Find all positive integers
for which the numbers in the set
can be colored red and blue, with the following condition being satisfied: The set
contains exactly
ordered triples
such that:
(i) the numbers
,
,
are of the same color,
and
(ii) the number
is divisible by
.
Author: Gerhard Wöginger, Netherlands
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![S = \{1,2, \ldots,n \}](/media/m/c/b/e/cbecd853eca16ec1cbb89c0453234d38.png)
![S \times S \times S](/media/m/8/1/b/81b7fe758c520d52436932696ea57bfd.png)
![2007](/media/m/3/f/4/3f405241bc274df2b17a4f30ef472364.png)
![\left(x, y, z\right)](/media/m/2/2/3/2230168a9bda3ede986fb7365bd4cec0.png)
(i) the numbers
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
![z](/media/m/d/2/4/d241a79f1fdd0ce9a8f3f91570ba5d62.png)
and
(ii) the number
![x + y + z](/media/m/d/8/3/d8357c7334ea1a7f6f240863c3c89751.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
Author: Gerhard Wöginger, Netherlands