Difference between revisions of "1972 IMO Problems/Problem 3"
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By induction, <math>f(m,n)</math> is integral for all <math>m,n \geq 0</math>. | By induction, <math>f(m,n)</math> is integral for all <math>m,n \geq 0</math>. | ||
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+ | Borrowed from http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln723.html |
Revision as of 10:44, 21 October 2014
Let and
be arbitrary non-negative integers. Prove that
is an integer. (
.)
Solution
Let . We intend to show that
is integral for all
. To start, we would like to find a recurrence relation for
.
First, let's look at :
Second, let's look at :
Combining,
.
Therefore, we have found the recurrence relation .
We can see that is integral because the RHS is just
, which we know to be integral for all
.
So, must be integral, and then
must be integral, etc.
By induction, is integral for all
.
Borrowed from http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln723.html