Difference between revisions of "2011 AIME I Problems/Problem 7"
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Revision as of 20:25, 4 July 2013
Contents
Problem 7
Find the number of positive integers for which there exist nonnegative integers , , , such that
Solution
Let . The problem then becomes finding the number of positive integer roots for which and are nonnegative integers. We plug in and see that . Now, we can say that for some polynomial with integer coefficients. Then if , . Thus, if , then . Now, we need to show that for all , . We try with the first few that satisfy this. For , we see we can satisfy this if , , , , , , , , because (based on the idea , leading to a chain of substitutions of this kind) $= 2^{2009} + 2^{2008} + 2^{2008} = 2^{2009} + 2^{2009} = '''2^{2010}'''$ (Error compiling LaTeX. ! Double superscript.). Thus is a possible value of . For other values, for example , we can use the same strategy, with , , , , and , because $=3^{1004} +3^{1004}+3^{1004} = '''3^{1005}'''$ (Error compiling LaTeX. ! Double superscript.). It's clearly seen we can use the same strategy for all . We count all positive satisfying , and see there are
Solution 2
One notices that if and only if there exist non-negative integers such that .
To prove the forward case, we proceed by directly finding . Suppose is an integer such that . We will count how many , how many , etc. Suppose the number of is non-zero. Then, there must be at least such since divides all the remaining terms, so must also divide the sum of all the terms. Thus, if we let for , we have, Well clearly, is greater than , so . will also divide every term, , where . So, all the terms, , where must sum to a multiple of . If there are exactly terms where , then we must have at least terms where . Suppose there are exactly such terms and for . Now, we have, One can repeat this process for successive powers of until the number of terms reaches 2011. Since there are terms after the th power, we will only hit exactly 2011 terms if is a factor of 2010. To see this,
\[m+j(m-1) = 2011 \Rightarrow m-1+j(m-1) &= 2010 \Rightarrow (m-1)(j+1) = 2010.\] (Error compiling LaTeX. ! Misplaced alignment tab character &.)
Thus, when (which is an integer since by assumption, there are exactly 2011 terms. To see that these terms sum to a power of , we realize that the sum is a geometric series:
\[1 + (m-1) + (m-1)m+(m-1)m^2 + \cdots + (m-1)m^j &= 1+(m-1)\frac{m^{j+1}-1}{m-1} = m^{j+1}.\] (Error compiling LaTeX. ! Misplaced alignment tab character &.)
Thus, we have found a solution for the case .
Now, for the reverse case, we use the formula Suppose has a solution. Subtract 2011 from both sides to get Now apply the formula to get where are some integers. Rearranging this equation, we find where . Thus, if is a solution, then .
So, there is one positive integer solution corresponding to each factor of 2010. Since , the number of solutions is .
See also
2011 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- AIME Problems and Solutions
- American Invitational Mathematics Examination
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.