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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
• 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
• Mock AIME 1 2007-2008 Problems/Problem 2
  ...system of two [[equation]]s with two unknowns, indicating that there is a unique solution. Solving by substitution or multiplying the top equation and subtr
  1 KB (182 words) - 17:25, 2 April 2008
• 2008 AIME II Problems/Problem 4
  ...negative integers <math>n_1 > n_2 > \cdots > n_r</math> and <math>r</math> unique integers <math>a_k</math> (<math>1\le k\le r</math>) with each <math>a_k</m
  1 KB (167 words) - 21:47, 21 September 2020
• Complete residue system
  ...which satisfy the following condition: Every integer is [[congruent]] to a unique member of the set modulo <math>n</math>.
  698 bytes (110 words) - 20:53, 1 January 2010
• Residue class
  ...]], a residue of an integer <math>a</math> in modulo <math>n</math> is the unique value of <math>0\leq r \leq n-1</math> such that <math>a=kn + r</math>. In
  670 bytes (115 words) - 19:25, 27 April 2008
• 2008 USAMO Problems/Problem 1
  ...of <math>a</math> such that <math>a^2 + a + 1</math> has arbitrarily many unique prime factors, by our second Lemma, and so it is possible for <math>P</math
  11 KB (1,964 words) - 03:38, 17 August 2019
• Free magma
  ...</math>, the integer <math>n</math> such that <math>x \in M_n(X)</math> is unique. This is called the ''length'' of <math>x</math>; it is sometimes denoted ...duction on <math>n</math> that for all integers <math>n</math>, there is a unique extension <math>f_n</math> of <math>f</math> to <math>\bigcup_{i=1}^n M_n(X
  4 KB (887 words) - 13:19, 6 July 2016
• Homogeneous set
  ...}</math>. Therefore each element of <math>N/H</math> is associated with a unique automorphism of the <math>G</math>-set <math>G/H</math>. ...th>, and <math>K</math> a subgroup of <math>H</math>. Then there exists a unique <math>G</math>-morphism <math>f : G/K \to G/S</math> for which <math>f(K) =
  7 KB (1,332 words) - 18:45, 9 September 2008
• Cycle (permutation)
  ...math>\sigma</math> be an element of <math>S_M</math>. Then there exists a unique set <math>C</math> of cycles of <math>S_M</math> with pairwise disjoint sup ...<math>\bar\sigma</math>, it follows from the lemma that <math>C</math> is unique. <math>\blacksquare</math>
  3 KB (616 words) - 22:13, 12 January 2017
• Orbit-stabilizer theorem
  ...</math> of <math>G</math> for which <math>\alpha(i)= x</math> constitute a unique [[coset |left coset]] modulo <math>\text{stab}(i)</math>. Thus
  1 KB (207 words) - 12:26, 9 April 2019
• Division Theorem
  For any positive integers <math> a </math> and <math> b </math>, there exist unique integers <math> q </math> and <math> r </math> such that <math> b = qa + r
  249 bytes (44 words) - 16:45, 10 November 2011
• Limit point
  ...[[reciprocal]]s of the [[positive integer]]s. Then <math>0</math> is the unique limit point of <math>S</math>.
  1 KB (220 words) - 17:50, 28 March 2009
• Division Algorithm
  ...ny integer <math>a</math> and positive integer <math>b</math>, there exist unique integers <math>q</math> and <math>r</math> such that <math>a = qb + r</math
  2 KB (311 words) - 02:25, 22 June 2009
• 1999 AHSME Problems/Problem 25
  There are unique integers <math>a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}</math> such that ...\equiv 514 \equiv 4 \pmod{6}</math>, and so forth. Continuing, we find the unique solution to be <math>(a_2, a_3, a_4, a_5, a_6, a_7) = (1,1,1,0,4,2)</math>
  4 KB (595 words) - 01:43, 4 October 2023
• 2001 IMO Shortlist Problems/A5
  Now we prove such a sequence is unique. We first claim that a sequence of positive integers <math>a_0, a_1,\dots, which contradicts the fact that they are equal. So there is a unique sequence satisfying the problem conditions.
  3 KB (477 words) - 17:42, 8 August 2019
• 2001 IMO Shortlist Problems/C2
  ...have that for some <math>n</math>-tuple of <math>c_i</math> there exists a unique permutation <math>x</math> of <math>s_n</math> such that <math>S(x)\equiv y
  2 KB (364 words) - 08:55, 31 August 2011
• 2001 IMO Shortlist Problems/C7
  ...r what choices are made at each stage, the final configuration obtained is unique. Describe that configuration in terms of <math>n</math>.
  3 KB (567 words) - 16:49, 5 January 2021
• 2008 IMO Problems/Problem 5
  ...h> to <math>j+n\in B</math>. Since each such set of flippings results in a unique <math>x</math>, all we want to count is the number of flippings. We can fli
  4 KB (677 words) - 01:10, 19 November 2023
• 2007 IMO Shortlist Problems/A4
  We will show that <math>f(x) = 2x</math> is the unique solution to this equation. To this end, let <math>g(x) = f(x) - x</math>. ...Since this function evidently satisfies the problem's condition, it is the unique solution, as desired. <math>\blacksquare</math>
  2 KB (375 words) - 16:44, 4 September 2008
• Opposite category
  ...phism <math>f:A\to B</math> in <math>\mathcal{C}</math> is associated to a unique morphism <math>f^{op}:B\to A</math> in <math>\mathcal{C}^{op}</math> (and e
  2 KB (438 words) - 17:07, 8 September 2008
• 2008 iTest Problems/Problem 80
  ...nction's degree is at most 3 and we know four points, we can construct the unique remainder function.
  3 KB (560 words) - 19:49, 23 November 2018
• Fermat's Two Squares Theorem
  ...math> or <math>p \equiv 1 \pmod{4}</math>; and that this representation is unique. ...]s <math>\mathbb{Z}[i]</math> has a [[Euclidean algorithm]]. Hence it has unique prime factorization, and is a [[principal ideal domain]] (i.e., every [[ide
  4 KB (612 words) - 12:10, 30 May 2019
• Absolute zero
  Because of its unique property, the absolute zero is defined as 0 for the [[Kelvin]] scale -- the
  753 bytes (108 words) - 15:35, 5 April 2024
• 2002 AMC 10A Problems
  The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The l
  11 KB (1,733 words) - 11:04, 12 October 2021
• 1998 AHSME Problems/Problem 19
  ...ath> has <math>2</math> solutions. Thus, there are <math>\boxed {4}</math> unique points such that the triangle has an area of <math>10</math>, or <math>C</m
  2 KB (368 words) - 11:08, 4 February 2017
• 2002 AMC 12A Problems/Problem 15
  The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The l As the unique mode is <math>8</math>, there are at least two <math>8</math>s.
  3 KB (519 words) - 21:34, 23 June 2023
• Transformations
  ...tive) plane to itself so that each point has a unique image and also has a unique point that was mapped to it. They include Translations, Reflections, Dilati
  238 bytes (39 words) - 22:17, 22 November 2015
• 2001 AMC 12 Problems/Problem 16
  If the numbers were all unique, the answer would be <math> 16! </math>. However, since 8 terms appear twic
  3 KB (418 words) - 23:48, 3 July 2022
• 2002 AMC 12A Problems/Problem 24
  ...(E)\ 2004}}</math> values of <math>z</math>, each of which correspond to a unique ordered pair <math>(a, b)</math>.
  4 KB (743 words) - 19:54, 14 March 2024
• 2009 AIME I Problems/Problem 8
  Consider the unique differences <math>2^{a + n} - 2^a</math>. Simple casework yields a sum of <
  3 KB (396 words) - 11:42, 23 January 2023
• 2009 AIME I Problems/Problem 4
  Unique solution: https://youtu.be/2Xzjh6ae0MU
  7 KB (1,117 words) - 00:23, 9 January 2023
• Polynomial ring
  ...ath> that is equivalent to <math>p</math> modulo <math>(x-a)</math>. This unique element is sometimes denoted <math>p(a)</math>. Thus we may associate each ...e ring of integers), this association of polynomials with functions ''is'' unique. In such contexts, the polynomials are often identified with their functio
  12 KB (2,010 words) - 00:10, 3 August 2020
• 2009 USAMO Problems/Problem 3
  b) Prove that such a tasteful tiling is unique.
