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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]]
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==Problems==
 
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===Introductory===
 
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*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>
 
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: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>
 
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:([[2024 AMC 8 Problems/Problem 15|Source]])
 
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===Intermediate===
 
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*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>.
 
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:([[2023 AIME I Problems/Problem 7|Source]])
 
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===Olympiad===
 
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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>.
 
−
:([[2011 IMO Problems/Problem 1|Source]])
 
  
 +
{{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 <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>
 +
| 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
 +
}}
  
 
{{stub}}
 
{{stub}}
 
[[Category:Definition]]
 
[[Category:Definition]]

Latest revision as of 11:13, 24 November 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 $F$,$L$,$Y$,$B$,$U$,$G$ represent distinct digits. Suppose $\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}$ is the greatest number that satisfies the equation \[8\cdot\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}=\underline{B}~\underline{U}~\underline{G}~\underline{B}~\underline{U}~\underline{G}.\] What is the value of $\underline{F}~\underline{L}~\underline{Y}+\underline{B}~\underline{U}~\underline{G}$?\[\textbf{(A)}\ 1089 \qquad \textbf{(B)}\ 1098 \qquad \textbf{(C)}\ 1107 \qquad \textbf{(D)}\ 1116 \qquad \textbf{(E)}\ 1125\]

(Source)

Intermediate

  • Call a positive integer $n$ extra-distinct if the remainders when $n$ is divided by $2, 3, 4, 5,$ and $6$ are distinct. Find the number of extra-distinct positive integers less than $1000$.

(Source)

Olympiad

  • Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i +a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$.

(Source)


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