19th USAMO 1990


1. 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?
2. Define f₁(x) = √(x² + 48) and f_n(x) = √(x² + 6f_n-1(x) ). Find all real solutions to f_n(x) = 2x.
3. Show that for any odd positive integer we can always divide the set {n, n+1, n+2, ... , 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 n = 1, arranging the numbers as 1, 2, 3, ... , 14 and 15, 16, 17, ... , 33, does not work, because 15 and 33 are not relatively prime.
4. How many positive integers can be written in base 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.
5. ABC is acute-angled. The circle diameter AB meets the altitude from C at P and Q. The circle diameter AC meets the altitude from B at R and S. Show that P, Q, R and S lie on a circle.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.