  2 KB (265 words) - 19:51, 6 May 2023
• Radical axis
  ...r radical axis is the perpendicular to <math>l</math> through point A, the unique point on <math>l</math> such that <math>pow(A, \omega_1) = pow(A, \omega_2) ''Lemma 2:'' There is an unique point P on line <math>O_1O_2</math> such that <math>pow(P, O_1) = pow(P, O_
  10 KB (1,797 words) - 02:05, 24 October 2023
• 1990 AJHSME Problems/Problem 20
  This assumes that incomes of all the families are unique. Otherwise, more than one family could have $98000 as income and the above
  1 KB (160 words) - 12:26, 13 June 2023
• 2009 USAMO Problems
  b) Prove that such a tasteful tiling is unique.
  4 KB (718 words) - 18:16, 17 September 2012
• 2009 USAMO Problems/Problem 5
  ...h>, and <math>G = PA \cap BQ</math> are collinear. However, there exists a unique point <math>Q</math> such that <math>HQ</math>, <math>AC</math>, and <math>
  6 KB (973 words) - 19:24, 18 October 2018
• Prime element
  ..., and hence <math>b</math> is a unit.) The converse of this holds in any [[unique factorization domain]], but it does not hold in a general integral domain.
  928 bytes (179 words) - 21:07, 23 August 2009
• 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
• 1989 IMO Problems/Problem 6
  ...any permutation in <math>A_n</math>. Now take <math>x_1</math> and for the unique value <math>k</math> such that <math>|x_1-x_k|=n</math>, put <math>x_1</mat
  11 KB (1,928 words) - 12:26, 26 July 2023
• 2010 AMC 12A Problems/Problem 25
  ...BD</math> is lengthened from the minimum to the maximum values, there is a unique value of <math>BD</math> somewhere in the middle such that the sum of oppos ...,8)</math>, and no matter how the sides are rearranged only <math>1</math> unique quadrilateral can be formed.
  8 KB (1,315 words) - 03:33, 23 January 2021
• Euclidean space
  .../math>. Similarly, the Euclidean space of dimension <math>n</math> is the unique (up to [[isomorphism]]) [[vector space]] of dimension <math>n</math> over <
  1,018 bytes (164 words) - 12:46, 13 March 2010
• Mock AIME 1 2010 Problems
  ...nced ternary}</math> representation of <math>N</math> is defined to be the unique sequence of integers <math>(b_0, b_1, \ldots, b_m)</math>, with <math>b_i \
  7 KB (1,297 words) - 01:29, 25 November 2016
• Identity matrix
  ...he [[matrix inverse|inverse]] <math>A^{-1}</math> of <math>A</math> is the unique matrix such that <math>AA^{-1} = A^{-1}A = I_n</math>.
  1 KB (178 words) - 00:00, 17 March 2010
• 2010 AIME I Problems/Problem 10
  ...that <math>(10^3)(a_3) + (10)(a_1) \leq 2010</math> there is a [[bijection|unique]] choice of <math>a_2</math> and <math>a_0</math> that makes the equality h ...ctly one way as <math>100a + 10c + e</math>, and this corresponds with one unique <math>100b + d + f</math>, so it remains to count the number of integers be
  7 KB (1,147 words) - 21:58, 23 January 2024
• Mock AIME 2 2010 Problems
  ...th>, where <math>\gcd(p, q, r) = 1</math> and <math>r > 0</math>, find the unique integer <math>k</math> between 0 and 982 inclusive such that <math>983</mat
  7 KB (1,150 words) - 09:10, 8 October 2018
• Mock AIME 2 2010 Answer Key
  ...th>, where <math>\gcd(p, q, r) = 1</math> and <math>r > 0</math>, find the unique integer <math>k</math> between 0 and 982 inclusive such that <math>983</mat
  36 KB (6,214 words) - 20:22, 13 July 2023
• 2010 USAMO Problems/Problem 3
  We observe that our solution is not unique, indeed for any <math>k>0</math>, even terms, respectively, of the unique sequence which satisfies our
  11 KB (1,889 words) - 13:45, 4 July 2013
• TenMarks Education
  ...rents and teachers face when trying to advance math achievement, given the unique learning needs of every child. Each of TenMarks’ interactive worksheets has 10 unique questions, which focuses on one concept at a time. While students practice,
  6 KB (841 words) - 02:41, 2 June 2010
• Getting Started With Python Programming
  set - an unordered list of unique data types - Ex: {1,2,3}
  13 KB (2,299 words) - 12:24, 2 March 2024
• 2011 AMC 12A Problems/Problem 16
  ...ces) of the pentagon needs to be established for the pair. The remaining 3 unique colors can be arranged 3! different ways on the remaining 3 vertices.
  9 KB (1,613 words) - 22:19, 5 June 2023
• 2021 AIME II Problems/Problem 5
  ...m <math>C</math> is fixed. We need obtuse <math>\triangle ABC</math> to be unique, so there can only be one possible location for <math>C.</math> As shown be
  14 KB (2,269 words) - 00:43, 2 January 2023
• 2011 AIME II Problems/Problem 14
  ...s of i, j, and k have been used, so every one of these numbers will have a unique triple for placement, as above with the number five. Thus, after 1, 2, 3, a ...of assignment from <math>a_j\rightarrow j\mod (2,3,5)</math> determines a unique string of <math>30</math> numbers. For example:
  10 KB (1,581 words) - 22:09, 27 August 2023
• 2011 USAMO Problems/Problem 2
  ...<math>s \not\equiv 0 \pmod 5</math>, if a winning vertex exists, it is the unique vertex with <math>r_i \equiv 0</math>. ...able, and with <math>s = 2011 \not\equiv 0 \pmod 5</math> we always have a unique such vertex, we are done.
  4 KB (761 words) - 02:36, 21 April 2015
• 2011 USAMO Problems/Problem 6
  ..., we can choose one of the remaining <math>9</math> subsets to produce the unique element. Hence, proved. <math>\blacksquare{}</math>
  7 KB (1,209 words) - 12:50, 25 August 2023
• Olympiad books
  ...wed by problems). '''Excellent''' book for number theory, and has it's own unique approach, Highly Suggested for Number Theory.
  17 KB (2,261 words) - 00:30, 22 April 2024
• 1998 AHSME Problems/Problem 1
  ...t block of their row. Therefore, <math>D</math>, the only block without a unique digit, must be the top-center block.
  2 KB (346 words) - 14:28, 5 July 2013
• 1997 USAMO Problems
  Prove that for any integer <math>n</math>, there exists a unique polynomial <math>Q</math> with coefficients in <math>\{0,1,...,9\}</math> s
  3 KB (507 words) - 14:31, 12 April 2023
• 1997 USAMO Problems/Problem 3
  Prove that for any integer <math>n</math>, there exists a unique polynomial <math>Q(x)</math> with coefficients in <math>\{0,1,...,9\}</math ...as to be within the set <math>{0,1,...,9}</math>, <math>a_0</math> must be unique according to <math>n</math>. The linear term is determined by <math>10a_1 +
  2 KB (357 words) - 21:15, 8 July 2021
• 1984 AHSME Problems/Problem 22
  ...h>, <math> b=-2ax_t </math>, so a unique <math> b </math>, and therefore a unique vertex, is determined for each point on the parabola. We therefore conclude
  2 KB (328 words) - 12:51, 5 July 2013
• 1997 AJHSME Problems/Problem 14
  ...in the middle, which is the third integer from the left. Since there is a unique mode of 8, both integers to the right of 5 must be 8s. Since the mean is 5,
  949 bytes (150 words) - 22:29, 17 December 2020
• 1997 AHSME Problems/Problem 28
  ...h>, and each pair of <math>(a_0,b_0)</math> that we get will generate four unique solutions: <math>(a_0, b_0), (b_0, a_0), (-a_0, -b_0), (-b_0, -a_0)</math>
  4 KB (679 words) - 17:21, 26 July 2021
• 1996 AHSME Problems/Problem 23
  Let <math>x, y</math>, and <math>z</math> be the unique lengths of the edges of the box. Each box has <math>4</math> edges of each
  1 KB (220 words) - 16:24, 28 June 2021
• 2011 AMC 8 Problems/Problem 12
  ...a certain side, then all placements of the other people can be considered unique. WLOG, assign Angie to be on the side. There are then <math>3!=6</math> tot
  1 KB (180 words) - 14:04, 4 January 2024
• 1985 AHSME Problems/Problem 30
  ...{n}}{2}\right\rfloor-51 = n.</cmath> To proceed, let <math>a</math> be the unique non-negative integer such that <math>a \leq \frac{\sqrt{n}}{2} < a+1</math>
  3 KB (563 words) - 03:57, 20 March 2024
• 1985 IMO Problems/Problem 6
  Hence, the <math>x_1</math> that satisfies this must be unique.
