2002 AMC 10P Problems/Problem 15

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Problem

What is the smallest integer $n$ for which any subset of $\{ 1, 2, 3, \; \dots \; , 20 \}$ of size $n$ must contain two numbers that differ by 8?

$\textbf{(A)} \; 2 \quad \textbf{(B)} \; 8 \quad \textbf{(C)} \; 12 \quad \textbf{(D)} \; 13 \quad \textbf{(E)} \; 15$

Solution

There are twelve pairs $\{ 1, 9 \}$ , $\{ 2, 10 \}$ , $\{ 3, 11 \}$ , $\{ 4, 11 \}$ , $\; \dots \; \{ 12, 20 \}$ in $\{ 1, 2, 3, \; \dots \; , 20 \}$ that differ by 8. If we take $n = 12$ , it could be that we selected one element from each pair for the subset: the condition may not be fulfilled. In order to select at least one pair, it is necessary to select $\textbf{(D)} \; 13$ elements (Pigeonhole principle).

Solution submitted by green_lotus

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2002 AMC 10P Problems/Problem 15

Problem

Solution