Download Questions - United Kingdom Mathematics Trust

Supported by
JMO mentoring scheme
Charitable Trust
January 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.
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What is the sum of 1 + 2 + 2 + 3 + 3 + 4 + 4 + 5 + 5 + … + 34 + 34 + 35 + 35 + 36?
2
How many positive integers n are there such that 8n + 50 is a multiple of 2n + 1?
3
(i) What is the area of a square whose diagonal is d?
(ii) What is the area of a regular octagon whose side is d?
4
Let E, F and G be the mid-points of the sides AB, BC and CD of a rectangle ABCD. The lines DE and
BG intersect the line AF in the points M and N (respectively). Find the ratio MN : AF.
5
How many different pairs of numbers can you choose from the set {1, 2, 3, …, 2012} such that their
sum is an even number?
6
ABCD is a regular tetrahedron. A plane passes through A parallel to D BCD (which is opposite A).
There are three more planes through B, C and D defined in a similar way. These four planes intersect
in pairs to give the edges of a larger tetrahedron; each vertex is the intersection of three of these
planes. What is the ratio of the volume of ABCD to the volume of the new tetrahedron?
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In this, and some subsequent papers, define a RAT to be a right angled triangle with integer value lengths of sides. Note
that if the sides have fractional values for their lengths, we can always obtain integer values by using a smaller unit. Thus
the set (0.15, 0.2, 0.25) using a particular unit becomes the set (3, 4, 5) if we use a unit twenty times smaller. You can
never do this in cases such as a right angled triangle with sides (1, 1, Ö 2).
(i) You are given a RAT with lengths a, b and c such that a² + b² = c² and such there is no common
factor between all three of a, b and c. Prove that just one of a and b is even and that c is odd.
(ii) If m and n are positive integers (m > n), prove that the set of values (m² - n², 2mn, m² + n²) can be
used to contruct a RAT.
In questions in subsequent papers we will call m and n key numbers for the triangle (abbreviated to KN).
A triangle has two vertices A (-3, 1) and B (3, -1). The third vertex C is at (2, k). Find all the values
of k such that the triangle is isosceles.
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4.
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Let n be the number of tests in total and a the average before the last two tests. Form and simplify equations.
What are the positive integers a and b such that 8n + 50 = a(2n + 1) + b?
Dissect the octagon into 4 triangles, 4 rectangles and a central square. What is the smaller side of one of the rectangles?
Extend the figure by rotating the rectangle and its elements about F through 180°.
Try writing out the pairs for a small set, e.g. {1, 2, 3, …, 10}.
Draw the larger tetrahedron first and study how ABCD fits inside it. Use note F on medians in the notes on the website.
(i) Note that the square of an even number must be a multiple of 4. What about squares of odd numbers?
(ii) Look at note B in the notes (www.ukmt.org.uk > Mentoring > Junior > Ideas in …) to help you with some algebra.
You are advised to draw an accurate diagram on graph paper to help you analyse this problem.
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