1990 USAMO Problems
Problems from the 1990 USAMO.
A certain state issues license plates consisting of six digits (from 0 through 9). The state requires that any two plates differ in at least two places. (Thus the plates and cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use.
A sequence of functions is defined recursively as follows: (Recall that is understood to represent the positive square root.) For each positive integer , find all real solutions of the equation .
Suppose that necklace has 14 beads and necklace has 19. Prove that for any odd integer , there is a way to number each of the 33 beads with an integer from the sequence so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a "necklace" is viewed as a circle in which each bead is adjacent to two other beads.)
Find, with proof, the number of positive integers whose base- representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by from some digit further to the left. (Your answer should be an explicit function of in simplest form.)
An acute-angled triangle is given in the plane. The circle with diameter intersects altitude and its extension at points and , and the circle with diameter intersects altitude and its extensions at and . Prove that the points lie on a common circle.
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