Putnam Practice Problems I (PPPI)

1.

If a0, a1,..., an are real numbers satisfying

a0/1 + a1/2 + ... + an/(n+1) = 0,

show that the equation

a0 + a1x + a2x2 + ... + anxn = 0

has at least one real root.

2.

Let there be given nine lattice points (points with integral coordinates) in three-dimensional Euclidean space. Show that there is a lattice point on the interior of one of the line segments joining two of these points.

3.

Find every twice-differentiable real-valued function f with domain the set of all real numbers and satisfying the functional equation

(f(x))2 - (f(y))2 = f(x+y)f(x-y)

for all real numbers x and y.