Distinct
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 ![\[8\cdot\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}=\underline{B}~\underline{U}~\underline{G}~\underline{B}~\underline{U}~\underline{G}.\]](//latex.artofproblemsolving.com/0/4/7/047513b257239e4538e713105a012a223b79ded7.png)
What is the value of 
?![\[\textbf{(A)}\ 1089 \qquad \textbf{(B)}\ 1098 \qquad \textbf{(C)}\ 1107 \qquad \textbf{(D)}\ 1116 \qquad \textbf{(E)}\ 1125\]](//latex.artofproblemsolving.com/6/6/4/6640f8c5138b8919fc7c546cefeba8f1a737cca6.png)
,
,
,
,
,
represent distinct digits. Suppose 
is the greatest number that satisfies the equation ![\[8\cdot\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}=\underline{B}~\underline{U}~\underline{G}~\underline{B}~\underline{U}~\underline{G}.\]](http://latex.artofproblemsolving.com/0/4/7/047513b257239e4538e713105a012a223b79ded7.png)
What is the value of 
?![\[\textbf{(A)}\ 1089 \qquad \textbf{(B)}\ 1098 \qquad \textbf{(C)}\ 1107 \qquad \textbf{(D)}\ 1116 \qquad \textbf{(E)}\ 1125\]](http://latex.artofproblemsolving.com/6/6/4/6640f8c5138b8919fc7c546cefeba8f1a737cca6.png)
(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 
.
extra-distinct if the remainders when 
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 .
of four distinct positive integers, we denote the sum 
by 
. Let 
denote the number of pairs 
with 
for which 
divides 
of four distinct positive integers which achieve the largest possible value of (Source)
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