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JMO mentoring scheme
June 2014 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
Find the value of x which fits the equation x(x - 4) + (x - 1)(x + 2) = 2x2 + x + 1.
2
How many numbers are there between 1 and 2014 (inclusive) which have an odd number of even
digits?
3
(a) By starting at the North Pole, travelling 10 km south, then 10 km east followed by 10 km north you
can return to the starting point without passing through any other point more than once. The path east
takes you round a ‘parallel’ of latitude, that is, a circle with its centre on the earth’s axis.
Approximately how far is this centre below the surface at the North Pole?
[Radius of earth ≈ 6370 km.]
(b) If the restriction about not passing through any position more than once is ignored, where else can
the starting point be on the surface of the earth? In this case, find the approximate position of the
starting point.
4
Last month (May) Emily celebrated her nth birthday. Her grandfather Jack celebrated his birthday
6 days later when he was n2 - 1 years old. Earlier in the year on March 25th, Jack was exactly 8 times
as old as Emily. When was Emily’s birthday and how old was she then?
[For simplicity assume all years have 365 days.]
5
BCEF is a square. M, A and D are the mid-points of the sides EF, BF and CE
respectively. O is the centre of the square. Two arcs pass through A and D.
One is a semi-circle with centre O; the other is a quarter-circle with centre M.
Prove that the area of the crescent shape between the two arcs is half the area
of the rectangle ABCD.
6
On a 75 by 75 ‘chessboard’, the rows and columns are numbered with coordinates from 1 to 75.
Alena wants to put a pawn on just those squares where one coordinate is a multiple of 3 and
the other coordinate is even.
N.B. A coordinate could be even and a multiple of 3.
How many pawns can she put on the chessboard?
A quadrilateral ABCD circumscribes a semi-circle. The diameter and
centre O are on AD. AO = OD. Prove that AO2 = AB ´ CD.
5
6
7
4
3
7
A parallel of latitude is a circle whose plane is at right-angles to the earth’s axis. It slices the earth into unequal parts looking
a bit like mushroom caps. If the parallel of latitude is along the equator, we obtain two hemispheres. Draw a diagram of the
earth seen from above the equator. For (b) consider starting near the South Pole.
Let Emily’s birthday be x days after March 25th. Form an equation expressed in days and find x in terms of n.
The value of n will become obvious.
The area of the quarter-circle defined by M, A and D is needed.
Why would splitting the boards into blocks of 6 by 6 squares be helpful?
It’s all about congruent and similar triangles, not all of them obvious.