Download KCLMS Mathematics Circle 2016

KCLMS Mathematics Circle
A mathematics circle describes a group of young mathematicians who meet regularly to engage
collaboratively in mathematical thinking.
Our mathematics circle meets once each fortnight, on Mondays from 5-7pm. The focus of the
meetings will be to develop skill in problem solving and in mathematical investigation. You’ll also
have fun making friends and having a regular setting for doing maths with your peers.
Here are some examples of the sorts of problems we’ll look at:
• Is 22225555 + 55552222 divisible by 7?
• A cake is cut with a knife poisoned on one side. Prove that however many cuts we make, there
will always be a piece of cake free of poison.
• What combinations of regular polygons will tessellate the plane?
Some of the problems we’ll look at will enable us to study ideas that are not only interesting in
their own right, but that are also helpful for solving Olympiad-style questions.
2016-17 Application Problems
We only have room for a small number of participants in our maths circle, and we need to know
that these participants are going to be interested in, and will benefit from, the sort of stuff we
intend to do.
To apply to join the mathematics circle, you need to attempt and solve as many of the problems
that follow as you can. These questions aren’t easy - they are designed to get you to think and to
try a few different things. Most of them will require you to persevere - they are hard enough that
a solution might not come to you at once.
What we are most interested in is not “the answer” but how you get to the answer. You will need
to write up your solutions and in that write up you will need to show us your train of thought. For
example, if you think the answer to 7d) is no, then you need to justify your answer - how would
you convince one of your friends that this is the correct answer, and that there isn’t a clever way
of covering the square with rectangles?
Here are some other pointers:
1. By all means discuss these problems with friends, but write up your own answers, in your
words (and your diagrams, if these help). Maths is not a spectator sport!
2. We certainly don’t expect you to try and solve all of them; pick a few that you like the look of
and put some concentrated thought into those. You will be more likely to make a breakthrough,
than if you jump around trying all the questions at random.
3. If you get stuck on (or breeze through) a certain question, you may want to play around with
it, and see whether tweaking things has any affect. It may help you to understand what is
going on; this is something that certainly applies to questions 3 and 7, amongst others.
4. Lastly, have fun! One of the reasons why we enjoy maths is because it is fun and we can
explore a lot of different things. It’s not always about what the correct answer is.
How and when to submit your ideas
You can send us your ideas by email or by post.
• If by email, scan your solutions and send them to [email protected] with the subject
MATHS CIRCLE APPLICATION
• If by post, send your solutions to:
Mathematics Circle Applications
King’s College London Mathematics School
80 Kennington Road
London
SE11 6NJ
The deadline for receiving applications is Friday 30 September.
The first session of the mathematics circle is Monday 31 October.
Question 1
Consider the number 32a35717b. Replace each of a and b with a (different) digit
so that the overall number is divisible by 72.
Question 2
The midpoints of all three sides of a triangle have been marked. The original
triangle is then erased, leaving only the three marked points. How can the original
triangle be recreated using only a compass and straightedge?
Question 3
(a) Numbers a, b and c have been written at the vertices of a triangle (one at each
vertex). They are such that each one is the average of its two neighbours.
Do all three numbers have to be the same?
(b) Numbers a1 , a2 , . . . , a10 have been written at the vertices of a decagon (one at
each vertex), such that each number is the average of its two neighbours.
Do all ten numbers have to be the same?
Question 4
An 8 × 8 grid is covered by 1 × 2 dominoes. Prove that no matter *how* the grid
has been covered, we can find two dominoes which form a 2 × 2 square.
Question 5
Given 20 integers, none of which are divisible by 5, prove that the sum of the 20th
powers of those 20 integers is divisible by 5.
Question 6
Consider the number line. Suppose that every integer point on it is coloured either
red or green (like in the diagram below, but continuing infinitely to the left and
right . . . ).
−5 −4 −3 −2 −1
0
1
2
3
4
5
Prove that it is possible to find three integers such that the following conditions
hold:
1. they all have the same colour, and
2. the middle one is the average of the other two (e.g. 1, 3, 5, or 5, 12, 19, or more
generally a − d, a, a + d for some integers a, d).
For example, in the above picture, the numbers −1, 2, 5 are all green and satisfy
both conditions. The numbers −4, 0, 4 are all red and also satisfy both conditions!
Question 7
For this question, it would be useful to remember that a chessboard has its squares
coloured in a particular pattern (white-black-white-black...). Go and look this up
if you are not sure!
(a) A 10 × 10 square has a 1 × 1 square removed from both the bottom-left and
the top-right hand corner (so there are 98 individual squares remaining).
Can it be completely covered by 49 1 × 2 rectangles?
(b) Alternatively, the 10 × 10 square can have a 1 × 1 square removed from the
bottom-left and bottom-right hand corners instead.
Can this be completely covered by 49 1 × 2 rectangles?
(c) Investigate further, and try to “classify” when removing two 1 × 1 squares
leaves an impossible-to-cover square...
(d) Can a 10 × 10 square be completely covered by 25 1 × 4 rectangles?
Question 8
A group of n people participate in a round-robin arm-wrestling tournament. Each
match ends in either a win or a loss (no draws!).
Explain how it is possible to label the players P1 , P2 , . . . , Pn in such a way that P1
defeated P2 , P2 defeated P3 , . . . and Pn−1 defeated Pn .
Question 9
(a) A straight piece of wood of length 2 metres is cut into five pieces. Each piece
is at least 17cm long. Prove that there are three of these pieces which can be
put together to form a triangle.
(b) What if we allow the minimum length of each cut piece to be 16cm - can we
always guarantee being able to find three pieces which form a triangle? Or is
there a way of cutting the wood to ensure no triangles can be assembled?
Question 10
In this question, if n is a positive integer, the notation n! means the number
n × (n − 1) × (n − 2) × . . . × 1.
For example, 1! = 1, 3! = 3 × 2 × 1 = 6 and 5! = 5 × 4 × 3 × 2 × 1 = 120.
Here’s the question: cross out one of the factors of the form n! in the product
(1!)(2!)(3!)(4!) . . . (99!)(100!)
so that the result is a perfect square.
Question 11
Fifteen 2 × 2 squares have been cut out from an 11 × 11 sheet of graph paper. All
the cuts were made along the grid lines. Prove that (at least) one more such square
can be cut from the remaining paper.
Note: if one of the pieces of the remaining paper is already a 2 × 2 square, then
that counts too!
Question 12
Vivian is thinking of a whole number between 1 and 16 inclusive. Angela can ask
Vivian any “yes” or “no” question. Vivian is allowed to lie in at most one of her
answers, but Angela does not know which (if any) answer Vivian will choose for
this. If Angela is allowed to ask seven questions, how can she find Vivian’s number?