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British Mathematical Olympiad 2007/8 British Mathematical Olympiad Round 2

The document is a practice test for the British Mathematical Olympiad consisting of 4 multi-part math problems. Students are given 3.5 hours to show their work and solutions. The top 20 scoring students will be invited to an intensive training session to further prepare the UK team of 6 for the International Mathematical Olympiad.

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Ajay Negi
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1K views1 page

British Mathematical Olympiad 2007/8 British Mathematical Olympiad Round 2

The document is a practice test for the British Mathematical Olympiad consisting of 4 multi-part math problems. Students are given 3.5 hours to show their work and solutions. The top 20 scoring students will be invited to an intensive training session to further prepare the UK team of 6 for the International Mathematical Olympiad.

Uploaded by

Ajay Negi
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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United Kingdom Mathematics Trust

United Kingdom Mathematics Trust

British Mathematical Olympiad


Round 2 : Thursday, 31 January 2008
2007/8 British Mathematical Olympiad
Time allowed Three and a half hours.
Each question is worth 10 marks. Round 2
Instructions • Full written solutions - not just answers - are
required, with complete proofs of any assertions
you may make. Marks awarded will depend on the
clarity of your mathematical presentation. Work 1. Find the minimum value of x2 + y 2 + z 2 where x, y, z are real numbers
in rough first, and then draft your final version
carefully before writing up your best attempt. such that x3 + y 3 + z 3 − 3xyz = 1.
Rough work should be handed in, but should be
clearly marked.
• One or two complete solutions will gain far more
credit than partial attempts at all four problems. 2. Let triangle ABC have incentre I and circumcentre O. Suppose that
• The use of rulers and compasses is allowed, but 6 AIO = 90◦ and 6 CIO = 45◦ . Find the ratio AB : BC : CA.
calculators and protractors are forbidden.
• Staple all the pages neatly together in the top left
hand corner, with questions 1,2,3,4 in order, and
the cover sheet at the front. 3. Adrian has drawn a circle in the xy-plane whose radius is a positive
integer at most 2008. The origin lies somewhere inside the circle. You
In early March, twenty students will be invited
are allowed to ask him questions of the form “Is the point (x, y) inside
to attend the training session to be held at
Trinity College, Cambridge (3-7 April). At the your circle?” After each question he will answer truthfully “yes” or
training session, students sit a pair of IMO-style “no”. Show that it is always possible to deduce the radius of the circle
papers and 8 students will be selected for further
training. Those selected will be expected to after at most sixty questions. [Note: Any point which lies exactly on
participate in correspondence work and to attend the circle may be considered to lie inside the circle.]
further training. The UK Team of 6 for this
summer’s International Mathematical Olympiad
(to be held in Madrid, Spain 14-22 July) will then
be chosen.
4. Prove that there are infinitely many pairs of distinct positive integers
Do not turn over until told to do so. x, y such that x2 + y 3 is divisible by x3 + y 2 .

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