1990 USAMO Problems

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Problems from the 1990 USAMO.

Problem 1

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 $\boxed{027592}$ and $\boxed{020592}$ cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use.

Solution

Problem 2

A sequence of functions $\, \{f_n(x) \} \,$ is defined recursively as follows: $\begin{align*}f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\f_{n + 1}(x) &= \sqrt {x^2 + 6f_n(x)} \quad \text{for } ...$ (Recall that $\sqrt {\makebox[5mm]{}}$ is understood to represent the positive square root.) For each positive integer $n$ , find all real solutions of the equation $\, f_n(x) = 2x \,$ .

Solution

Problem 3

Suppose that necklace $\, A \,$ has 14 beads and necklace $\, B \,$ has 19. Prove that for any odd integer $n \geq 1$ , there is a way to number each of the 33 beads with an integer from the sequence $\{ n, n + 1, n + 2, \dots, n + 32 \}$ 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.)

Solution

Problem 4

Find, with proof, the number of positive integers whose base- $n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\pm 1$ from some digit further to the left. (Your answer should be an explicit function of $n$ in simplest form.)

Solution

Problem 5

An acute-angled triangle $ABC$ is given in the plane. The circle with diameter $\, AB \,$ intersects altitude $\, CC' \,$ and its extension at points $\, M \,$ and $\, N \,$ , and the circle with diameter $\, AC \,$ intersects altitude $\, BB' \,$ and its extensions at $\, P \,$ and $\, Q \,$ . Prove that the points $\, M, N, P, Q \,$ lie on a common circle.

Solution

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1990 USAMO Problems

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Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See also