JMO mentoring scheme
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December 2012 paper
Generally earlier questions are easier and later questions more difficult.
Some questions are devised to help you learn aspects of mathematics which you may not meet in school.
Hints are upside down at the bottom of the page; fold the page back to view them when needed.
1
Yesterday Lucian Schwindler, the dodgy money changer, exchanged 2 rotts for 3 weilers. Today he
exchanged 2 weilers for 1 rott and 3 hunds at the same rate. How many hunds are there in a weiler?
2
Can you draw 8 straight lines (of infinite length) so that no two lines have the same number of
intersection points with other lines? [E.g if one line intersects with 6 of the other lines, then no other line may
intersect with 6 of the other lines.]
3
The vertices of a regular 15 sided polygon are
connected as in the figure. What is the obtuse angle
between the chords AE and CG at the point P?
4
We wind a length of sticky plastic red tape round a
cylindrical white pole which has diameter 4 cm. The
tape makes an angle of 45 ° with the dotted line
down the pole (parallel to the axis of the cylinder)
as shown in the figure. In this way we obtain two
spirals down the pole, one red and one white. These
spirals have the same width. What is the width of
the plastic tape?
5
Adam writes a Fibonnacci type sequence of 10 integers (which may be positive or negative) where
each number, apart from the first two numbers, is the sum of the previous two numbers. The first
number is 34 and the last number is 0. What is the sum of the numbers?
6
One Saturday a tennis coach holds a training session for the six school pupils that he expects will form
the team for the next competition in the county school doubles league. He is interested to see how each
player adapts to playing alongside one of her team-mates. He pairs the players off at random to form
three ‘strings’. He puts the first string against the second to play just one set, then likewise, he pairs
the second string against the third and finally the third against the first in the morning.
In the afternoon he changes them all round. No pupil plays with the same partner as she did in the
morning. Given the choice he made in the morning, in how many ways can he arrange the pupils in the
afternoon so that this happens?
7
One side of the two sides which form the right angle of a right-angled triangle has length 40 mm. The
circle which can be drawn inside the right-angled triangle to touch all the sides has radius 10 mm.
What is the length of the hypotenuse?
8
You are given D ABC of any shape with points P, Q and R on the sides BC, CA and AB respectively.
The ratios BP : BC, CQ : CA and AR : AB are all the same, namely 1 : 4. Find the area of D PQR as a
fraction of the area of D ABC.
8.
3
4
5
6
7
Join XC and XE.
Imagine the post as a hollow paper cylinder cut down the dotted line then unrolled to lie on a flat surface.
Let the first term be a and the second term be b. Work out formulas for the other eight numbers in terms of a and b.
It can be helpful to represent the players as points round a hexagon and think about the joins you are allowed to make.
The sides of the triangle are tangents to the circle. If you draw two tangents from a point outside a circle, the lengths of the
tangents are equal. One of such lengths could be made x and then you can form an equation to solve for x.
Join, for example, AP and consider the area of ÐABP as a fraction of the area of ÐABC and then the area of ÐBPR as a
fraction of the area of ÐABP.