Week 5
October 19, 2015
1. Construct an equilateral triangle on side of a triangle. Prove that the
circumcentres of these three triangles are the vertices of an equilateral
triangle.
2. If I cram infinitely many people into a room and randomly put red
and blue hats on their heads (I have lots of both), and then force
them to all stand in a line facing the same direction, so that they can
only see the infinite hats in front of them. Then I get them to all
simultaneously guess the colour of their hat. Assuming, once again,
that my eardrums do not burst, can infinitely many of them be right?
3. So we all know 1 + 12 + 13 + · · · → ∞ Show that the sum of
any n with a 9 in their decimal expansion is finite.
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n
omitting
4. Show that we can actually omit all n containing any given digit in
their decimal expansion, and the sum will be finite.
5. n students get up and start wandering around the class at random
before sitting down a minute later in a nearby seat. Only one person
ends up in any given seat. What is the probability that at least one
student ends up back in the seat they started in? (Proof by experiment
accepted and encouraged)
6. Proof that, given any 10 numbers in the list 10,11,12,. . . ,99, we can
find two disjoint subsets with the same sum. What is the largest
number we can change 99 to so that this still holds?
7. So, two wizards sit on a bus. Not, like Potterverse wizards, but
oldschool-pointy-hat-grey-beard wizards. They have an unusual conversation:
A: I have a positive integral number of children, whose ages are
positive integers. The sum of their ages is the number of the bus
and the product is my age.
B: If I knew your age and how many children you had, could I work
out their ages?
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A: No.
B: Ha! I finally know how old you are!
How old is he?( I’ve forgotten the bus number. Sozzles)
8. So we all know that there is a 50% chance of two people having the
same birthday in a group of 23. How many people are needed for a
75% chance? A 90% chance? An x% chance?
9. Beat me, or a friend, at Nim. Or explain why you can’t. (Me being
amazing at Nim is not an explanation)
10. Find an exciting proof of Pythagoras.
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