Competition problem — Number theory

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Number theory Difficulty 3.5 AMC 10/12 Find the answer

Let $R$ be a set of nine distinct integers. Six of the elements are $2$ , $3$ , $4$ , $6$ , $9$ , and $14$ . What is the number of possible values of the median of $R$ ?

Pick one

Solution

First, we find that the minimum value of the median of $R$ will be $3$ .
We then experiment with sequences of numbers to determine other possible medians.
Median: $3$
Sequence: $-2, -1, 0, 2, 3, 4, 6, 9, 14$
Median: $4$
Sequence: $-1, 0, 2, 3, 4, 6, 9, 10, 14$
Median: $5$
Sequence: $0, 2, 3, 4, 5, 6, 9, 10, 14$
Median: $6$
Sequence: $0, 2, 3, 4, 6, 9, 10, 14, 15$
Median: $7$
Sequence: $2, 3, 4, 6, 7, 8, 9, 10, 14$
Median: $8$
Sequence: $2, 3, 4, 6, 8, 9, 10, 14, 15$
Median: $9$
Sequence: $2, 3, 4, 6, 9, 14, 15, 16, 17$
Any number greater than $9$ also cannot be a median of set $R$ .
Therefore, the answer is $7\implies \textbf{(D)}.$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.