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B M O B M O: Ritish Athematical Lympiad Ritish Athematical Lympiad

This document contains instructions and 4 problems for the British Mathematical Olympiad competition. Students are asked to show their work and write full solutions in 3 1/2 hours. The top 20 scoring students will be invited to a training session in March. At the end, the top 7 students will be selected for the UK team to compete in the International Mathematical Olympiad in July.

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Ajay Negi
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138 views1 page

B M O B M O: Ritish Athematical Lympiad Ritish Athematical Lympiad

This document contains instructions and 4 problems for the British Mathematical Olympiad competition. Students are asked to show their work and write full solutions in 3 1/2 hours. The top 20 scoring students will be invited to a training session in March. At the end, the top 7 students will be selected for the UK team to compete in the International Mathematical Olympiad in July.

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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BRITISH MATHEMATICAL OLYMPIAD BRITISH MATHEMATICAL OLYMPIAD

Round 2 : Thursday, 16 February 1995

Time allowed Three and a half hours.


Each question is worth 10 marks.
Instructions • Full written solutions - not just answers - are 1. Find all triples of positive integers (a, b, c) such that
required, with complete proofs of any assertions ³ 1 ´³ 1 ´³ 1´
you may make. Marks awarded will depend on the 1+ 1+ 1+ = 2.
a b c
clarity of your mathematical presentation. Work
in rough first, and then draft your final version
carefully before writing up your best attempt.
Rough work should be handed in, but should be 2. Let ABC be a triangle, and D, E, F be the midpoints of
clearly marked. BC, CA, AB, respectively.
• One or two complete solutions will gain far more Prove that 6 DAC = 6 ABE if, and only if, 6 AF C = 6 ADB.
credit than partial attempts at all four problems.
• The use of rulers and compasses is allowed, but
calculators and protractors are forbidden. 3. Let a, b, c be real numbers satisfying a < b < c, a + b + c = 6
• Staple all the pages neatly together in the top left and ab + bc + ca = 9.
hand corner, with questions 1,2,3,4 in order, and
the cover sheet at the front. Prove that 0 < a < 1 < b < 3 < c < 4.

In early March, twenty students will be invited 4. (a) Determine, with careful explanation, how many ways 2n
to attend the training session to be held at
people can be paired off to form n teams of 2.
Trinity College, Cambridge (30 March – 2 April).
On the final morning of the training session, (b) Prove that {(mn)!}2 is divisible by (m!)n+1 (n!)m+1 for all
students sit a paper with just 3 Olympiad-style positive integers m, n.
problems. The UK Team - six members plus
one reserve - for this summer’s International
Mathematical Olympiad (to be held in Toronto,
Canada, 13–23 July) will be chosen immediately
thereafter. Those selected will be expected
to participate in further correspondence work
between April and July, and to attend a
short residential session 2–6 July before leaving
for Canada.

Do not turn over until told to do so.

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