Difference between revisions of "Distinct"
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| − | + | ==Definition== | |
| + | '''Distinct''' is a commonly used word in [[mathematics competitions]] meaning different. | ||
| + | ==Examples== | ||
| + | *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 [[polygons]] are polygons which are not [[congruent (geometry)|congruent]] to each other. | ||
| + | *Distinct objects are objects which are [[distinguishability|distinguishable]] | ||
| + | ==Problems== | ||
| + | ===Introductory=== | ||
| + | *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 | ||
| + | <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> | ||
| + | :([[2024 AMC 8 Problems/Problem 15|Source]]) | ||
| + | ===Intermediate=== | ||
| + | *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>. | ||
| + | :([[2023 AIME I Problems/Problem 7|Source]]) | ||
| + | ===Olympiad=== | ||
| + | *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]]) | ||
| + | |||
| + | {{stub}} | ||
Revision as of 17:05, 3 September 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
,
,
,
,
,
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
?![\[\textbf{(A)}\ 1089 \qquad \textbf{(B)}\ 1098 \qquad \textbf{(C)}\ 1107 \qquad \textbf{(D)}\ 1116 \qquad \textbf{(E)}\ 1125\]](//latex.artofproblemsolving.com/6/6/4/6640f8c5138b8919fc7c546cefeba8f1a737cca6.png)
- (Source)
?![\[\textbf{(A)}\ 1089 \qquad \textbf{(B)}\ 1098 \qquad \textbf{(C)}\ 1107 \qquad \textbf{(D)}\ 1116 \qquad \textbf{(E)}\ 1125\]](http://latex.artofproblemsolving.com/6/6/4/6640f8c5138b8919fc7c546cefeba8f1a737cca6.png)
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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