Problem 287

We find the smallest such integer greater than 30 000 and the largest such integer less than 30 000 and then determine which is closest to 30 000.

Let $M$ be the smallest integer greater than 30 000 that is formed using the digits 2, 3, 5, 7, and 8, each exactly once.

Since $M$ is greater than 30 000, its ten thousands digit is at least 3.

To make $M$ as small as possible (but greater than 30 000), we set its ten thousands digit to 3.

To make $M$ as small as possible, its thousands digit should be as small as possible, and thus equals 2.

Continuing in this way, its hundreds, tens and ones digits are 578. Thus, $M = 32\,578$.

Let $m$ be the largest integer less than 30 000 that is formed using the digits 2, 3, 5, 7, and 8, each exactly once.

Since $m$ is less than 30 000, its ten thousands digit is less than 3 and must thus be 2.

To make $m$ as large as possible (but less than 30 000), its thousands digit should be as large as possible, and thus equals 8.

Continuing in this way, its hundreds, tens and ones digits are 7, 5 and 3, respectively. Thus, $m = 28\,753$.

Since $M - 30\,000 = 2578$ and $30\,000 - m = 1247$, then $m$ is closer to 30 000.

Thus, $N = m = 28\,753$. The tens digit of $N$ is 5.