  9 KB (1,788 words) - 00:02, 30 January 2021
• 2012 AMC 12A Problems/Problem 20
  Every number, including <math>2012</math>, has a unique representation by the sum of powers of two, and that representation can be
  2 KB (298 words) - 20:59, 5 February 2019
• 2012 AMC 12B Problems/Problem 19
  ...is true for the three dots that are near <math>(1,1,1).</math> There is a unique <math>x</math> for which the rectangle drawn in red becomes a square. This
  5 KB (815 words) - 21:59, 19 September 2023
• Epsilon Camp
  ...campers, a parent or guardian must attend with the child. Epsilon Camp is unique in that it includes a parent program, which aims to support parents in meet
  4 KB (538 words) - 15:27, 6 January 2023
• Mock AIME I 2012 Problems/Problem 12
  ...so that the above two sets contain finitely many elements. But there is a unique such pair of roots of unity, making <math>a = </math> <math>-1/2 \pm i\sqrt
  1 KB (196 words) - 01:56, 25 November 2016
• 1978 USAMO Problems/Problem 2
  The point is obviously unique, because the two maps have different scales (but if P and Q where two fixed ...ral similarity is a dilation combined with a rotation, and that there is a unique spiral similarity taking any segment to another segment, provided the four
  4 KB (712 words) - 21:57, 12 November 2023
• 2012 AMC 8 Problems
  The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, and <math>x</math> are all
  13 KB (1,835 words) - 08:51, 8 March 2024
• 2012 AMC 8 Problems/Problem 11
  The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, and <math>x</math> are all We know the unique mode must be <math>6</math>, so the mean must be the same number <math>6</m
  3 KB (446 words) - 12:25, 3 January 2024
• 2013 AMC 12A Problems/Problem 25
  ...ords, when <math>b^2 > a-1</math>, the equation <math>f(z)=a+bi</math> has unique solution <math>z</math> in the region <math>\operatorname{Im}(z)>0</math>;
  4 KB (766 words) - 14:22, 8 October 2023
• 2013 AMC 10B Problems/Problem 19
  We need to find the unique root, or <math>-\frac {b} {2a} </math> (discriminant is 0). The unique root is <math>-\frac {b} {2a} </math> as discriminant has to be 0.
  5 KB (969 words) - 19:14, 15 August 2023
• 2013 AMC 12B Problems/Problem 25
  We do casework on the (unique) even constant term <math>d\in\{\pm2,\pm10,\pm50\}</math> in our product. F
  5 KB (750 words) - 01:10, 31 December 2022
• 2013 AIME II Problems/Problem 6
  For <math>k=31</math>, one can verify that <math>N = 282</math> is the unique integer satisfying the inequalities.
  5 KB (826 words) - 09:29, 1 January 2023
• 2013 AIME II Problems/Problem 12
  ...<math>79</math> integers in this interval, each of which corresponds to a unique complex number on the circle of radius <math>20</math> with positive imagin ...<math>51</math> integers in this interval, each of which corresponds to a unique complex number on the circle of radius <math>13</math> with positive imagin
  8 KB (1,393 words) - 19:00, 24 May 2023
• 2013 USAMO Problems/Problem 2
  Lemma 1: Any unique configuration of <math>K</math> such that [Work In Progress]
  14 KB (2,076 words) - 20:29, 10 July 2023
• Rotation
  ...triangle of side length <math>s</math>, with the property that there is a unique point <math>P</math> inside the triangle such that <math>AP=1</math>, <math
  3 KB (432 words) - 23:22, 13 January 2021
• 1983 IMO Problems/Problem 4
  ...mon with the other side and the point which projects orthogonally onto the unique blue point on the third side). We can now assume all three sides contain at
  2 KB (373 words) - 23:40, 29 January 2021
• 1951 AHSME Problems/Problem 27
  therefore 1. does not give any unique criteria.
  2 KB (321 words) - 15:48, 10 September 2018
• 1960 AHSME Problems
  ...ath>b</math> are real numbers, the equation <math>3x-5+a=bx+1</math> has a unique solution <math>x</math> [The symbol <math>a \neq 0</math> means that <math> <math>\textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value}\qquad \\
  21 KB (3,242 words) - 21:27, 30 December 2020
• 2014 AMC 12A Problems/Problem 15
  ...<math>e=9-b</math>, <math>f=9-c</math>. Notice how every palindrome has a unique compliment, and that the sum of a palindrome and its complement is <math>99
  4 KB (612 words) - 22:42, 2 August 2021
• 2014 AMC 12A Problems/Problem 23
  ...ngs to notice here. First, <math>\frac{1}{99}</math> has a very simple and unique decimal expansion, as shown. Second, for <math>\frac{0.\overline{01}}{99}</
  3 KB (453 words) - 09:35, 26 May 2023
• 2014 AMC 12B Problems
  A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list?
  13 KB (2,066 words) - 14:08, 1 November 2022
• 2014 AMC 10B Problems
  ...integers has a mean of <math>10</math>, a median of <math>9</math>, and a unique mode of <math>8</math>. What is the largest possible value of an integer in
  13 KB (2,011 words) - 21:54, 8 November 2022
• 2014 AMC 10B Problems/Problem 18
  ...integers has a mean of <math>10</math>, a median of <math>9</math>, and a unique mode of <math>8</math>. What is the largest possible value of an integer in ..., 8, 8, 9, f, g, h, i, j</math>. In this case, since <math>8</math> is the unique mode, the rest of the integers have to be distinct. So we minimize <math>a,
  2 KB (369 words) - 17:44, 30 January 2021
• 2014 AMC 12B Problems/Problem 11
  ...integers has a mean of <math>10</math>, a median of <math>9</math>, and a unique mode of <math>8</math>. What is the largest possible value of an integer in ..., 8, 8, 9, f, g, h, i, j</math>. In this case, since <math>8</math> is the unique mode, the rest of the integers have to be distinct. So we minimize <math>a,
  2 KB (277 words) - 10:13, 3 March 2015
• 1993 AHSME Problems/Problem 24
  First let’s try to find the number of possible unique combinations. I’ll denote shiny coins as 1 and dull coins as 0. ...- 1 1 - - where the dashes can be replaced by 0s. This makes the number of unique combinations 7 choose 3 (if you’re counting w.r.t. shiny coins) OR 7 choo
  3 KB (558 words) - 00:25, 23 December 2020
• 2007 iTest Problems/Problem 36
  ...= 0</math>, then expression leads to <math>x^4 = 0</math> and only has one unique solution, so discard <math>b = 0</math> as a solution. The rest of the val
  2 KB (276 words) - 15:49, 20 June 2018
• 2014 IMO Problems/Problem 1
  ...h> be an infinite sequence of positive integers, Prove that there exists a unique integer <math>n\ge1</math> such that ...tonic decreasing. Therefore, because <math>f(0) > 0</math>, there exists a unique <math>N</math> such that <math>f(N-1) > 0 \ge f(N)</math>. In other words,
  3 KB (611 words) - 11:16, 8 July 2023
• 2014 IMO Problems
  ...h> be an infinite sequence of positive integers, Prove that there exists a unique integer <math>n\ge1</math> such that
  3 KB (529 words) - 19:35, 14 March 2021
• 1988 AHSME Problems/Problem 28
  ...n <math>\frac{1}{2}</math> and <math>1</math>, meaning that there is not a unique value of <math>p</math>, so the answer is <math>\boxed{\text{D}}</math>.
