Find all positive integers $n$ such that the set
${n,n+1,n+2,n+3,n+4,n+5}$
can be split into to disjoint subsets such that the products of elements in these subsets are the same.
[This probolem can be solved without using any theorem! ]
IMO 1970
- Kazi_Zareer
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Re: IMO 1970
Try yourself
We cannot solve our problems with the same thinking we used when we create them.
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Re: IMO 1970
By contradiction we assume that there exist a n such that that satisfy this proposition. there is a number in those there must be a number which is divisible by 5. so that , we have another number which is divisible by 5. suppose those number are n , n+5 . now n+1, n+2, n+3, n+4 none of them will not be divisible by any p>5 because we can't have two such prime divisor into another's. two of them must be even. so that they must be power of 3 . two consecutive even number can't be power of 3 unless they are 1,3 . so we have such a answer n=0. but this is not positive integer. so that there does not exist such an integer n that satisfy the proposition