Prove the property mentioned at the end of Section 1.3.3: that all balanced functions lack the...

Prove the property mentioned at the end of Section 1.3.3:
that all balanced functions lack the highest-degree term A1A2 ... An in their
Reed–Muller expansions. Demonstrate (with the help of a counterexample) that
the inverse theorem (i.e., that all functions lacking the highest-degree
Reed–Muller term are balanced) is false.