  2 KB (240 words) - 14:50, 27 February 2018
• 2016 AMC 12A Problems
  .... In fact, for this value of <math>a</math> the value of <math>b</math> is unique. What is the value of <math>b?</math>
  15 KB (2,348 words) - 17:20, 19 January 2024
• 2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 3
  The product ages is 72. The largest age is unique. The smallest age is unique.
  2 KB (239 words) - 20:42, 25 July 2016
• 2016 AMC 10A Problems/Problem 14
  ...336</math>, inclusive. Since each value of <math>n</math> corresponds to a unique way to write the sum, we get <math>336 + 1 = \boxed{\textbf{(C)}337}</math>
  4 KB (604 words) - 04:38, 2 November 2023
• Searching the community
  Note: On the previous website, forums were associated with a unique ID that could be used in searching, similar to user ID numbers, but this is ...'ve used is usually quite easy, and since the keywords are generally quite unique, searching for them will often narrow results down to posts using [[LaTeX]]
  21 KB (3,334 words) - 23:53, 23 March 2022
• Algebraic number theory
  ==Unique factorization== ...unit (ring theory)|units]]. The first order of business is to explore when unique factorization holds.
  10 KB (1,646 words) - 15:04, 28 May 2020
• 1977 AHSME Problems/Problem 28
  ...12})</math> is divided by <math>g(x)</math>. Then <math>r(x)</math> is the unique polynomial such that
  3 KB (579 words) - 11:56, 21 April 2024
• 2017 AIME I Problems
  ...+ 99i,</math> where <math>i = \sqrt{-1}</math>. Let <math>z</math> be the unique complex number with the properties that <math>\frac{z_3 - z_1}{z_2 - z_1} \
  7 KB (1,163 words) - 16:43, 2 June 2022
• 2016 AMC 12A Problems/Problem 24
  .... In fact, for this value of <math>a</math> the value of <math>b</math> is unique. What is this value of <math>b</math>?
  3 KB (568 words) - 17:14, 20 April 2024
• Coplanar
  ...oplanar, and if the 3 points are distinct and non-collinear, they define a unique plane.
  228 bytes (39 words) - 13:45, 30 April 2016
• 1990 IMO Problems/Problem 4
  ...math>f(\frac{1}{q}) = p</math> where <math>p</math> and <math>q</math> are unique primes. Thus our 2 sets <math>S_1</math> and <math>S_2</math> is simply 2 s
  5 KB (1,045 words) - 11:01, 30 September 2022
• 1996 USAMO Problems/Problem 6
  ...erate a subset <math>S</math> such that all non-negative integers n have a unique solution <math>a + 2b = n</math>.
  3 KB (584 words) - 07:56, 16 April 2018
• 2016 AMC 8 Problems/Problem 25
  ...d. For your own benefit, look at all of the solutions, as they employ many unique techniques to get to the final answer.
  13 KB (2,115 words) - 16:21, 18 April 2021
• Mock AIME 4 Pre 2005/Problems
  ...in its interior such that <math>ABP</math> is equilateral. There exists a unique pair <math>\{m, n\}</math> of relatively prime positive integers such that ...h>BC = 27, CA = 32,</math> and <math>AB = 35.</math> <math>P</math> is the unique point such that the
  7 KB (1,094 words) - 15:39, 24 March 2019
• 2017 AMC 10A Problems/Problem 24
  ...are roots of <math>g(x)</math>. Using the fact that every polynomial has a unique factorization into its roots, and since the leading coefficient of <math>f(
  10 KB (1,708 words) - 23:16, 7 October 2023
• 2017 AMC 12A Problems/Problem 17
  ...{12})}</math>, where <math>0 \leq k<24</math> that produce <math>24</math> unique results.
  3 KB (459 words) - 04:19, 2 February 2021
• 2017 AMC 12A Problems/Problem 23
  ...are roots of <math>g(x)</math>. Using the fact that every polynomial has a unique factorization into its roots, and since the leading coefficient of <math>f(
  10 KB (1,861 words) - 10:47, 17 October 2021
• 2017 AMC 10A Problems/Problem 20
  The number <math>n</math> can be viewed as having some unique digits in the front, following by a certain number of nines. We can then ev
  5 KB (798 words) - 14:35, 14 October 2022
• 2017 AMC 10B Problems/Problem 18
  ...<math>4</math> fixed colorings. Hence, by Burnside's lemma, the number of unique colorings up to reflections and rotations is
  7 KB (1,057 words) - 23:27, 27 August 2022
• 2017 AMC 12B Problems/Problem 13
  ...<math>2\cdot2=4</math>. The other two reflections also have <math>4</math> unique fixed diagrams. Therefore, applying Burnside's Lemma, we get <math>(60+4+4+
  3 KB (455 words) - 13:55, 15 February 2021
• 2017 AIME I Problems/Problem 10
  ...3=78+99i,</math> where <math>i=\sqrt{-1}.</math> Let <math>z</math> be the unique complex number with the properties that <math>\frac{z_3-z_1}{z_2-z_1}~\cdot
  13 KB (2,252 words) - 15:46, 6 January 2024
• 2017 AIME I Problems/Problem 15
  ...oint be at <math>O(0,0)</math>. Notice the 2 other points will determine a unique equilateral triangle. Let 2 points be on the <math>x</math>-axis (<math>B</
  22 KB (3,622 words) - 17:11, 6 January 2024
• 2017 AIME II Problems/Problem 9
  Note that there is no case with Gg, Gg because all 49 cards are unique! ...a column with 2 colored cells. Call the unique row, "Row A", and Call the unique column, "Column B".
  11 KB (1,964 words) - 23:03, 29 January 2024
• 2017 AIME II Problems/Problem 11
  ...them away from <math>\omega_1</math>. Therefore, one path can have only 1 unique vertex that has no arcs leading to it. There are 5 ways to choose the verte ...ed away from them and points to <math>\omega_1</math>. Therefore, only one unique vertex can have all arcs leading to it. There are 5 ways to choose the vert
  16 KB (2,867 words) - 17:51, 6 January 2024
• 2017 AIME II Problems/Problem 13
  ...\frac{s-2}{2}</math> solutions for <math>(x,y)</math> which would create a unique type of isosceles triangle. We subtract two since <math>y</math> cannot be ...triangles in the polygon with <math>s+1</math> sides and <math>s+1</math> unique points for each type of triangle. Therefore, <math>\frac{s}{2}\cdot{(s+1)}-
  12 KB (2,069 words) - 20:45, 29 January 2024
• 2017 AIME II Problems/Problem 14
  Case 3: We use 3 unique pairs, one that is not important.
  6 KB (1,015 words) - 18:02, 31 December 2023
• A* Summer Math Camp
  A* Summer Math Camp is a unique opportunity for interested and talented 4th - 12th grade students to improv
  1 KB (177 words) - 19:31, 15 December 2017
• 2020 AMC 12A Problems
  There is a unique positive integer <math>n</math> such that<cmath>\log_2{(\log_{16}{n})} = \l There exists a unique strictly increasing sequence of nonnegative integers <math>a_1 < a_2 < …
  14 KB (2,073 words) - 15:15, 21 October 2021
• 2018 AMC 12B Problems
  A list of <math>2018</math> positive integers has a unique mode, which occurs exactly <math>10</math> times. What is the least number
  14 KB (2,118 words) - 15:36, 28 October 2021
• 2018 AMC 10B Problems
  A list of <math>2018</math> positive integers has a unique mode, which occurs exactly <math>10</math> times. What is the least number
  15 KB (2,237 words) - 23:08, 15 November 2023
• 2018 AMC 10B Problems/Problem 14
  A list of <math>2018</math> positive integers has a unique mode, which occurs exactly <math>10</math> times. What is the least number
  2 KB (237 words) - 15:07, 14 June 2023
• 2018 USAJMO Problems/Problem 5
  ...have used the fact that every nonzero residue modulo <math>p</math> has a unique multiplicative inverse. Therefore, there is exactly one <math>k\in {1, 2, .
