Problem:
A polynomial
P(x)=c2004​x2004+c2003​x2003+⋯+c1​x+c0​
has real coefficients with c2004â€‹î€ =0 and 2004 distinct complex zeros zk​=ak​+bk​i, 1≤k≤2004 with ak​ and bk​ real, a1​=b1​=0, and
k=1∑2004​ak​=k=1∑2004​bk​
Which of the following quantities can be a nonzero number?
Answer Choices:
A. c0​
B. c2003​
C. b2​b3​…b2004​
D. ∑k=12004​ak​
E. ∑k=12004​ck​
Solution:
Since z1​=0, it follows that c0​=P(0)=0. The nonreal zeros of P must occur in conjugate pairs, so ∑k=12004​bk​=0 and ∑k=12004​ak​=0 also. The coefficient c2003​ is the sum of the zeros of P, which is
k=1∑2004​zk​=k=1∑2004​ak​+ik=1∑2004​bk​=0
Finally, since the degree of P is even, at least one of z2​,…,z2004​ must be real, so at least one of b2​,…,b2004​ is 0 and consequently b2​b3​…b2004​​=0. Thus the quantities in (A), (B), (C), and (D) must all be 0 .
Note that the polynomial
P(x)=x(x2​)(x3​)(x2003​)x+k=2∑2003​k)
satis es the given conditions, and ∑k=12004​ck​=P(1)î€ =0.
The problems on this page are the property of the MAA's American Mathematics Competitions