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Final Minus Two (Posted on 2015-01-24) Difficulty: 3 of 5
Start with the set S = {0, 2014}. Then, repeatedly, expand S as follows.

Place into S any integer that is a root of a polynomial the coefficients of which are in S.

Prove that the negative number −2 eventually appears in S.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution How to add -2 Comment 2 of 2 |
Construct the equation 2014x + 2014 = 0.
Then x=-1 is added to S.

Construct (-1)*x + 2014 = 0.
Then x=-2014 is added to S.

Construct 2014x + (-2014) = 0
Then x=1 is added to S.

2014 in binary is 11111011110.  Construct 1*x^10 + 1*x^9 + 1*x^8 + 1*x^7 + 1*x^6 + 1*x^4 + 1*x^3 + 1*x^2 + 1*x + (-2014) = 0.
Then x=2 is added to S.

Finally, construct x + 2 = 0.
Then x=-2 is added to S.

  Posted by Brian Smith on 2016-07-01 09:38:28
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