Question

Find the number of polynomials f(x) that satisfy all of the
following conditions:
f(x) is a monic polynomial,
f(x) has degree 1000,
f(x) has integer coefficients,
f(x) divides
f(2x^3+x)

Answer 1

http://www.google.com/search?q=Find+the+number+of+polynomials+f(x)+that+satisfy+all+of+the+following+conditions%3A&oq=Find+the+number+of+polynomials+f(x)+that+satisfy+all+of+the+following+conditions%3A&aqs=chrome.0.57j62.4880j0&sourceid=chrome&ie=UTF-8#sclient=psy-ab&q=brilliant+Find+the+number+of+polynomials+f(x)+that+satisfy+all+of+the+following+conditions:&oq=brilliant+Find+the+number+of+polynomials+f(x)+that+satisfy+all+of+the+following+conditions:&gs_l=serp.3...11592.13491.0.13851.10.10.0.0.0.0.220.1571.0j9j1.10.0...0.0.0..1c.1.17.psy-ab.YdKvWIF7Svo&pbx=1&bav=on.2,or.r_cp.r_qf.&bvm=bv.48705608,d.eWU&fp=6ca84d98d48f74bf&biw=1440&bih=706

Answer 2

Step 1; Factor f(x) = (x - r_1)...(x - r_1000)

Step 2: Hint hint: something about roots and circle centered at r_i of radius 2 |r_i|^3

Step 3: triangle inequality --> implication: same argument.

Step 4: Verify

This is a brilliant question, therefore i will not give full answer. i hope the hints are enough to set one on the right track.