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• Associative property
  ...are not equal for all choices of <math>a, b, c</math> (in particular, they differ whenever <math>c \neq 0</math>).
  1 KB (222 words) - 16:09, 15 August 2006
• Chinese Remainder Theorem
  ...orem. The converse follows because <math>a</math> and <math>b</math> must differ by a multiple of <math>m</math> and <math>n</math>, and <math>\mbox{lcm}(m, ...r]]. Moreover, in the case when the problem is solvable, any two solutions differ by some [[common multiple]] of <math>m_1,\ldots,m_n</math>.'' (the extended
  6 KB (1,022 words) - 14:57, 6 May 2023
• 2003 AMC 12A Problems/Problem 1
  ...e see that each of the <math>n</math>th term in the series of even numbers differ by <math>1</math>. For example, take series <math>O</math> and <math>E_1</m
  3 KB (436 words) - 20:31, 28 December 2021
• 2002 IMO Shortlist Problems/A2
  Thus for any <math>n </math>, we have two <math>a_i </math> that differ by more than <math> \frac{(n-1)^2}{n(n+1)} </math>. But this becomes arbi
  3 KB (448 words) - 17:58, 20 August 2008
• 2007 AIME I Problems/Problem 4
  ...linear, <math>a(t)</math>, <math>b(t) </math>, and <math>c(t) </math> must differ by a multiple of <math>\pi </math>. Note that <math> a(t) - b(t) = \frac{t
  1 KB (228 words) - 18:46, 11 March 2021
• 2004 IMO Shortlist Problems/A3
  ...laystyle n </math>. We also note that if <math> \displaystyle x,y </math> differ by an integer, then <math> \displaystyle t(x) = t(y) </math>.
  6 KB (958 words) - 22:15, 9 June 2007
• Sum and difference of powers
  An amazing thing happens when <math>x</math> and <math>y</math> differ by <math>1</math>, say, <math>x = y+1</math>. Then <math>x-y = 1</math> and
  3 KB (450 words) - 06:25, 6 May 2024
• 2006 AIME II Problems/Problem 8
  ...rotation and reflection. Thus, to make two triangles distinct, they must differ either in their center triangle or in the collection of colors which make u
  4 KB (695 words) - 10:37, 4 November 2023
• 1990 USAMO Problems
  ...g of six digits (from 0 through 9). The state requires that any two plates differ in at least two places. (Thus the plates <math>\boxed{027592}</math> and <m
  3 KB (386 words) - 20:47, 3 July 2013
• 1990 USAMO Problems/Problem 1
  ...g of six digits (from 0 through 9). The state requires that any two plates differ in at least two places. (Thus the plates <math>\boxed{027592}</math> and <m ...plates which agree on the first <math>n-1</math> digits; these plates thus differ only on one digit, the last one.
  2 KB (292 words) - 14:28, 20 November 2019
• 2007 AMC 10A Problems/Problem 23
  ...nteger]]s, with <math>m \ge n</math>, have the property that their squares differ by <math>96</math>?
  2 KB (352 words) - 14:20, 3 July 2023
• 2007 AMC 10A Problems
  ...integers, with <math>m \ge n</math>, have the property that their squares differ by <math>96</math>?
  13 KB (2,058 words) - 17:54, 29 March 2024
• 2008 AMC 12B Problems/Problem 21
  In other words, the two chosen <math>x</math>-coordinates must differ by no more than <math>\sqrt{3}</math>. To find this probability, we divide
  6 KB (1,008 words) - 11:46, 24 December 2020
• 2008 AIME I Problems/Problem 4
  ...cutive odd numbers, we note that <math>y</math> and <math>x+42</math> must differ by an even number. We can use casework with the even numbers, starting with
  4 KB (732 words) - 22:17, 28 November 2023
• KGS math club
  ...mmunication channel, how can they (1) check whether their length-two lists differ; and if they do, (2) let each other know who the murderer is? They must not
  22 KB (3,358 words) - 15:17, 18 July 2017
• 2008 iTest Problems
  ...rom the alphabet <math>\{0,1,2,3,4,5,6\}</math> if neighboring digits must differ by exactly <math>2</math>.
