Difference between revisions of "1990 USAMO Problems"
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<cmath> \begin{align*} | <cmath> \begin{align*} | ||
f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\ | f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\ | ||
| − | f_{n + 1}(x) = \sqrt {x^2 + 6f_n(x)} \quad \text{for } n \geq 1. | + | f_{n + 1}(x) &= \sqrt {x^2 + 6f_n(x)} \quad \text{for } n \geq 1. |
\end{align*} </cmath> | \end{align*} </cmath> | ||
(Recall that <math>\sqrt {\makebox[5mm]{}}</math> is understood to represent the positive square root.) For each positive integer <math>n</math>, find all real solutions of the equation <math>\, f_n(x) = 2x \,</math>. | (Recall that <math>\sqrt {\makebox[5mm]{}}</math> is understood to represent the positive square root.) For each positive integer <math>n</math>, find all real solutions of the equation <math>\, f_n(x) = 2x \,</math>. | ||
Revision as of 00:10, 12 January 2008
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 
and 
cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use.
Problem 2
A sequence of functions 
is defined recursively as follows:
(Recall that ![$\sqrt {\makebox[5mm]{}}$](http://latex.artofproblemsolving.com/e/5/f/e5f47dc6b447e054a3d1806dfb40f12d3932582f.png)
is understood to represent the positive square root.) For each positive integer 
, find all real solutions of the equation 
.
Problem 3
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
![\[\{ n, n + 1, n + 2, \dots, n + 32 \}\]](http://latex.artofproblemsolving.com/6/5/b/65b01c0b81481b6702e38fa4fe22c248b5dca207.png)
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.)
Problem 4
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.)
Problem 5
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.
