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• Mathematics competitions
  Math competitions differ widely in difficulty and content. Most test a broader range of mathematics
  698 bytes (88 words) - 14:36, 24 January 2024
• Prime number
  Two primes that differ by exactly 2 are known as [[twin primes]]. The following are the first few
  6 KB (985 words) - 12:38, 25 February 2024
• Mathematical Kangaroo
  ...bsites have released their problems from past years. The various versions differ from each other in terms of names and grammar usage. Most problems remain
  6 KB (936 words) - 15:38, 22 February 2024
• Twin Prime Conjecture
  ...re [[infinite]]ly many pairs of [[twin prime]]s, i.e. pairs of primes that differ by <math>2</math>. This version stated that there are infinitely many pairs of primes that differ by a finite number. The number Yitang chose was 7,000,000. Terence Tao and
  2 KB (308 words) - 02:27, 1 May 2024
• Partition of a rectangle into squares problem
  ...iplication by <math>q</math>. So, we may assume that all these coordinates differ from integers by, at most, <math>\frac 15</math>. Now let <math>a</math> an
  2 KB (333 words) - 13:50, 26 June 2006
• Modular arithmetic/Intermediate
  ...d <math>b</math> from the same set, then <math>a</math> and <math>b</math> differ by a multiple of <math>3</math>, and therefore <math>a \equiv b \pmod{3}.</
  14 KB (2,317 words) - 19:01, 29 October 2021
• Limit
  If the left hand and right hand limits at a certain point differ, than the limit does not exist at that point. For example, if we consider t
  7 KB (1,325 words) - 13:51, 1 June 2015
• 2002 AMC 12A Problems
  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>?
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2006 AMC 12A Problems/Problem 24
  ...ath> cannot combine and simplify, as their exponent of <math>x</math> will differ. Therefore, we simply add the number of terms produced from each addend. By
  8 KB (1,332 words) - 17:37, 17 September 2023
• 2005 AIME II Problems/Problem 13
  ...\; n_1-24 \mid 10.</cmath>We look for two divisors of <math>10</math> that differ by <math>7</math>; we find that <math>\{2, -5\}</math> and <math>\{5, -2\}<
  4 KB (642 words) - 14:55, 12 August 2019
• 2019 AIME I Problems/Problem 2
  ...e <math>P</math> is the probability that <math>B</math> and <math>J</math> differ by <math>1</math> (no zero, because the two numbers are distinct). There ar ...h>, the numbers differ by less than 2, so the probability that the numbers differ by at least 2 is <math>1-\tfrac{19}{190}=\tfrac9{10}</math>. Then <math>B-J
  5 KB (830 words) - 22:15, 28 December 2023
• 1989 AIME Problems
  ...}_{},\ldots,1989\}</math> such that no two members of <math>S^{}_{}</math> differ by <math>4^{}_{}</math> or <math>7^{}_{}</math>. What is the largest number
  7 KB (1,045 words) - 20:47, 14 December 2023
• 2003 AIME I Problems
  ...t three terms form a geometric progression, and the first and fourth terms differ by <math>30.</math> Find the sum of the four terms.
  6 KB (965 words) - 16:36, 8 September 2019
• 2003 AIME II Problems
  Two positive integers differ by <math>60.</math> The sum of their square roots is the square root of an
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1989 AIME Problems/Problem 13
  ...th>\{1,2,3,\ldots,1989\}</math> such that no two members of <math>S</math> differ by <math>4</math> or <math>7</math>. What is the largest number of [[elemen
  2 KB (274 words) - 04:07, 17 December 2023
• 1993 AIME Problems/Problem 4
  ...does not work, because if <math>c-d = 2</math>, then <math>a+b</math> must differ by 2 as well, but an odd number <math>93</math> can only result from two nu
  8 KB (1,343 words) - 16:27, 19 December 2023
• 2000 AIME I Problems/Problem 6
  ...ot easier since now we just have to find all pairs <math>(a,b)</math> that differ by 2.
  6 KB (966 words) - 21:48, 29 January 2024
• 2003 AIME I Problems/Problem 8
  ...ree terms form a [[geometric progression]], and the first and fourth terms differ by <math>30</math>. Find the sum of the four terms. Since the first and fourth terms differ by <math>30</math>, we have that <math>\frac{(a + 2d)^2}{a + d} - a = 30</m
  5 KB (921 words) - 23:21, 22 January 2023
• 2003 AIME II Problems/Problem 10
  Two positive integers differ by <math>60</math>. The sum of their square roots is the square root of an
  1 KB (218 words) - 14:14, 25 June 2021
• 2000 AMC 12 Problems/Problem 8
  We know that 1 and 5 differ by 4, 5 and 13 differ by 8, and 13 and 25 differ by 12. Hence the differences are 4, 8, and 12, resp. And the differences of
  7 KB (988 words) - 15:14, 10 April 2024
• 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) - 17:47, 26 March 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

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