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2002 AMC 12A Problems/Problem 13
  Two different positive numbers <math>a</math> and <math>b</math> each differ from their reciprocals by <math>1</math>. What is <math>a+b</math>?
  1 KB (189 words) - 20:43, 25 August 2022
• 2000 AMC 10 Problems/Problem 20
  ...uppose that some two of <math>A</math>, <math>M</math>, and <math>C</math> differ by at least <math>2</math>. Then this triple <math>(A,M,C)</math> is not o
  2 KB (447 words) - 18:12, 22 April 2023
• Simple harmonic motion
  ...e mathematically represented by the following cosine (or sine, which would differ by a phase constant) function:
  926 bytes (145 words) - 06:07, 23 February 2023
• Change of base formula
  ...he functions <math>f(x) = \log_a x</math> and <math>g(x) = \log_b x</math> differ by a constant factor, <math>f(x) = (\log_a b) g(x)</math> for all <math>x >
  2 KB (412 words) - 15:32, 22 February 2024
• 2010 AIME II Problems/Problem 3
  ...al, there are <math>20-n</math> pairs of integers <math>(a, b)</math> that differ by <math>n</math> because we can let <math>b</math> be any integer from <ma
  2 KB (303 words) - 15:17, 12 May 2020
• 2010 USAMO Problems/Problem 3
  Now, when the indices differ by <math>1</math> we have ensured equality (and
  11 KB (1,889 words) - 13:45, 4 July 2013
• 2011 AMC 10B Problems/Problem 25
  ...math>). All additional triangles will differ by one as the solutions above differ by one so this process can be repeated indefinitely until the side lengths
  4 KB (620 words) - 03:15, 3 November 2022
• 2011 AMC 10A Problems/Problem 19
  ...9,784,841,900,961</cmath>We see that <math>484</math> and <math>784</math> differ by <math>300</math>, and we can confirm that <math>484</math> is the correc
  3 KB (545 words) - 20:54, 21 August 2023
• 2003 AMC 8 Problems
  ...ookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
  16 KB (2,236 words) - 12:02, 19 February 2024
• 2003 AMC 8 Problems/Problem 8
  ...ookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
  2 KB (320 words) - 00:47, 5 July 2013
• 2003 AMC 8 Problems/Problem 9
  ...ookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
  2 KB (304 words) - 19:14, 28 August 2016
• 2003 AMC 8 Problems/Problem 10
  ...ookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
  2 KB (245 words) - 00:47, 5 July 2013
• 2012 AMC 10A Problems/Problem 20
  ...o choose from). However, we must subtract the ways where two white squares differ by a rotation of 90 degrees about the middle of the square. There are a tot ...ion of 90 degrees and minus when there are 3 squares such that two of them differ by rotation of 90 and 1 of them differs by rotation of 180, because of the
  6 KB (1,001 words) - 22:19, 29 December 2023
• 1953 AHSME Problems
  ...les. It is <math>10</math> feet wide. The circumference of the two circles differ by about:
  21 KB (3,123 words) - 14:24, 20 February 2020
• 1989 AHSME Problems/Problem 22
  ...th> shapes (circle, hexagon, square, triangle). How many blocks in the set differ from the 'plastic medium red circle' in exactly <math>2</math> ways? (The ' ...he 'plastic medium red circle' in one of its qualities <math>(1)</math> or differ from it in <math>k</math> different ways <math>(kx)</math>. Choosing the ma
  3 KB (507 words) - 20:48, 6 December 2021
• 1996 USAMO Problems/Problem 2
  ...ses in such a way that the maximum and minimum sum belonging to each class differ by no more than a factor of 2.''' Using the previous notation, we first pro
  4 KB (823 words) - 11:40, 29 September 2019
• 2014 AIME II Problems/Problem 10
  ...degrees apart from <math>3 \theta_z</math> , so <math>\theta_w</math> must differ from <math>\theta_z</math> by some multiple of 120 degrees.
