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Difference between revisions of "Mock AIME 4 2006-2007 Problems/Problem 3"

(hope I did this right.. + sol)
m (→‎Solution: {})
 
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:''The following solution is non-rigorous''.
 
:''The following solution is non-rigorous''.
  
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We would normally expect 2007 terms after multiplying out all of the binomials, but our goal is minimize the number of non-zero terms. We could get rid of some terms by applying repeated [[difference of squares]]. In other words, we let <math>r_1 = -r_2, \ldots, r_2005 = -r_2006</math>. Then our polynomial reduces to <math>(x^2 - r_1^2)(x^2 - r_3^2)\cdots (x^2 - r_{2005}^2)</math>. This is the product of <math>1003</math> binomials, which gives us <math>1004</math> terms (with nonzero coefficients). Since <math>1004 = 2^2 \cdot 251</math> (assuming the constant term counts as a coefficient), our answer is <math>251</math>.   
+
We would normally expect 2007 terms after multiplying out all of the binomials, but our goal is minimize the number of non-zero terms. We could get rid of some terms by applying repeated [[difference of squares]]. In other words, we let <math>r_1 = -r_2, \ldots, r_{2005} = -r_{2006}</math>. Then our polynomial reduces to <math>(x^2 - r_1^2)(x^2 - r_3^2)\cdots (x^2 - r_{2005}^2)</math>. This is the product of <math>1003</math> binomials, which gives us <math>1004</math> terms (with nonzero coefficients). Since <math>1004 = 2^2 \cdot 251</math> (assuming the constant term counts as a coefficient), our answer is <math>251</math>.   
  
 
Note that we could not apply difference of cubes etc, since that would require complex roots.
 
Note that we could not apply difference of cubes etc, since that would require complex roots.

Latest revision as of 15:44, 8 October 2007

Problem

Find the largest prime factor of the smallest positive integer $n$ such that $r_1, r_2, \ldots , r_{2006}$ are distinct integers such that the polynomial $(x-r_{1})(x-r_{2})\cdots (x-r_{2006})$ has exactly $n$ nonzero coefficients.

Solution

The following solution is non-rigorous.

We would normally expect 2007 terms after multiplying out all of the binomials, but our goal is minimize the number of non-zero terms. We could get rid of some terms by applying repeated difference of squares. In other words, we let $r_1 = -r_2, \ldots, r_{2005} = -r_{2006}$. Then our polynomial reduces to $(x^2 - r_1^2)(x^2 - r_3^2)\cdots (x^2 - r_{2005}^2)$. This is the product of $1003$ binomials, which gives us $1004$ terms (with nonzero coefficients). Since $1004 = 2^2 \cdot 251$ (assuming the constant term counts as a coefficient), our answer is $251$.

Note that we could not apply difference of cubes etc, since that would require complex roots.

See also

Mock AIME 4 2006-2007 (Problems, Source)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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