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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
2015 AMC 12B Problems/Problem 22
...nce the elements in a 2-cycle cannot differ by 1, 3, or 5 mod 6, they must differ by 2 or 4 mod 6, i.e. they must be of the same parity. But since we have th

9 KB (1,623 words) - 18:48, 30 October 2022
2015 USAMO Problems
...e equivalent, but <math>\{1, 1, 2, 3\}</math> and <math>\{1, 2, 3\}</math> differ.)

4 KB (629 words) - 13:49, 22 November 2023
Algebraic number theory
...that no two of these irreducibles are <b>associates</b>; i.e. they do not differ only by [[unit (ring theory)|units]]. Fortunately, it is easy to verify tha

10 KB (1,646 words) - 15:04, 28 May 2020
2015 AMC 8 Problems/Problem 11
...> consonants), <math>20</math> choices for the third letter (since it must differ from the second letter), and <math>10</math> choices for the digit. This l

2 KB (319 words) - 23:58, 16 May 2023
2015 AMC 8 Problems/Problem 10
...first two, and <math>7</math> choices for the fourth number, since it must differ from all three. This means there are <math>9 \times 9 \times 8 \times 7=\b

1 KB (165 words) - 11:22, 23 November 2023
2016 AMC 12A Problems/Problem 25
...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 ...ill show the hundreds place at the end. In order for the hundreds place to differ by 2, the difference between two perfect squares needs to be at least <math

8 KB (1,224 words) - 18:28, 16 November 2022
1953 AHSME Problems/Problem 11
...les. It is <math>10</math> feet wide. The circumference of the two circles differ by about:

814 bytes (124 words) - 12:23, 22 April 2020
Mock AIME 3 2010 Problems
...ith integer side lengths almost isosceles if its two shortest side lengths differ by 1. The

14 KB (2,904 words) - 18:24, 16 May 2017
2017 AMC 12A Problems/Problem 16
...hts. Because the bases of these two triangles (that have the same heights) differ by a factor of 2, the area of <math>\Delta CBP</math> must be twice that of

13 KB (1,982 words) - 17:12, 20 December 2022
2020 AMC 12B Problems
...in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the

16 KB (2,417 words) - 01:03, 28 April 2022
2018 AIME II Problems/Problem 15
Because <math>f(n)</math> and <math>f(n+1)</math> can differ by at most 3 for <math>n=0,1,2,3,4,5</math>, <math>f</math> can decrease at

9 KB (1,607 words) - 10:27, 4 February 2022
2008 iTest Problems/Problem 87
...rom the alphabet <math>\{0,1,2,3,4,5,6\}</math> if neighboring digits must differ by exactly <math>2</math>. ...numbers" are 1 and 5, and the middle number is 3. Since neighboring digits differ by exactly 2, the outer numbers and the middle number must alternate. Denot

3 KB (451 words) - 20:25, 23 November 2018
2020 AMC 10B Problems
...in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the

16 KB (2,497 words) - 21:14, 12 November 2023
2019 AIME I Problems/Problem 12
The arguments of the two complex numbers differ by <math>90^\circ</math> if the ratio of the numbers is a pure imaginary nu

8 KB (1,534 words) - 22:17, 28 December 2023
What is the definition of Pure Mathematics?
Two primes that differ by exactly 2 are known as twin primes. The following are the first few pai

35 KB (5,882 words) - 18:08, 28 June 2021
1975 AHSME Problems/Problem 22
...math> and <math>q</math> and <math>p=q+1</math>. The only two primes which differ by <math>1</math> are <math>2,3</math> so <math>p=3</math> and <math>q=2</m

1 KB (226 words) - 17:48, 19 January 2021
2020 AMC 10B Problems/Problem 25
...in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the

16 KB (2,560 words) - 20:26, 11 November 2023
2020 USOJMO Problems/Problem 3
...ordinates, and similarly call it x-oriented or y-oriented if the endpoints differ in x- or y-coordinates, respectively.

5 KB (769 words) - 19:14, 6 October 2023
2022 AIME II Problems
...e order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are a

8 KB (1,397 words) - 09:18, 15 August 2022
2021 Fall AMC 10B Problems
...y <math>3</math> degrees, at which time the temperatures in the two cities differ by <math>2</math> degrees. What is the product of all possible values of <m

16 KB (2,450 words) - 00:13, 12 November 2023
1977 IMO Problems
...ndecomposable in <math>V_n</math> in more than one way. (Expressions which differ only in order of the elements of <math>V_n</math> will be considered the sa

3 KB (466 words) - 16:54, 29 January 2021
1977 IMO Problems/Problem 3
...ndecomposable in <math>V_n</math> in more than one way. (Expressions which differ only in order of the elements of <math>V_n</math> will be considered the sa

2 KB (368 words) - 16:43, 29 January 2021
2021 AIME II Problems/Problem 11
...larly, if they differ by 3, there will be only 1 possible set, and if they differ by 1, 3 possible sets. ...so the condition is met. Now, this means that the multiple of 6 and 7 must differ by 1.

15 KB (2,211 words) - 10:59, 4 February 2024
Jadhav Triads
...square roots of the numbers which are also the product of numbers which '''differ by 2.''' It is found that the square root of the product may '''differ to maximum of 0.3 to minimum of 0.1''' from actual middle number, but this

1 KB (188 words) - 10:34, 17 April 2021
2002 AMC 10P Problems/Problem 15
...ots \; , 20 \}</math> of size <math>n</math> must contain two numbers that differ by 8? ...; \{ 12, 20 \}</math> in <math>\{ 1, 2, 3, \; \dots \; , 20 \}</math> that differ by 8. If we take <math>n = 12</math>, it could be that we selected one elem

900 bytes (122 words) - 11:24, 9 March 2021
2022 AIME II Problems/Problem 5
...e order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are a

3 KB (481 words) - 11:08, 28 January 2024
2022 AIME II Problems/Problem 8
...nd <math>c</math>, since they make either <math>a</math> or <math>b</math> differ by at least 1.

13 KB (2,136 words) - 22:39, 31 December 2023
2021 IMO Problems/Problem 1
...<math>\sqrt{1 + n} - 1</math> and <math>\sqrt{1 + \frac{n}{2}} + 1</math> differ by at least <math>1</math>, then we can guarantee that there is an integer

5 KB (955 words) - 07:32, 13 April 2024
2021 Fall AMC 12B Problems/Problem 3
...y <math>3</math> degrees, at which time the temperatures in the two cities differ by <math>2</math> degrees. What is the product of all possible values of <m

2 KB (353 words) - 13:51, 30 December 2022
2021 Fall AMC 12B Problems
...y <math>3</math> degrees, at which time the temperatures in the two cities differ by <math>2</math> degrees. What is the product of all possible values of <m

14 KB (2,191 words) - 19:57, 12 November 2023

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