  2 KB (455 words) - 13:53, 21 April 2018
• 2018 USAMO Problems/Problem 4
  ...have used the fact that every nonzero residue modulo <math>p</math> has a unique multiplicative inverse. Therefore, there is exactly one <math>k\in {1, 2, . Alternatively, every pair <math>(a_i, a_j)</math> has a unique <math>k</math> s.t <math>(a_i + ik) - (a_j + jk) \equiv 0</math> by the Lem
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• 2016 JBMO Problems/Problem 3
  <math>2016 = 2^5 \cdot 3^2 \cdot 7</math> is the unique prime factorization.
  4 KB (639 words) - 15:14, 28 November 2023
• 1960 AHSME Problems/Problem 14
  ...ath>b</math> are real numbers, the equation <math>3x-5+a=bx+1</math> has a unique solution <math>x</math> [The symbol <math>a \neq 0</math> means that <math>
  866 bytes (150 words) - 19:01, 17 May 2018
• 1960 AHSME Problems/Problem 22
  <math>\textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value}\qquad \\ \textbf{(C)}\ b = 0, a \text{ } \text{has a unique non-zero value}\qquad \\
  1 KB (223 words) - 19:06, 17 May 2018
• Star League Summer Camp
  Star League 3- Week Summer Math Camp is a unique opportunity for interested and talented 4th - 12th grade students to improv
  1 KB (210 words) - 17:11, 22 May 2018
• Star League Summer Camps - RESIDENTIAL & DAY CAMPS
  Star League Summer Math Camp is a unique opportunity for interested and talented 4th - 12th grade students to improv
  2 KB (315 words) - 16:44, 25 February 2020
• Star League Math Camps - RESIDENTIAL & DAY CAMPS
  Star League 3- Week Summer Math Camp is a unique opportunity for interested and talented 4th - 12th grade students to improv
  3 KB (413 words) - 17:41, 22 May 2018
• Star League Math Camps
  Star League 3- Week Summer Math Camp is a unique opportunity for interested and talented 4th - 12th grade students to improv
  3 KB (413 words) - 17:42, 22 May 2018
• Star League USACO Camps - RESIDENTIAL & DAY CAMPS
  Star League 3- Week Summer USACO Camp is a unique opportunity for interested and talented 6th - 12th grade students to improv
  3 KB (395 words) - 17:49, 22 May 2018
• Wolfram High School Summer Camp
  ..., hackathons, and tech innovation. All students will create and complete a unique project to be published by the end of the program.
  941 bytes (135 words) - 12:56, 26 November 2019
• 2018 AIME I Problems/Problem 9
  ...ke there are <math>10</math> values for <math>a</math> and <math>17</math> unique values for each <math>?</math>, giving a total of <math>170</math>, but tha
  7 KB (1,092 words) - 23:32, 5 February 2022
• 1964 AHSME Problems/Problem 13
  ...gruent external tangents to the triangle each, there are really only three unique lengths, and two of these three lengths each make up a side. Thus, we can
  2 KB (306 words) - 18:26, 23 July 2019
• 2019 AIME II Problems/Problem 10
  There is a unique angle <math>\theta</math> between <math>0^{\circ}</math> and <math>90^{\cir ...tan untrue. This must also be true in order for <math>\theta</math> to be unique.)
  7 KB (1,109 words) - 00:40, 28 January 2024
• 2019 USAJMO Problems/Problem 2
  .... Similarly, <math>g(x)</math> maps each residue mod <math>|a|</math> to a unique residue mod <math>|b|</math>, so <math>|b|\ge|a|</math>. Therefore, <math>|
  4 KB (816 words) - 18:53, 7 April 2021
• 2019 USAJMO Problems/Problem 3
  By monotonicity, we can see that the point <math>P</math> is unique. Therefore, if we find another point <math>P'</math> with all the same prop
  4 KB (654 words) - 20:13, 12 April 2021
• 2019 USAMO Problems/Problem 4
  ...>n + j + 1</math> distinct elements and so in this process we only add one unique element to sets among <math>C_{j+1}</math>. There are <math>2n - (n+j) = n-
  7 KB (1,288 words) - 19:17, 26 April 2023
• 2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 2
  ...ust be <math>9</math> or <math>1</math>. However, since all the digits are unique, <math>T</math> = <math>9</math>.
  2 KB (385 words) - 12:27, 21 August 2023
• 1983 AHSME Problems/Problem 24
  Notice also that each of these values of <math>a</math> will give a '''unique''' corresponding value of <math>b</math>, since <math>\frac{4a-8}{a-4} = 4
  2 KB (421 words) - 15:20, 10 April 2021
• 2019 AIME I Problems/Problem 5
  Unique solution: https://youtu.be/I-8xZGhoDUY
  4 KB (668 words) - 15:51, 25 February 2021
• 2019 AIME II Problems
  There is a unique angle <math>\theta</math> between <math>0^\circ</math> and <math>90^\circ</
  7 KB (1,254 words) - 14:45, 21 August 2023
• 2019 AIME I Problems/Problem 10
  simply asks for the absolute value of the sum of the product of the distinct unique roots of the polynomial taken two at a time or
  10 KB (1,689 words) - 17:02, 8 February 2024
• What is the definition of Pure Mathematics?
  ...ne unique way to factor a positive integer into a product of primes. This unique prime factorization plays an important role in solving many kinds of number
  35 KB (5,882 words) - 18:08, 28 June 2021
• 2006 iTest Problems/Problem 37
  ...tarrow x=\frac{z+y}{2}</math>. As we are told that <math>y^2</math> has a unique value, we can use either option.
  3 KB (520 words) - 18:27, 14 January 2020
• 2020 AIME I Problems/Problem 2
  There is a unique positive real number <math>x</math> such that the three numbers <math>\log_
  6 KB (1,040 words) - 18:31, 29 January 2023
• 2020 AIME I Problems/Problem 9
  ...integers from <math>0</math> to <math>20</math>, they will correspond to a unique solution, forming a 1-1 correspondence between the numbers <math>b_1</math>
  6 KB (1,036 words) - 00:59, 25 February 2021
• 2019 IMO Problems/Problem 1
  ...ve for any '''integral''' constant c, and that this family of equations is unique.
  4 KB (680 words) - 01:42, 4 May 2024
• 2023 AIME I Problems/Problem 10
  ...nteger strictly between <math>-1000</math> and <math>1000</math>. For that unique <math>a</math>, find <math>a+U</math>. For <math>UB \in \left[ - 1000, 1000 \right]</math>, we get a unique <math>a = 1349</math>.
  10 KB (1,578 words) - 09:48, 24 April 2024
• 2020 AIME II Problems/Problem 1
  ...frac{20^{20}}{x}</math>. This reduces the problem to finding the number of unique perfect square factors of <math>20^{20}</math>.
  2 KB (346 words) - 13:59, 25 February 2024
• 2008 Indonesia MO Problems
  ...ccupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communicat
  3 KB (478 words) - 14:09, 23 June 2021
• 2020 AMC 10A Problems/Problem 21
  There exists a unique strictly increasing sequence of nonnegative integers <math>a_1 < a_2 < …
  6 KB (773 words) - 21:33, 5 April 2024
• 2020 AMC 12A Problems/Problem 10
  There is a unique positive integer <math>n</math> such that<cmath>\log_2{(\log_{16}{n})} = \l
  5 KB (717 words) - 02:32, 30 January 2023
• 2020 AMC 12A Problems/Problem 24
  ...triangle of side length <math>s</math>, with the property that there is a unique point <math>P</math> inside the triangle such that <math>AP=1</math>, <math
  16 KB (2,509 words) - 17:49, 8 February 2024
• 2020 AIME I Problems
  There is a unique positive real number <math>x</math> such that the three numbers <math>\log_
  7 KB (1,257 words) - 17:51, 5 February 2022
• 2024 AIME II Problems
  <math>\bullet</math> The unique mode of the list is <math>9</math>. ...th>x</math>-axis and <math>Q</math> on the <math>y</math>-axis. There is a unique point <math>C</math> on <math>\overline{AB},</math> distinct from <math>A</
  8 KB (1,236 words) - 23:11, 12 March 2024
• 2022 AMC 10A Problems
  ...l the integers from <math>1</math> to <math>n</math> inclusive. There is a unique integer <math>h</math> such that
  18 KB (2,662 words) - 02:08, 9 March 2024
• 2023 AIME I Problems/Problem 11
  Using casework on where the consecutive element pair is, there is a unique consecutive element pair that satisfies the conditions. It is easy to see t
  15 KB (2,414 words) - 06:57, 26 November 2023
• Ordinal
  .../math>, statements like "For every <math>x\in\mathcal{F}</math> there is a unique smallest <math>y>x</math> and <math>y\in\mathcal{F}</math>."
