1.

Consider polynomial forms ax²-bx+c with integer coefficients which have two distinct zeros in the open interval 0 < x < 1. Exhibit with a proof the least positive integer value of a for which such a polynomial exists.

2.

With each subset X of a set is associated a second subset f(X). The association is such that whenever X contains Y then f(X) contains f(Y). Show that for some set A, f(A) = A.

3.

Let 0 < x₁ < 1 and x_n+1 = x_n(1-x_n), n = 1,2,3... Show that

4.

For f(x) a positive, monotone decreasing function defined in 0 < = x < = 1, prove that

5.

Let f(x) = a₁sin(x) + a₂sin(2x) +...+ a_nsin(nx), where a₁,a₂,...,a_n are real numbers and where n is a positive integer. Given that |f(x)| < = |sin(x)| for all real x, prove that

6.