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Difference between revisions of "2015 AIME II Problems/Problem 5"

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==Solution==
 
==Solution==
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Call the given grid "Grid A". Consider Grid B, where the vertices of Grid B fall in the centers of the squares of Grid A; thus, Grid B has dimensions <math>(n-1) \times (n-1)</math>. There is a one-to-one correspondence between the edges of Grid B and the number of adjacent pairs of unit squares in Grid A. The number of edges in Grid b is <math>2n(n-1)</math>, and the number of ways to pick two squares out of grid A is <math>\dbinom{n^2}{2}</math>. So, the probability that the two chosen squares are adjacent is <math>\frac{2n(n-1)}{\dbinom{n^2}{2}} = \frac{2n(n-1}{\frac{n^2(n^2-1)}{2}} = \frac{4}{n(n+1)}</math>. We wish to find <math>n</math> such that <math>\frac{4}{n(n+1)} < \frac{1}{2015}</math>, and the first such <math>n</math> is <math>\boxed{90}</math>.
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Call the given grid "Grid A". Consider Grid B, where the vertices of Grid B fall in the centers of the squares of Grid A; thus, Grid B has dimensions <math>(n-1) \times (n-1)</math>. There is a one-to-one correspondence between the edges of Grid B and the number of adjacent pairs of unit squares in Grid A. The number of edges in Grid b is <math>2n(n-1)</math>, and the number of ways to pick two squares out of grid A is <math>\dbinom{n^2}{2}</math>. So, the probability that the two chosen squares are adjacent is <math>\frac{2n(n-1)}{\binom{n^2}{2}} = \frac{2n(n-1}{\frac{n^2(n^2-1)}{2}} = \frac{4}{n(n+1)}</math>. We wish to find the smallest positive integer <math>n</math> such that <math>\frac{4}{n(n+1)} < \frac{1}{2015}</math>, and by inspection the first such <math>n</math> is <math>\boxed{90}</math>.

Revision as of 21:44, 26 March 2015

Problem

Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$.

Solution

Call the given grid "Grid A". Consider Grid B, where the vertices of Grid B fall in the centers of the squares of Grid A; thus, Grid B has dimensions $(n-1) \times (n-1)$. There is a one-to-one correspondence between the edges of Grid B and the number of adjacent pairs of unit squares in Grid A. The number of edges in Grid b is $2n(n-1)$, and the number of ways to pick two squares out of grid A is $\dbinom{n^2}{2}$. So, the probability that the two chosen squares are adjacent is $\frac{2n(n-1)}{\binom{n^2}{2}} = \frac{2n(n-1}{\frac{n^2(n^2-1)}{2}} = \frac{4}{n(n+1)}$. We wish to find the smallest positive integer $n$ such that $\frac{4}{n(n+1)} < \frac{1}{2015}$, and by inspection the first such $n$ is $\boxed{90}$.

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