Difference between revisions of "Distinct"
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*Distinct [[polygons]] are polygons which are not [[congruent (geometry)|congruent]] to each other. | *Distinct [[polygons]] are polygons which are not [[congruent (geometry)|congruent]] to each other. | ||
*Distinct objects are objects which are [[distinguishability|distinguishable]] | *Distinct objects are objects which are [[distinguishability|distinguishable]] | ||
− | = | + | {{Exampleprob|intro=Let the letters <math>F</math>,<math>L</math>,<math>Y</math>,<math>B</math>,<math>U</math>,<math>G</math> represent distinct digits. Suppose <math>\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}</math> is the greatest number that satisfies the equation |
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<cmath>8\cdot\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}=\underline{B}~\underline{U}~\underline{G}~\underline{B}~\underline{U}~\underline{G}.</cmath> | <cmath>8\cdot\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}=\underline{B}~\underline{U}~\underline{G}~\underline{B}~\underline{U}~\underline{G}.</cmath> | ||
− | :What is the value of <math>\underline{F}~\underline{L}~\underline{Y}+\underline{B}~\underline{U}~\underline{G}</math>?<cmath>\textbf{(A)}\ 1089 \qquad \textbf{(B)}\ 1098 \qquad \textbf{(C)}\ 1107 \qquad \textbf{(D)}\ 1116 \qquad \textbf{(E)}\ 1125</cmath> | + | :What is the value of <math>\underline{F}~\underline{L}~\underline{Y}+\underline{B}~\underline{U}~\underline{G}</math>?<cmath>\textbf{(A)}\ 1089 \qquad \textbf{(B)}\ 1098 \qquad \textbf{(C)}\ 1107 \qquad \textbf{(D)}\ 1116 \qquad \textbf{(E)}\ 1125</cmath>|inter=Call a positive integer <math>n</math> extra-distinct if the remainders when <math>n</math> is divided by <math>2, 3, 4, 5,</math> and <math>6</math> are distinct. Find the number of extra-distinct positive integers less than <math>1000</math>.|oly=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>.|introsource=2024 AMC 8 Problems/Problem 15|intersource=2023 AIME I Problems/Problem 7|olysource=2011 IMO Problems/Problem 1}} |
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{{stub}} | {{stub}} | ||
[[Category:Definition]] | [[Category:Definition]] |
Revision as of 15:41, 30 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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