  5 KB (811 words) - 14:16, 7 June 2020
• Taylor polynomial
  ...ial''' of a [[function]] <math>f(x)</math> about <math>x = a</math> is the unique [[polynomial]] of degree <math>n</math> whose value and first <math>n</math
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• College Application Assistance
  ...for you! Our scholarship program, worth over $8,000, helps students craft unique and compelling themes and high-quality college application essays.
  571 bytes (76 words) - 03:44, 7 September 2020
• IvyZen Scholarship Program
  IvyZen has helped students craft unique and compelling themes covering a diverse area of interests such as social e
  1 KB (194 words) - 13:25, 11 August 2023
• 2020 CMC 12B Problems/Problem 19
  ...uadrilateral such that <math>AB=4, BC=4, CD=3, DA=7</math>. There exists a unique point <math>P</math> inside quadrilateral <math>ABCD</math> such that the a
  619 bytes (95 words) - 17:16, 7 September 2020
• 2021 AIME I Problems/Problem 3
  ...2^{a_2}+2^{b_1}</math>. We use the fact that every positive integer has a unique binary representation: <b><i>Note that each of the remaining ordered pairs generates one unique desired positive integer.</i></b>
  7 KB (1,174 words) - 08:21, 13 May 2023
• 2021 AIME I Problems/Problem 4
  ...1 \leq k_1 \leq 31</math>. Each value of <math>k_1</math> corresponds to a unique value of <math>k_2</math>, so there are <math>31</math> solutions in this c
  3 KB (532 words) - 21:04, 18 December 2023
• Octans Math Tournament
  ...st Math Tournament is an international online mathematics team event. This unique competition has been designed in order to test your mathematical problem so
  2 KB (300 words) - 03:19, 28 August 2021
• Cantor Dust Math Tournament
  ...st Math Tournament is an international online mathematics team event. This unique competition has been designed in order to test your mathematical problem so
  2 KB (293 words) - 03:17, 28 August 2021
• 2021 AMC 12A Problems
  ...k</math> and <math>y_k</math> are real. Let <math>\mathcal E</math> be the unique ellipse that passes through the points <math>(x_1,y_1),(x_2,y_2),(x_3,y_3), ...visor function.) Let<cmath>f(n)=\frac{d(n)}{\sqrt [3]n}.</cmath>There is a unique positive integer <math>N</math> such that <math>f(N)>f(n)</math> for all po
  15 KB (2,383 words) - 09:49, 25 June 2023
• 2023 AIME II Problems
  ...such that <math>M</math> is on <math>\overline{AP}.</math> There exists a unique point <math>Q</math> on segment <math>\overline{AM}</math> such that <math>
  8 KB (1,370 words) - 21:34, 28 January 2024
• 2022 AIME I Problems/Problem 14
  ...ath> is the ring of integers of <math>\mathbb Q[\sqrt{-3}]</math> and is a unique factorization domain. Notice that <math>N(x-y\omega) = (x-y\omega)(x-y\omeg ...must divide <math>73^2</math>. One can then easily check that this is the unique solution.
  16 KB (2,730 words) - 02:56, 4 January 2023
• 2020 CIME I Problems
  There exists a unique positive real number <math>x</math> satisfying <cmath>x=\sqrt{x^2+\frac{1}{
  7 KB (1,188 words) - 18:00, 31 August 2020
• 2020 CIME I Problems/Problem 4
  There exists a unique positive real number <math>x</math> satisfying <cmath>x=\sqrt{x^2+\frac{1}{
  753 bytes (128 words) - 20:43, 27 December 2021
• 2022 AMC 12B Problems
  Camila writes down five positive integers. The unique mode of these integers is <math>2</math> greater than their median, and the ...ial <math>x^2 + 1</math>, the remainder is <math>2x + 1</math>. There is a unique polynomial of least degree with these two properties. What is the sum of th
  15 KB (2,233 words) - 13:02, 10 November 2023
• 2022 AMC 12B Problems/Problem 17
  Second, we observe that there is a unique benchmark matrix, as shown below: .../math> and exactly <math>1</math> column sum is <math>4</math>, there is a unique entry in the array such that it, and every other entry in the same row or c
  7 KB (1,071 words) - 00:19, 10 July 2023
• 2021 Fall AMC 10A Problems
  ...eal numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial <math>\tilde{p}(x)</math> for which the sum of the roots is
  14 KB (2,162 words) - 21:33, 2 November 2023
• 2021 Fall AMC 10A Problems/Problem 18
  ...ll be the unique one), and <math>4</math> ways to choose which section the unique crop is planted on. This gives us a total of <math>4 \cdot 8 = \underline{3
  11 KB (1,867 words) - 00:42, 17 July 2023
• 2021 Fall AMC 12B Problems/Problem 20
  ...le, all cornerings as indistinguishable, and of course the opposite one is unique from all the others. Thus, we can list out like so (keeping in mind that th
  10 KB (1,641 words) - 22:17, 26 September 2023
• 2021 AMC 12A Problems/Problem 21
  ...k</math> and <math>y_k</math> are real. Let <math>\mathcal E</math> be the unique ellipse that passes through the points <math>(x_1,y_1),(x_2,y_2),(x_3,y_3),
  11 KB (1,858 words) - 18:33, 1 August 2022
• 2021 AMC 12A Problems/Problem 25
  ...visor function.) Let<cmath>f(n)=\frac{d(n)}{\sqrt [3]n}.</cmath>There is a unique positive integer <math>N</math> such that <math>f(N)>f(n)</math> for all po
  7 KB (1,083 words) - 10:17, 5 February 2024
• 2020 INMO Problems/Problem 3
  ...o a switch, the digits we choose for <math>a</math> and <math>b</math> are unique). But then, for any digit <math>d \in \{0,1,\cdots, 9\}</math>, consider <m
  4 KB (713 words) - 03:22, 22 September 2023
• 2021 USAMO Problems
  ...positive integer divisor <math>d</math> of <math>s</math>, there exists a unique element <math>t \in S</math> satisfying <math>\text{gcd}(s, t) = d</math>.
  4 KB (677 words) - 13:48, 22 November 2023
• 2021 AMC 12B Problems
  Let <math>Q(z)</math> and <math>R(z)</math> be the unique polynomials such that<cmath>z^{2021}+1=(z^2+z+1)Q(z)+R(z)</cmath>and the de
  15 KB (2,302 words) - 12:31, 27 October 2023
• Maximal element
  ...lements are not necessarily unique in general posets, but they are usually unique in totally ordered sets such as the [[real number|real numbers]].
