Difference between revisions of "Distinct"
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==Definition== | ==Definition== | ||
'''Distinct''' is a commonly used word in [[mathematics competitions]] meaning different. | '''Distinct''' is a commonly used word in [[mathematics competitions]] meaning different. | ||
− | ==Examples== | + | ===Examples=== |
*Distinct [[number|numbers]] are numbers which are not [[equal]] to each other. | *Distinct [[number|numbers]] are numbers which are not [[equal]] to each other. | ||
*Distinct [[set|sets]] are sets which are not equal to each other. | *Distinct [[set|sets]] are sets which are not equal to each other. | ||
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*Given any set <math>A = \{a_1, a_2, a_3, a_4\}</math> of four distinct positive integers, we denote the sum <math>a_1 +a_2 +a_3 +a_4</math> by <math>s_A</math>. Let <math>n_A</math> denote the number of pairs <math>(i, j)</math> with <math>1 \leq i < j \leq 4</math> for which <math>a_i +a_j</math> divides <math>s_A</math>. Find all sets <math>A</math> of four distinct positive integers which achieve the largest possible value of <math>n_A</math>. | *Given any set <math>A = \{a_1, a_2, a_3, a_4\}</math> of four distinct positive integers, we denote the sum <math>a_1 +a_2 +a_3 +a_4</math> by <math>s_A</math>. Let <math>n_A</math> denote the number of pairs <math>(i, j)</math> with <math>1 \leq i < j \leq 4</math> for which <math>a_i +a_j</math> divides <math>s_A</math>. Find all sets <math>A</math> of four distinct positive integers which achieve the largest possible value of <math>n_A</math>. | ||
:([[2011 IMO Problems/Problem 1|Source]]) | :([[2011 IMO Problems/Problem 1|Source]]) | ||
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{{stub}} | {{stub}} | ||
+ | [[Category:Definition]] |
Latest revision as of 20:26, 6 October 2024
Definition
Distinct is a commonly used word in mathematics competitions meaning different.
Examples
- Distinct numbers are numbers which are not equal to each other.
- Distinct sets are sets which are not equal to each other.
- Distinct polygons are polygons which are not congruent to each other.
- Distinct objects are objects which are distinguishable
Problems
Introductory
- Let the letters ,,,,, represent distinct digits. Suppose is the greatest number that satisfies the equation
- What is the value of ?
- (Source)
Intermediate
- Call a positive integer extra-distinct if the remainders when is divided by and are distinct. Find the number of extra-distinct positive integers less than .
- (Source)
Olympiad
- Given any set of four distinct positive integers, we denote the sum by . Let denote the number of pairs with for which divides . Find all sets of four distinct positive integers which achieve the largest possible value of .
- (Source)
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