Problem Solving Set 8

11 July 2012

1. Let C be a nonempty closed bounded subset of the real line and f : C → C be a non-decreasing continuous function. Show that there exists a point p ∈ C such that f(p) = p. (A set is closed if its complement is a union of open intervals. A function g is non-decreasing if g(x) ≤ g(y) for all x ≤ y.)

2. Modified Let P (x) be a polynomial with integer coefficients of the form

P (x) = x^{n} + a_{n−1}x^{n−1} + ⋯ + a_{2}x^{2} + a_{1}x + a_{0}.

Show that if the degree n is odd, the constant term a_{0} is odd, and P (x) has an odd number of odd coefficients, then P (x) has at least one irrational real root.