2002 AMC 10A Problems/Problem 21

Problem

The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is

$\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15$

Solution

Given $n^2$ tiles, a step removes $n$ tiles, leaving $n^2 - n$ tiles behind. Now, $(n - 1)^2 = n^2 - n + (1 - n) < n^2 - n < n^2$ , so in the next step $n - 1$ tiles are removed. This gives $(n^2 - n) - (n - 1) = n^2 - 2n + 1 = (n - 1)^2$ , another perfect square, and the process repeats.

Thus each two steps we cycle down a perfect square, and in $(10 - 1)\times 2 = 18$ steps, we are left with $1$ tile.

2002 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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2002 AMC 10A Problems/Problem 21

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