2002 AMC 10P Problems/Problem 15

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 $\{ 1, 9 \}$, $\{ 2, 10 \}$, $\{ 3, 11 \}$, $\{ 4, 12 \}$, $\; \dots \; \{ 12, 20 \}$ 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. By the Pigeonhole principle, in order to select at least one pair, it is necessary to select $\boxed{\textbf{(D)} 13}$ elements.

~green_lotus

Solution 2

Use casework to find that the answer is $\boxed{\textbf{(D) } 13}$.

See Also

2002 AMC 10P (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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