Supported by
JMO mentoring scheme
Charitable Trust
December 2011 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
(a) Luke writes all the consecutive even numbers from 2 to 2012 (inclusive) on a board. Louise then
erases all the numbers that are multiples of three. How many numbers are left?
(b)
How many digits does the answer to 111222333444555666777888999 ¸ 37 have?
2
DABC is scalene and strictly acute angled, that is, all sides are of different lengths and it has no right
or obtuse angles.
D is a point on BA (extended past A if necessary). It is also on the perpendicular bisector of BC.
Prove that Ð ADC = 2 ´ Ð ABC or that Ð ADC + 2 ´ Ð ABC = 180°.
3
Show that the tens digit of every power of 3 is even.
4
Major Tom has landed on a planet populated by purple cats, who always tell the truth, and black cats,
who always lie. In the pitch blackness, he meets 5 cats. The first cat says: “I am purple.” The second
cat says: “At least 3 of us are purple.” The third cat says: “The first cat is black.” The fourth cat says:
“At least 3 of us are black.” The fifth cat says: “We’re all black.” How many of the 5 cats are purple?
5
How many pairs of integers (x, y), x > 1 and y > 1, are there such that x2 + y = xy + 1 ?
6
A blacksmith is building a fence consisting of uprights 18 cm apart. Instead of welding a bar
across the top (like the dotted line), he makes individual arcs of circles like the one shown in
the diagram. (He continues this pattern along the fence.) The highest point of the arc is 3Ö3
cm above the dotted line. All the pieces of metal lie in the same vertical plane.
How long is the piece of metal used to make one of the circular arcs?
7
(i)
(ii)
8
We are given a 50 by 50 chessboard, divided into 2500 squares which can be coloured black or white.
Initially they are all white. At any step, a whole row or column of squares can reverse colour. Thus if
32 of the squares in a row are white and the other 18 are black, a step will change them to 32 black and
18 white. We aim to perform a sequence of steps so that we finish with exactly y black squares where
y is the year number. Show that it was possible to do this last year but that it is not possible this year or
next year.
Prove that if n is even, a n +1 + 1 = (a + 1)(a n - a n -1 + a n -2 - a n -3 + … + 1).
Which numbers of the form m m +1 + 1 are prime if m is a positive integer?
You should prove your result.
Rearrange the equation. Remember that (x - 1)(x + 1) = x2 - 1.
Spot the statements that contradict each other.
Multiply out in part (i) and use it in part (ii).
What happens if you do just a column step and a row step? Use some algebra.
5
6
7
8
The result you prove will depend on whether AB > AC or AC > AB.
Consider the remainders after a few powers are divided by 20.
Draw lines from the centre of the circle to the ends and top of the arc. You should find some simple angles.
2
3
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