 10A Problems/Problem 4
  ...| = 2-x</math>, so we must solve <math>x - 1 = 2 - x</math>, which has the unique solution <math>x = \frac32</math>.
  2 KB (289 words) - 18:01, 16 January 2021
• Chinese Remainder Theorem
  ...to the [[intersection]] of a unique residue class mod <math>m</math> and a unique residue class mod <math>n</math>, and the intersection of each residue clas
  6 KB (1,022 words) - 14:57, 6 May 2023
• Euclid's proof of the infinitude of primes
  ...ndamental Theorem of Arithmetic]] states that all positive integers have a unique prime factorization. Therefore, <math>N</math> must have a prime factor (po
  1 KB (179 words) - 11:57, 14 August 2022
• ELMO
  ...common mistakes (often this rubric only includes the most common solution; unique ideas are determined on a case-by-case basis). The sheer amount of ideas th
  5 KB (773 words) - 19:16, 17 June 2022
• Mock AIME 4 2006-2007 Problems/Problem 8
  Note that in fact, the answer is not unique because a many numbers can be represented as a [[binomial coefficient]] in
  2 KB (293 words) - 16:20, 8 October 2007
• Fundamental Theorem of Arithmetic
  ...[[expression]] for <math>n</math> (called its [[prime factorization]]) is unique, up to rearrangement of the factors. ...erribly interesting, but it does prove that every [[Euclidean domain]] has unique prime factorization.
  2 KB (376 words) - 23:28, 4 August 2022
• 2020 AMC 10A Problems
  There exists a unique strictly increasing sequence of nonnegative integers <math>a_1 < a_2 < …
  13 KB (1,968 words) - 18:32, 29 February 2024
• 2002 USAMO Problems/Problem 3
  ...<math>x_i \neq x_j </math> for all <math>i \neq j </math>, there exists a unique monic real polynomial <math>P(x) </math> of degree <math>n </math> such th ''Proof 1.'' By the [[Lagrange Interpolation Formula]], there exists a unique real polynomial <math>Q(x) </math> of degree less than <math>n </math> such
  4 KB (688 words) - 13:38, 4 July 2013
• Lagrange Interpolation Formula
  ...h> and any complex numbers <math> y_0, \ldots, y_n </math>, there exists a unique [[polynomial]] <math>P(x) </math> of [[degree of a polynomial | degree]] le ...s useful for many olympiad problems, especially since such a polynomial is unique.
  2 KB (398 words) - 03:50, 20 November 2023
• 2007 AIME I Problems
  For each positive integer <math>p</math>, let <math>b(p)</math> denote the unique positive integer <math>k</math> such that <math>|k-\sqrt{p}| < \frac{1}{2}<
  7 KB (1,218 words) - 15:28, 11 July 2022
• 2007 AIME I Problems/Problem 11
  ...[[positive]] [[integer]] <math>p</math>, let <math>b(p)</math> denote the unique positive integer <math>k</math> such that <math>|k-\sqrt{p}| < \frac{1}{2}<
  3 KB (562 words) - 20:02, 30 December 2023
• 2007 USAMO Problems
  ...1=n</math> and, for each <math>k>1</math>, letting <math>a_k</math> be the unique integer in the range <math>0\le a_k\le k-1</math> for which <math>a_1+a_2+\
  3 KB (539 words) - 13:42, 4 July 2013
• 2007 USAMO Problems/Problem 1
  ...= n</math> and, for each <math>k>1</math>, letting <math>a_k</math> be the unique integer in the range <math>0 \le a_k \le k-1</math> for which <math>a_1 + a
  6 KB (1,204 words) - 20:06, 23 August 2023
• 2004 IMO Shortlist Problems/N2
  ...th> such that the equation <math> \displaystyle \psi(x) = ax </math> has a unique solution. ...Thus the canonical solution to the equation <math>\psi(x) = ax </math> is unique if and only if <math>a </math> is a power of 2. Q.E.D.
  6 KB (1,007 words) - 09:10, 29 August 2011
• Dirichlet convolution
  ...{mn} </math>. Furthermore, for each <math> d \mid mn </math>, there exist unique <math>d_m, d_n </math> such that <math> d_m \mid m </math>, <math> d_n \mid
  3 KB (613 words) - 21:40, 21 June 2009
• 2004 IMO Shortlist Problems/A3
  ...servations, we can see that our function <math> \displaystyle s </math> is unique up to sign.
