Prove that if you pick 10 numbers from 1 to 1000...?

Question

Prove that if you pick 10 numbers from 1 to 1000, then there exists a pair where the larger of the pair is at most twice the other.

I'm imagining this has to deal with either a proof by contrapositive or contradiction. In the case of contradiction I would assume that there exists a set of 10 numbers from 1 to 1000 then for all pairs the larger of the pair is greater than twice the smaller. But I cant seem to find a flaw with that. I also tried creating a set of 10 where the numbers were evenly spaced out as far as possible (100, 200, 300) but it holds.

Any help would be great, thanks.

Answer 1

we suppose that the numbers are x1,x2,x3,,,x10 and we suppose that they are arranged in ascending order
so for the statement to be wrong we need all x(n)>2x(n-1)

so x10>2x9
x9>2x8
x8>2x7
x7>2x6
,,,,
,,
x3>2x2
x1>2x1

by multiplying all these inequalities
so
x10 x9 x8 ,,,,x3 x2> 2^10 x9 x8 ....x2 x1
x10>1024 x1
so as the minimum is x1=1
so x10>1024 & the numbers ranges from 1 to 1000
so the statement is true

Answer 2

If this could happen, then it must be able to happen when 1 is the smallest of the ten numbers, so let's assume that.

Smallest number: 1
Now, the next smallest number must be at least 1•2+1=3, so assume that it is 3
Second: 3
Now the next must be at least 3•2+1=7
Third: 7
Fourth 7•2+1=15
Fifth: 15•2+1=31
Sixth: 31•2+1=63
Seventh: 63•2+1=127
Eighth: 127•2+1=255
Ninth: 255•2+1=511
Tenth: 511•2+1=1023>1000.

Thus this cannot happen.

Answer 3

I don't have an exact answer to your question, but I'm fairly certain you can use exponents to prove it.

2 to the power of 10 is 1024. That might be a good basis for forming a proof. Or, if you start with 1, double it, then double again, and so forth, 10 times, you get 1024.

Answer 4

Use powers of 2; ten powers of 2 is 1024. If you massage that just so, you've got your proof.

Answer 5

The smallest that the least variety might want to be is a million. The smallest that the subsequent least variety might want to be is two. The smallest that the subsequent least variety might want to be is 4 proceed this and also you discover that the smallest that the 10th variety might want to be is 1024.

Answer 6

I can answer about anything on this board, but one area I don't even touch is math. I just happened to read your question though and it was interesting.