Let f(z) be the degree n polynomial z^n + 2*z^(n-1) + 3*z^(n-2) + ... + (n-1)*z^2 + n*z + n+1.

Prove all n roots of f(z) have a magnitude greater than 1; i.e. if f(z)=0 then |z|>1.

A

pentagram is typically constructed by taking the diagonals of a regular pentagon.

This common pentagram has five angles measured at the vertices, each of which equals 36 degrees. In total all five angles sum to 180 degrees.

Generalize to make an irregular pentagram by taking the diagonals of a convex irregular pentagon.

This irregular pentagram also has five angles measured at the five vertices. Show that the sum of these five angles equals 180 degrees.

A certain road has a path of a perfect circle with a single entrance/exit. A woman enters the road and walks the full circumference at a constant speed without stopping or changing direction. During her time on the road, N cars, each at its own random time during the duration of the walk, enter the circle. Each car proceeds, on the shortest path, to its own randomly selected stopping point on the circle. If cars travel 10 times as fast as the woman walks, answer the following:

1) For N=1, what is the probability that the woman “encounters” a car?

Definition: An “encounter” is when a moving car either overtakes the woman in the same direction or passes her while going in the opposite direction. If a car is stationary, there can be no encounter.

2) What is N such that there is at least a 75% chance of encountering a car?

3) For N=20, what is the expected number of encounters?

On my jogging days I start at 6:00 sharp and follow a paved road, heading strictly North.

At some point this same-level road turns West, but I go on on a path heading North going uphill till I reach an antenna site located on top of the hill.

I rest there for 10 minutes exactly then return following the same route in opposite direction.
I arrive home at 08:10.

My average speeds are: on the paved road 6 mph, uphill 4.8 and downhill 8,

What is the distance between my home and the point of return?