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• 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

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