  6 KB (958 words) - 22:15, 9 June 2007
• 2006 Cyprus MO/Lyceum/Problem 4
  Notice that if <math>f(x) = 0</math>, then <math>x</math> has the unique root of <math>-\frac{\frac{9}{2\sqrt{\alpha}}}{\sqrt{\alpha}} = \frac{-9}{2
  1 KB (216 words) - 10:46, 27 April 2008
• Chicken McNugget Theorem
  <b>Lemma</b>. For any integer <math>N</math>, there exists unique <math>(a_{N},b_{N}) \in \mathbb{Z} \times \{0,1,\ldots,m-1\}</math> such th
  17 KB (2,748 words) - 19:22, 24 February 2024
• Parity
  There exist unique positive integers <math>x</math> and <math>y</math> that satisfy the equati Indeed, by solving, we find <math>(x,y) = (18,62)</math> is the unique solution.
  4 KB (694 words) - 22:00, 12 January 2024
• 2005 AMC 12A Problems/Problem 11
  ...ath> depends on the value of both <math>A</math> and <math>C</math> and is unique for each <math>(A,C)</math>. Thus our answer is <math>9 \cdot 5 \cdot 1 = 4
  2 KB (266 words) - 00:59, 19 October 2020
• 2006 AIME II Problems/Problem 8
  ...the center triangle. Note that given any <math>3</math> colors, there is a unique way to assign them to the corner triangles. We have <math>6</math> differen
  4 KB (695 words) - 10:37, 4 November 2023
• 2005 PMWC Problems/Problem I2
  ...math>\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1</math>. The solution with unique values is <math>a = 2, b = 3, c = 6</math>. If we use this format to guess
  938 bytes (136 words) - 08:56, 6 August 2019
• 1990 USAMO Problems/Problem 2
  ...that for all nonnegative integers <math>n</math>, <math>x=4</math> is the unique solution to the equation <math>f_n(x) = 2x</math>. <math>\blacksquare</mat
  2 KB (322 words) - 19:14, 18 July 2016
• 1999 AHSME Problems
  There are unique integers <math>a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}</math> such that
  13 KB (1,945 words) - 18:28, 19 June 2023
• 1960 IMO Problems/Problem 5
  ...(0,1,1)</math>. For every point <math>P</math> in this region, there exist unique points <math>X</math> and <math>Y</math> such that <math>P</math> is the mi ...,2/3,0)</math>. For every point <math>Z</math> in this region, there exist unique points <math>X</math> and <math>Y</math> such that <math>Z\in XY</math> and
  2 KB (402 words) - 23:28, 18 July 2016
• Excircle
  For any triangle, there are three unique excircles. This follows from the fact that there is one, if any, circle suc
  5 KB (843 words) - 03:02, 1 July 2020
• Brocard point
  ...h> such that <math>\angle PAB=\angle PCA=\angle PBC</math>. It is also the unique point <math>P</math> inside <math>\triangle ABC</math> such that the sum of
  626 bytes (98 words) - 20:13, 27 January 2024
• Construction
  ...verse of a point P with respect to circle C. In other words, construct the unique point <math>P'</math> on ray <math>CP</math> such that <math>CP * CP'</math
  3 KB (443 words) - 20:52, 28 August 2014
• 1974 USAMO Problems/Problem 1
  Assume for the sake of contradiction that is possible for unique integers <math>a,b,c</math>. Let <math>P(x)=d_1x^n+d_2x^{n-1}+\cdots+d_n.</
  7 KB (1,291 words) - 20:30, 27 April 2020
• Isogonal conjugate
  It is clear that each point <math>P</math> has the unique isogonal conjugate point.
  54 KB (9,416 words) - 08:40, 18 April 2024
• Partition (combinatorics)
  The ''empty partition'' (with no parts) is the unique partition of <math>0</math>, so <math>P(0) = 1</math>. The unique partition of <math>1</math> is <math>1</math>, so <math>P(1) = 1</math>.
  10 KB (1,508 words) - 14:24, 17 September 2017
• Symmetric group
  '''Theorem 4.''' There exists a unique homomorphism <math>\epsilon</math> from <math>\mathfrak{S}_M</math> to the ...y its transpositions (Theorem 1), it follows that <math>\epsilon</math> is unique. <math>\blacksquare</math>
  10 KB (1,668 words) - 15:33, 25 May 2008
• 2008 AIME I Problems
  There exist unique positive integers <math>x</math> and <math>y</math> that satisfy the equati
  9 KB (1,536 words) - 00:46, 26 August 2023
• 2008 AIME II Problems
  There exist <math>r</math> unique nonnegative integers <math>n_1 > n_2 > \cdots > n_r</math> and <math>r</mat
  7 KB (1,167 words) - 21:33, 12 August 2020
• 2008 AIME I Problems/Problem 4
  There exist unique positive integers <math>x</math> and <math>y</math> that satisfy the equati Indeed, by solving, we find <math>(x,y) = (18,62)</math> is the unique solution.
  4 KB (732 words) - 22:17, 28 November 2023

View (previous 100 | next 100) (20 | 50 | 100 | 250 | 500)