Difference between revisions of "1989 USAMO Problems/Problem 1"
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If we let <math>n=1989</math>, we see that <math>(a,b,c,d) = (1989,1989,1990, 2\cdot 1989)</math> is a suitable solution. <math>\blacksquare</math> | If we let <math>n=1989</math>, we see that <math>(a,b,c,d) = (1989,1989,1990, 2\cdot 1989)</math> is a suitable solution. <math>\blacksquare</math> | ||
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+ | Notice that it is also possible to use induction to prove the equations relating <math>T_n</math> and <math>U_n</math> with <math>S_n</math>. | ||
{{alternate solutions}} | {{alternate solutions}} | ||
− | == See | + | == See Also == |
{{USAMO box|year=1989|before=First question|num-a=2}} | {{USAMO box|year=1989|before=First question|num-a=2}} | ||
* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356633#p356633 Discussion on AoPS/MathLinks] | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356633#p356633 Discussion on AoPS/MathLinks] | ||
+ | {{MAA Notice}} | ||
[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] |
Latest revision as of 19:10, 18 July 2016
Problem
For each positive integer , let
Find, with proof, integers
such that
and
.
Solution
We note that for all integers ,
It then follows that
If we let , we see that
is a suitable solution.
Notice that it is also possible to use induction to prove the equations relating and
with
.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1989 USAMO (Problems • Resources) | ||
Preceded by First question |
Followed by Problem 2 | |
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.