1989 USAMO Problems: Difference between revisions
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[[1989 USAMO Problems/Problem 5 | Solution]] | [[1989 USAMO Problems/Problem 5 | Solution]] | ||
== See | == See Also == | ||
{{USAMO box|year=1989|before=[[1988 USAMO]]|after=[[1990 USAMO]]}} | |||
Revision as of 13:49, 17 September 2012
Problem 1
For each positive integer 
, let
.
Find, with proof, integers 
such that 
and 
.
Problem 2
The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.
Problem 3
Let 
be a polynomial in the complex variable 
, with real coefficients 
. Suppose that 
. Prove that there exist real numbers 
and 
such that 
and 
.
Problem 4
Let 
be an acute-angled triangle whose side lengths satisfy the inequalities 
. If point 
is the center of the inscribed circle of triangle and point 
is the center of the circumscribed circle, prove that line 
intersects segments 
and 
.
Problem 5
Let 
and 
be real numbers such that
Determine, with proof, which of the two numbers, or , is larger.
See Also
| 1989 USAMO (Problems • Resources) | ||
| Preceded by 1988 USAMO |
Followed by 1990 USAMO | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
