Difference between revisions of "1990 USAMO Problems"
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| − | Problems from the '''1990 [[USAMO]]'''. | + | Problems from the '''1990 [[United States of America Mathematical Olympiad | USAMO]]'''. |
==Problem 1== | ==Problem 1== | ||
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==Problem 5== | ==Problem 5== | ||
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An acute-angled triangle <math>ABC</math> is given in the plane. The circle with diameter <math>\, AB \,</math> intersects altitude <math>\, CC' \,</math> and its extension at points <math>\, M \,</math> and <math>\, N \,</math>, and the circle with diameter <math>\, AC \,</math> intersects altitude <math>\, BB' \,</math> and its extensions at <math>\, P \,</math> and <math>\, Q \,</math>. Prove that the points <math>\, M, N, P, Q \,</math> lie on a common circle. | An acute-angled triangle <math>ABC</math> is given in the plane. The circle with diameter <math>\, AB \,</math> intersects altitude <math>\, CC' \,</math> and its extension at points <math>\, M \,</math> and <math>\, N \,</math>, and the circle with diameter <math>\, AC \,</math> intersects altitude <math>\, BB' \,</math> and its extensions at <math>\, P \,</math> and <math>\, Q \,</math>. Prove that the points <math>\, M, N, P, Q \,</math> lie on a common circle. | ||
[[1990 USAMO Problems/Problem 5 | Solution]] | [[1990 USAMO Problems/Problem 5 | Solution]] | ||
| − | == See | + | == See Also == |
| − | + | {{USAMO box|year=1990|before=[[1989 USAMO]]|after=[[1991 USAMO]]}} | |
| − | + | {{MAA Notice}} | |
Latest revision as of 20:47, 3 July 2013
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.
See Also
| 1990 USAMO (Problems • Resources) | ||
| Preceded by 1989 USAMO |
Followed by 1991 USAMO | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
