1.

Evaluate

2.

Let p(x) = 2+4x+3x²+5x³+3x⁴+4x⁵+2x⁶. For k with 0 < k < 5, define

For which k is I_k smallest?

3.

Consider all lines which meet the graph of y = 2x⁴+7x³+3x-5 in four distinct points, say (x_i,y_i), i = 1,2,3,4. Show that

is independent of the line and find its value.

4.

Let z = x+iy be a complex number with x and y rational and with |z| = 1. Show that the number |z²ⁿ-1| is rational for every integer n.

5.

Evaluate the double series

6.

Let

be a polynomial of degree n with integral coefficients. If a₀, a_n and f(1) are odd, prove that f(x) = 0 has no rational roots.