  288 bytes (44 words) - 00:20, 7 November 2020
• 2020 AMC 8 Problems/Problem 16
  ...i.e. <math>2(A+B+C+D+E+F)+B=47</math>. Since <math>A,B,C,D,E,F</math> are unique integers between <math>1</math> and <math>6</math>, we obtain <math>A+B+C+D
  3 KB (550 words) - 16:35, 26 January 2024
• 1972 AHSME Problems/Problem 33
  ...simplifies to <math>10+ \dfrac{90a-81}{a+b+9}</math>. Since each digit is unique, we set <math>b</math> to <math>8</math>, leaving us with <math>10 + \dfrac
  979 bytes (163 words) - 23:58, 21 December 2020
• 2021 AMC 10A Problems/Problem 11
  ...ves us with <math>\boxed{\textbf{(E)} ~8},</math> the answer choice with a unique residue modulo <math>3.</math>
  3 KB (527 words) - 11:05, 16 June 2023
• Mock AIME 2 Pre 2005 Problems/Problem 9
  ...lement with degree <math>k_{1997}</math>. Since each sum of powers of 3 is unique, there is only one; namely, take the x element for every binomial with a de
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• 2022 AMC 12A Problems
  Let <math>h_n</math> and <math>k_n</math> be the unique relatively prime positive integers such that <cmath>\frac{1}{1}+\frac{1}{2}
  15 KB (2,250 words) - 00:32, 9 March 2024
• 2021 CIME I Problems/Problem 3
  ...the digits of <math>n</math> are reversed. It is given that there exists a unique set of <math>3</math>-digit positive integers <math>\{p, r(p)\}\neq\{273, 3
  483 bytes (83 words) - 16:26, 16 January 2021
• 2021 AMC 12B Problems/Problem 20
  Let <math>Q(z)</math> and <math>R(z)</math> be the unique polynomials such that<cmath>z^{2021}+1=(z^2+z+1)Q(z)+R(z)</cmath>and the de
  8 KB (1,315 words) - 11:43, 24 October 2023
• Laurent Series
  Is such a series unique, however? It certainly should be, or else I would not be writing this all d This shows that the series is unique, done <math>\square</math>
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• 2021 USAJMO Problems
  ...positive integer divisor <math>d</math> of <math>s</math>, there exists a unique element <math>t \in S</math> satisfying <math>\text{gcd}(s, t) = d</math>.
  3 KB (513 words) - 16:47, 5 August 2023
• 2021 USAJMO Problems/Problem 5
  ...positive integer divisor <math>d</math> of <math>s</math>, there exists a unique element <math>t \in S</math> satisfying <math>\text{gcd}(s, t) = d</math>.
  2 KB (390 words) - 18:39, 12 May 2021
• MehtA+ Machine Learning Bootcamp
  ...proaching the problem statement in a unique way. The final project will be unique to each group, and students will work on the project independently or in co ...that our students have the potential to keep an open-mind, adapt and apply unique solutions to new challenges.
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• 2022 AIME II Problems/Problem 8
  ...h> that met the second condition in the problem, there would have been one unique answer, <math>\boxed{080}</math>. ~burkinafaso ~sethl The problem is the same as asking how many unique sets of values of <math>\lfloor\frac{n}{4}\rfloor</math>, <math>\lfloor\fra
  13 KB (2,136 words) - 22:39, 31 December 2023
• 2023 IOQM Problems
  ...For example 6 is beautiful since 6 = 3 · 2 · 1 = 3 + 2 + 1 , and this is unique. But 8 is not beautiful since 8 = 4 + 2 + 1 + 1 = 4 · 2 · 1 · 1 as well
  7 KB (1,225 words) - 03:05, 4 May 2024
• G285 2021 Summer Problem Set
  ...ndomly be assigned a label such that teams can be empty, and each label is unique in the group. Find the number of ways that the molecules can be arranged by ...h>. If distinct permutations of the same set <math>P</math> are considered unique, find the remainder when <cmath>\sum_{p=1}^{1000}\sum_{k=1}^{1000} S_{p,k}<
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• 2022 AMC 10B Problems
  Camila writes down five positive integers. The unique mode of these integers is <math>2</math> greater than their median, and the ...ial <math>x^2 + 1</math>, the remainder is <math>2x + 1</math>. There is a unique polynomial of least degree with these two properties. What is the sum of th
  15 KB (2,224 words) - 13:10, 20 February 2024
• G285 2021 Summer Problem Set Problem 3
  ...ndomly be assigned a label such that teams can be empty, and each label is unique in the group. Find the number of ways that the molecules can be arranged by
  1 KB (152 words) - 23:18, 28 June 2021
• 2023 AIME I Problems/Problem 9
  ...gers in <math>\{-20,-19,-18,\ldots,18,19,20\},</math> such that there is a unique integer <math>m \not= 2</math> with <math>p(m) = p(2).</math> ...> has two solutions, one being equal to <math>2,</math> but the other is a unique solution not equal to <math>2.</math>
  5 KB (960 words) - 20:40, 27 December 2023
• 2021 USAMO Problems/Problem 4
  ...positive integer divisor <math>d</math> of <math>s</math>, there exists a unique element <math>t \in S</math> satisfying <math>\gcd(s,t)=d</math> (the eleme
  429 bytes (78 words) - 13:59, 3 March 2023
• 2021 Fall AMC 12A Problems/Problem 23
  ...eal numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial <math>\tilde{p}(x)</math> for which the sum of the roots is Finally, the unique such polynomial is <cmath>\tilde{p}(x)=\left(x-\frac14\right)^2-\frac14,</c
  26 KB (4,697 words) - 22:19, 2 January 2024
• 2021 Fall AMC 12A Problems
  ...eal numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial <math>\tilde{p}(x)</math> for which the sum of the roots is
  15 KB (2,452 words) - 19:37, 7 June 2023
• Mustang Math Tournament
  The Mystery Mare round is a MMT unique round. The round structure is changed yearly and is not announced until the ...ivities all have some correlation to math but aim to show other, more fun, unique, or creative sides to math. Past activities have included Estimathons and P
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• 2022 IMO Problems/Problem 2
  ...( y \neq x \) (directly from AM-GM) and equal to 2 (with equality) for the unique value \( y = x \). ...\) while applying the same property for \( x \mapsto g(x) \) gives another unique \( y := g(g(x)) \) such that \( g(x)f(y) + yf(g(x)) \leq 2 \). Therefore, w
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• 2022 IMO Problems/Problem 6
  There are <math>2n(n-1)</math> such adjacent pairs (each with at least one unique uphill path). In addition, the cell containing the value 1 will always be a ...manner, the pathway of finite descent from each pair of adjacent cells is unique, and all terminate at the "1" valley, therefore the above-derived minimum i
  2 KB (406 words) - 19:17, 30 January 2024
• Differential equation
  ...itial condition <math>f(0) = 1</math>, <math>f(x) = e^x</math> becomes the unique solution.
  6 KB (969 words) - 20:33, 15 December 2023
• Newton's method
  ...has two real roots <math>\pm \sqrt{r}</math>, where <math>r</math> is the (unique) positive root of <math>x^2 - 3x - 2</math>. However, if we attempt to find ...v 0\pmod{p^n}</math>. If <math>k=v(p'(x))<n/2</math>, then there exists a unique root <math>\xi</math> of <math>p\in\mathbb{Z}_p[x]</math> such that <math>\
  13 KB (2,298 words) - 23:34, 28 May 2023
• 2023 AIME I Problems
  ...gers in <math>\{-20,-19,-18,\ldots,18,19,20\},</math> such that there is a unique integer <math>m \not= 2</math> with <math>p(m) = p(2).</math> ...nteger strictly between <math>-1000</math> and <math>1000</math>. For that unique <math>a</math>, find <math>a+U</math>.
  7 KB (1,154 words) - 12:54, 20 February 2024
• 2022 AMC 12A Problems/Problem 23
  Let <math>h_n</math> and <math>k_n</math> be the unique relatively prime positive integers such that <cmath>\frac{1}{1}+\frac{1}{2}
  5 KB (770 words) - 08:52, 29 March 2024
• 2022 AMC 10A Problems/Problem 19
  ...l the integers from <math>1</math> to <math>n</math> inclusive. There is a unique integer <math>h</math> such that
  4 KB (604 words) - 20:38, 29 October 2023
• 2022 AMC 10A Problems/Problem 24
  ..._3, a_4, a_5\}</math>, if there is at least one element in the set that is unique (i.e. there is at least one digit in the set that is found nowhere else in
  12 KB (1,898 words) - 12:48, 26 March 2024
• 2022 AMC 10A Problems/Problem 25
  ...find the residue of <math>r+s+t-3</math> mod <math>13</math> and find the unique answer choice that fits, without actually finding <math>r, s, t</math>.
