2011 AIME I Problems/Problem 7
Problem 7
Find the number of positive integers for which there exist nonnegative integers , , , such that
Solution
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. Unknown error_msg)
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. Unknown error_msg)
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 |