Difference between revisions of "1972 IMO Problems/Problem 3"
Mathboy100 (talk | contribs) (→Solution) |
Mathboy100 (talk | contribs) (→Solution) |
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<cmath>\frac{(2m)!(2n)!}{m!n!(m+n)!}</cmath> | <cmath>\frac{(2m)!(2n)!}{m!n!(m+n)!}</cmath> | ||
<cmath> = \frac{(2m)!}{m!n!} \cdot \frac{(2n!)}{(m+n)!}</cmath> | <cmath> = \frac{(2m)!}{m!n!} \cdot \frac{(2n!)}{(m+n)!}</cmath> | ||
− | <cmath> = \ | + | <cmath> = 2m \cdot (2m - 1) \dotsm (m+1) \cdot \frac{1}{n!} \cdot (m+n+1) \cdot (m+n+2) \dotsm (2n)</cmath> |
− | + | <cmath> = \frac{(m + 1) \cdot (m + 2) \dotsm (2m) \cdot (m+n+1) \cdot (m + n + 2) \dotsm (2n)}{n!}</cmath> | |
− | <cmath> = \frac{( | + | <cmath> = \frac{(n + 1) \cdot (n + 2) \dotsm (2n)}{n!} \cdot \frac{(m + 1) \cdot (m + 2) \dotsm (n)}{(2m + 1) \cdot (2m + 2) \dotsm (m + n)}</cmath> |
− | <cmath> = \frac{ | ||
== Solution 2 == | == Solution 2 == |
Revision as of 00:23, 7 December 2022
Let and
be arbitrary non-negative integers. Prove that
is an integer. (
.)
Contents
Solution
If , then
which is integral.
Suppose that . Then, we have
Solution 2
Denote the given expression as . 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
Note that is just
, which is an integer for all
. Then
so
is an integer, and therefore
must be an integer, etc.
By induction, is an integer for all
.
Borrowed from http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln723.html
Solution 3
Let p be a prime, and n be an integer. Let be the largest positive integer
such that
WTS: For all primes ,
We know
Lemma 2.1: Let be real numbers. Then
Proof of Lemma 2.1: Let and
On the other hand,
It is trivial that (Triangle Inequality)
Apply Lemma 2.1 to the problem: and we are pretty much done.
Note: I am lazy, so this is only the most important part. I hope you can come up with the rest of the solution. This is my work, but perhaps someone have come up with this method before I did.
See Also
1972 IMO (Problems) • Resources | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All IMO Problems and Solutions |