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>. | ||
Revision as of 17:14, 15 January 2008
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 |