Problem:
Consider the polynomial
P(x)=k=0∏10(x2k+2k)=(x+1)(x2+2)(x4+4)⋯(x1024+1024).
The coefficient of x2012 is equal to 2a. What is a ?
Answer Choices:
A. 5
B. 6
C. 7
D. 10
E. 24
Solution:
A factor in the product defining P(x) has degree 2012 if and only if the sum of the exponents in x is equal to 2012. Because there is only one way to write 2012
as a sum of distinct powers of 2, namely the one corresponding to its binary expansion 2012=111110111002, it follows that the coefficient of x2012 is equal to 20⋅21⋅25=26.
Note: In general, if 0≤n≤2047 and n=∑j∈A2j for A⊆{0,1,2,…,10}, then the coefficient of xn is equal to 2a where a=(112)−∑j∈Aj.
The problems on this page are the property of the MAA's American Mathematics Competitions