Week 3
October 5, 2015
1. Split 100 apples between two bags so that one bag contains exactly
twice as many as the other.
2. Do it again, but without cutting or in any way damaging the apples
3. 25 girls and 25 boys sit around a table. Prove it is always possible to
find a person both of whose neighbours are girls
4. On my birthday cake, there are n candles. I blow out k of them, where
k is a random number between 1 and n. I then try again and again,
taking the number of still-flaming candles as my new n each time. On
average, how many attempts will it take me before the candles are all
out, and I can have cake?
5. There are 2n people at a party. Each person has an even number
of friends at the party. Prove that there are 2 people with an even
number of common friends. Assume friendship is mutual.
6. Find all integer solutions to xy = y x .
7. Find tan−1 (1) + tan−1 (2) + tan−1 (3).
8. Each of Alice and Bob have a hat put on their head. Each hat is either
red or blue. Alice and Bob must simultaneously guess what colour hat
they think they are wearing. Is there a strategy so that one of them
must be right?
9. Draw a pentagram, and, using only two extra lines, make 10 triangles.(
The triangles must be the smallest shapes. That is, there can be no
lines inside the triangles)
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