  6 KB (1,045 words) - 13:08, 21 January 2024
• 1995 USAMO Problems/Problem 4
  ...1,\ldots, x-d)</math> from <math>kQ(x)</math>; hence <math>q_x</math> must differ by a multiple of <math>\frac{x(x-1)\cdots (x-d)}{kd!(d-1)!\cdots 1!}</math>
  3 KB (502 words) - 15:27, 11 May 2018
• 1971 IMO Problems/Problem 5
  2. Two vector sums are at unit distance from one another if and only if they differ in presence of exactly one summand (i.e. one and only one coefficient <math
  4 KB (749 words) - 14:09, 29 January 2021
• 1962 AHSME Problems/Problem 36
  ...<math>x-8</math> and <math>x-10</math> are both powers of two; since they differ by two, it must be the case that <math>(x-8,x-10) = (4,2)</math> or <math>(
  729 bytes (120 words) - 15:01, 15 June 2018
• 2015 USAMO Problems/Problem 6
  ...e equivalent, but <math>\{1, 1, 2, 3\}</math> and <math>\{1, 2, 3\}</math> differ.)
  3 KB (560 words) - 21:08, 15 December 2018
• 2014 AMC 12B Problems
  ...consecutively on the circle that sum to <math>n</math>. Arrangements that differ only by a rotation or a reflection are considered the same. How many diffe
  13 KB (2,066 words) - 14:08, 1 November 2022
• 2014 AMC 10B Problems
  ...consecutively on the circle that sum to <math>n</math>. Arrangements that differ only by a rotation or a reflection are considered the same. How many differ
  13 KB (2,011 words) - 21:54, 8 November 2022
• 2014 AMC 10B Problems/Problem 24
  ...consecutively on the circle that sum to <math>n</math>. Arrangements that differ only by a rotation or a reflection are considered the same. How many differ
  7 KB (1,133 words) - 12:46, 12 March 2022
• 2014 AMC 12B Problems/Problem 10
  ...5</math>, so <math>a</math> and <math>c</math> must either be the equal or differ by <math>5</math>. We can quickly conclude that the former is impossible, s
  2 KB (321 words) - 19:46, 17 January 2021
• 2014 AMC 12B Problems/Problem 18
  ...consecutively on the circle that sum to <math>n</math>. Arrangements that differ only by a rotation or a reflection are considered the same. How many differ
  2 KB (331 words) - 04:43, 12 January 2021
• 1992 AHSME Problems/Problem 14
  The equations differ at <math>x=-2</math>. The graph of <math>I</math> would contain the point <
  938 bytes (164 words) - 22:45, 18 January 2018
• 1970 AHSME Problems/Problem 14
  ...th> and <math>q</math> are positive numbers. If the roots of this equation differ by 1, then <math>p</math> equals
  783 bytes (128 words) - 01:30, 24 February 2023
• 1970 AHSME Problems
  ...th> and <math>q</math> are positive numbers. If the roots of this equation differ by <math>1</math>, then <math>p</math> equals
  15 KB (2,366 words) - 07:52, 26 December 2023
• 2013 UNCO Math Contest II Problems
  ...at most six digits? (Consider two nonterminating decimals distinct if they differ in any digit. Nonterminating means that the digits are not eventually all z
  11 KB (1,648 words) - 09:55, 20 December 2021
• 2013 UNCO Math Contest II Problems/Problem 9
  ...at most six digits? (Consider two nonterminating decimals distinct if they differ in any digit. Nonterminating means that the digits are not eventually all z
  1 KB (205 words) - 03:29, 13 January 2019
• 2009 UNCO Math Contest II Problems
  elements and such that no two elements of <math>A</math> differ by <math>1</math> when
  5 KB (753 words) - 20:35, 8 February 2024
• 2009 UNCO Math Contest II Problems/Problem 10
  elements and such that no two elements of <math>A</math> differ by <math>1</math> when
  666 bytes (98 words) - 01:59, 13 January 2019
• 2016 AMC 12A Problems
  ...this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not w
  15 KB (2,348 words) - 17:20, 19 January 2024

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