Surname's on an island

On an isolated island each person has
1/5 probability of having 0 kids
1/5 probability of having 1 kid
1/5 probability of having 2 kids
1/5 probability of having 3 kids
1/5 probability of having 4 kids

Heterosexual partners are chosen at random from the population within the given generational cohort.

It is an old fashioned place where the men pass on their surname to their children,
the women don't.

Initially, the surname Klimpt is not very common. Just 1.00% of the total population.

After 100 generations what is the probability there there will be at least one person called Klimpt in the youngest generation if:
      (i) the initial generational cohort had 400 members
or   (ii) the initial generational cohort had 4,000 members

Express your answer as a percentage with two digits after the decimal point.

Assume that the starting population is half men, half women and that there is a 50-50 chance that each child born is a boy or girl.

Hint: you are probably going to need a computer for this one.

The Prince's Ancestors

Part A:
Once upon a time, there lived a fine young prince. He was next in line to the throne in a dynasty that stretched back for centuries. He had the official records that showed that his father's father's father's... 20 generations back had effectively created a country by unifying warring tribes.

Though it should be noted that his male line ancestors were not always very good husbands, and indeed their wives sometimes had lovers when the king was away fighting wars.

At each generation is was estimated that there was a 10% chance that the father's name on the birth certificate was not in fact the biological father.

What is probability that the man that the prince thinks is his father's father's father ... 20 generations back on the male line is indeed his true biological male line ancestor?

Background maths level required: high-school.
Difficulty level: fairly straight-forward.

Part B: 
Over the course of every generation, immigration caused a 10% population increase.

The true figures for the number of biological children each person had are as follows:
20% of people had no children that survived
20% of people had 1 child
20% of people had 2 children
20% of people had 3 children
20% of people had 4 children

Assume that when people paired off to reproduce, their partner was effectively just randomly chosen from their generation of population.

At the time of the founding king, his generation accounted for 1 million people, though the full population of the country was higher if we include the older people who were alive at the time.

What is the probability that the founding king was indeed an ancestor of the young prince?
Please give your answer as a percentage with two digits after the decimal point.

Level of maths required: some basic university maths is probably required.
Level of difficulty: for someone who can write a bit of code, fairly straight-forward

A witch with a compass

Wilma the Witch was given a new electronic compass that could tell her which way was north. It pointed to true north rather than magnetic north.

So she looked at her compass, found out which direction was north-east, put the compass away in her pocket and set out on her broom in a north-easterly direction. She continued as straight as she could over the surface of the earth for 1,000 km, without adjusting left or right.

She then took out her compass and found that even though she had started out by heading north-east and had continued going straight she was no longer exactly on a north-east bearing.

So she now found out which way was east and headed east. This time, for a change, she keep looking at her compass and diligently maintained a constant bearing, ( i.e. she kept going exactly east ) and continued for 6,000 km.

Then she turned to the south-east and headed on a constant south-east bearing. After a while she found herself back at the starting point. She noted that she had crossed every line of longitude exactly once.

She decided to continue on a constant south-east bearing, she crossed the equator and eventually found herself at the south pole.

She then stopped.

How far did she travel in total?

You can assume that the earth is a sphere and that the distance from the equator to either pole it is 10,000 km.
Please also ignore any distance covered due to altitude gain traversing mountains.

Level of background mathematics required: some undergraduate university maths
Question difficulty level: a tad tricky.

Average number of matches won

In Wimbledon's women's singles tournament there are 128 players. It is a knock-out tournament. I.e. after each round the losers are excluded from the next round.

The first round losers will have a win percentage of 0%.

The overall winner will have a 100% win percentage.

Part A: What's the average win percentage, (assuming the weighting of each player is equal)?

Part B: What's the average win percentage if each player is weighted proportional to how many matches she played?

Level of background mathematics required: high-school.

Difficulty level: straight-forward.

Maths puzzle: a bit of trig

Here's a little mathematics puzzle.

Suppose points A, B, C are all on a circle
and we have a point P outside the circle such that
the line that passes through A and P is a tangent to the
circle ( at A )
 the line that passes through B and P is a tangent to the
 circle ( at B)

also the line segments PA and BC are parallel.

 Suppose |AP| = 3
 and |BC| = 2

 What is |AC| ?

Background level of mathematics required: high-school

Difficulty level: by high-school standards it is a bit tricky.

In a while I'll post the answer ...