1. Let http://alt1.artofproblemsolving.com/...5d8f2ac8a3.gif where http://alt2.artofproblemsolving.com/...a40bbc4a47.gif is an integer. Prove that http://alt1.artofproblemsolving.com/...df107fcc1f.gif cannot be expressed as the product of two polynomials, each of which has all its coefficients integers and degree at least 1.

2. Let http://alt1.artofproblemsolving.com/...1b6737d4f7.gif be a polynomial with integer coefficients satisfying http://alt1.artofproblemsolving.com/...407bcda799.gif. Show that http://alt1.artofproblemsolving.com/...1b6737d4f7.gif has no integer zeros.

3. Let http://alt2.artofproblemsolving.com/...ec2edd79d4.gif be a polynomial with real coefficients. Show that there exists a nonzero polynomial http://alt2.artofproblemsolving.com/...c4b94514fd.gif with real coefficients such that http://alt1.artofproblemsolving.com/...844f680995.gif has terms that are all of a degree divisible by http://alt2.artofproblemsolving.com/...7c8cf1d503.gif.