USAMO 1990

A license plate has six digits from 0 to 9 and may have leading zeros. If two plates must always differ in at least two places, what is the largest number of plates that is possible?

Define $f_{1}(x)=\sqrt{ }\left(x^{2}+48\right)$ and $f_{n}(x)=\sqrt{ }\left(x^{2}+6 f_{n-1}(x)\right)$. Find all real solutions to $f_{n}(x)=2 x$.

Show that for any odd positive integer we can always divide the set $\{\mathrm{n}, \mathrm{n}+1, \mathrm{n}+2, \ldots, \mathrm{n}+32\}$ into two parts, one with 14 numbers and one with 19 , so that the numbers in each part can be arranged in a circle, with each number relatively prime to its two neighbours. For example, for $\mathrm{n}=1$, arranging the numbers as $1,2,3, \ldots$, 14 and $15,16,17, \ldots, 33$, does not work, because 15 and 33 are not relatively prime.

How many positive integers can be written in base $\mathrm{n}$ so that (1) the integer has no two digits the same, and (2) each digit after the first differs by one from an earlier digit? For example, in base 3 , the possible numbers are $1,2,10,12,21,102,120,210$.

$\mathrm{ABC}$ is acute-angled. The circle diameter $\mathrm{AB}$ meets the altitude from $\mathrm{C}$ at $\mathrm{P}$ and $\mathrm{Q}$. The circle diameter $\mathrm{AC}$ meets the altitude from B at R and S. Show that P, Q, R and S lie on a circle.