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• 2022 AMC 10B Problems/Problem 21
  is <math>2x+1</math>. There is a unique polynomial of least degree with these two properties. What is the sum of
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• 2022 AMC 10B Problems/Problem 18
  ...there will only be two or fewer equations for three variables, making one unique solution impossible. Therefore, we must choose <math>3</math> different equ
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• 2022 AMC 10B Problems/Problem 7
  Each unordered pair gives a unique value of <math>k.</math> Therefore, there are <math>\boxed{\textbf{(B) }8}<
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• 2022 AMC 10B Problems/Problem 10
  Camila writes down five positive integers. The unique mode of these integers is <math>2</math> greater than their median, and the ...ven. We minimize this sum so that the arithmetic mean, the median, and the unique mode are minimized. Let <math>a=1</math> and <math>b=3,</math> from which <
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• How to Learn Turkish Language Fast
  ...es you can appreciate a wide cluster of Turkey’s artistic works in their unique structure. Might articulate a few words diversely, yet the distinctions are ...es you can appreciate a wide cluster of Turkey’s artistic works in their unique structure. Learning the Turkish language is not difficult; it might take a
  6 KB (871 words) - 14:50, 30 November 2022
• 2018 EGMO Problems
  ...perpendicular to <math>AB</math> intersects the line <math>MQ</math> at a unique point <math>N</math>.
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• 2023 AIME II Problems/Problem 15
  ...st check that <math>\gcd(23, 2^n) = 1</math> hence we are always seeking a unique modular inverse of <math>23</math>, <math>b_n</math>, such that <math>a_n \ Now that we know that <math>b_n</math> is unique, we proceed to recast this problem in binary. This is convenient because <m
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• 2023 AIME II Problems/Problem 12
  ...such that <math>M</math> is on <math>\overline{AP}.</math> There exists a unique point <math>Q</math> on segment <math>\overline{AM}</math> such that <math> Notice that the problem statement tells us that point Q is <math>\textit{unique.}</math> EVERY piece of information in the problem statement is intentional
  12 KB (1,900 words) - 18:14, 28 January 2024
• 2023 USAMO Problems/Problem 3
  ...atches the blank tile's color of <math>C</math>, there is no more than one unique configuration <math>C'</math> that can be produced from <math>C</math> usin
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• 2023 AMC 10A Problems
  Let <math>P(x)</math> be the unique polynomial of minimal degree with the following properties:
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• 2022 SSMO Speed Round Problems
  ...positive perfect square. Let <math>T</math> be the set which contains all unique remainders when the elements in <math>S</math> are divided by <math>2022</m
  5 KB (824 words) - 21:35, 31 May 2023
• UserLynneLogan23
  In my experience as a diabetologist, each patient is unique. Therefore, it's crucial to follow a comprehensive and individualized appro
  4 KB (651 words) - 13:39, 5 July 2023
• 2023 AMC 12A Problems
  Let <math>f</math> be the unique function defined on the positive integers such that<cmath>\sum_{d\mid n}d\c There is a unique sequence of integers <math>a_1, a_2, \cdots a_{2023}</math> such that
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• 1992 IMO Problems/Problem 5
  Let <math>a_{i}</math> be the number of unique projected points from each <math>Z_{i}</math> to the <math>yz</math>-plane Let <math>b_{i}</math> be the number of unique projected points from each <math>Z_{i}</math> to the <math>xz</math>-plane
  3 KB (560 words) - 00:43, 17 November 2023
• User/LEDStrip
  ...a dimmer specifically designed for LEDs. LED dimmers are optimized for the unique electrical characteristics of LED lights, ensuring optimal performance and
  6 KB (964 words) - 00:29, 24 October 2023
• 2023 IOQM/Problem 29
  since 6 = 3 · 2 · 1 = 3 + 2 + 1 , and this is unique. But 8 is not beautiful since
  631 bytes (89 words) - 10:48, 25 October 2023
• 2023 AMC 10A Problems/Problem 21
  Let <math>P(x)</math> be the unique polynomial of minimal degree with the following properties:
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• 2023 AMC 10A Problems/Problem 12
  Each of these cases are unique, so there are a total of <math>1+2+1+2+1+1+2+1+2+1=\boxed{\textbf{(B) } 14.
  5 KB (830 words) - 21:02, 27 April 2024
• 2023 AMC 12A Problems/Problem 25
  There is a unique sequence of integers <math>a_1, a_2, \cdots a_{2023}</math> such that
  6 KB (960 words) - 01:27, 11 November 2023
• 2023 AMC 12A Problems/Problem 22
  Let <math>f</math> be the unique function defined on the positive integers such that <cmath>\sum_{d\mid n}d\
  7 KB (1,200 words) - 01:03, 13 November 2023
• 2023 AMC 10B Problems/Problem 16
  ...e that "unique" constrains this to be a finite number), we can construct a unique downno. Similarly, we can also construct an upno, but the selection can not
  8 KB (1,195 words) - 12:03, 26 February 2024
• 2023 AMC 12B Problems/Problem 17
  ...side must correspond to the <math>120^\circ</math> angle, the triangle is unique. By analytic geometry, we construct the following plot.
  4 KB (611 words) - 09:52, 4 December 2023
• Mock AIME 6 2006-2007 Problems/Problem 2
  ...gle of the intersection is <math>22.5^\circ (m-k)</math> Since all of the unique values of <math>(m-k)</math> will be <math>1,2,3,\cdots 7</math>, then let
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• 1991 OIM Problems/Problem 6
  But the given points do not give us one unique triangle. It gives us two because the blue circle also intersects the blac
  5 KB (854 words) - 09:41, 23 December 2023
• 2005 OIM Problems/Problem 6
  ...t the arrangement of the circles that gives a coloring does not need to be unique. For example, if all points are red, any arrangement of circles can give th
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• HowtoHangStringLights
  String lights come in four different bulb types, each with its own unique characteristics.
  9 KB (1,431 words) - 04:40, 21 December 2023
• 2024 AIME I Problems/Problem 11
  ...s. There are <math>8</math> configurations here as <math>b_1</math> can is unique.
  9 KB (1,304 words) - 14:08, 6 March 2024
• Catenary
  A catenary has several mathematical properties unique to it. By the virtue of its mechanical description, its analysis finds seve
  454 bytes (73 words) - 01:58, 25 December 2023
• 2024 AIME II Problems/Problem 2
  <math>\bullet</math> The unique mode of the list is <math>9</math>. ...number that satisfy this case is 4,8 and 5,7 not 6 and 6 because 9 is the unique mode. it also states the median is an integer, so the pair is 5 and 7 and t
  5 KB (900 words) - 00:10, 3 April 2024
• 2024 AIME II Problems/Problem 14
  ...A \right)</math>, if a solution of <math>z</math> exists, then it must be unique. ...h ordered partition for <math>b'</math>, the solution of <math>z</math> is unique.
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• 2024 AIME II Problems/Problem 6
  ...or the number of sets bob have listed to be 2024, we want to find a sum of unique powers of two that can achieve this. 2024 is equal to <math>2^{10}+2^9+2^8+
  2 KB (370 words) - 18:07, 5 March 2024
• 2024 AIME II Problems/Problem 12
  ...drant with \(P\) on the \(x\)-axis and \(Q\) on the \(y\)-axis. There is a unique point \(C\) on \(\overline{AB}\), distinct from \(A\) and \(B\), that does In Equation (1), there exists a unique <math>x \in \left( 0, 1 \right)</math>, denoted as <math>x_C</math> (<math>
  10 KB (1,655 words) - 00:31, 11 April 2024
• 2007 JBMO Problems/Problem 3
  ...points are collinear, any combination of three points is associated with a unique nondegenerate triangle, so there exist <math>{13\choose3}=286</math> triang
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• 2005 iTest Problems
  <math>ITEST + AHSIMC = 6666CS</math>. Each letter represents a unique digit from <math>0</math> to <math>9</math>. How many solutions of the form
  25 KB (3,738 words) - 10:53, 23 April 2024
• Power of a point theorem
  There are three unique cases for this theorem. Each case expresses the relationship between the le Given lines <math> BP </math> and <math> CP </math> originate from two unique points on the [[circumference]] of a circle (<math> B </math> and <math> C
  6 KB (1,010 words) - 02:38, 7 May 2024