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

1. Let M and N be 9-digit integers such that replacing any digit of M with the corresponding digit of N results in a number divisible by 7. Prove that replacing any digit of N with the corresponding digit of M also results in a number divisible by 7. Find the smallest number of digits d > 9 for which this property holds. 2. Given an acute-angled triangle ABC with altitude CF and median BM such that BM = CF and the angles they cut off are equal, prove the triangle is equilateral. 3. Find the number of polynomials of degree 5 with coefficients from {1,2,3,4,5,6,7,8} that are divisible by x^

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Ajay Negi
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87 views1 page

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

1. Let M and N be 9-digit integers such that replacing any digit of M with the corresponding digit of N results in a number divisible by 7. Prove that replacing any digit of N with the corresponding digit of M also results in a number divisible by 7. Find the smallest number of digits d > 9 for which this property holds. 2. Given an acute-angled triangle ABC with altitude CF and median BM such that BM = CF and the angles they cut off are equal, prove the triangle is equilateral. 3. Find the number of polynomials of degree 5 with coefficients from {1,2,3,4,5,6,7,8} that are divisible by x^

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, 27 February 1997

Time allowed Three and a half hours. 1. Let M and N be two 9-digit positive integers with the
Each question is worth 10 marks. property that if any one digit of M is replaced by the digit
Instructions • Full written solutions - not just answers - are of N in the corresponding place (e.g., the ‘tens’ digit of M
required, with complete proofs of any assertions replaced by the ‘tens’ digit of N ) then the resulting integer is
you may make. Marks awarded will depend on the a multiple of 7.
clarity of your mathematical presentation. Work Prove that any number obtained by replacing a digit of N by
in rough first, and then draft your final version the corresponding digit of M is also a multiple of 7.
carefully before writing up your best attempt.
Find an integer d > 9 such that the above result concerning
Rough work should be handed in, but should be
divisibility by 7 remains true when M and N are two d-digit
clearly marked.
positive integers.
• One or two complete solutions will gain far more
credit than partial attempts at all four problems.
• The use of rulers and compasses is allowed, but
2. In the acute-angled triangle ABC, CF is an altitude, with F
calculators and protractors are forbidden.
on AB, and BM is a median, with M on CA. Given that
• Staple all the pages neatly together in the top left
BM = CF and 6 M BC = 6 F CA, prove that the triangle
hand corner, with questions 1,2,3,4 in order, and
ABC is equilateral.
the cover sheet at the front.

In early March, twenty students will be invited 3. Find the number of polynomials of degree 5 with distinct
to attend the training session to be held at coefficients from the set {1, 2, 3, 4, 5, 6, 7, 8} that are divisible
Trinity College, Cambridge (10-13 April). On by x2 − x + 1.
the final morning of the training session, students
sit a paper with just 3 Olympiad-style problems.
The UK Team - six members plus one reserve
- for this summer’s International Mathematical 4. The set S = {1/r : r = 1, 2, 3, . . .} of reciprocals of the
Olympiad (to be held in Mar del Plata, Argentina, positive integers contains arithmetic progressions of various
21-31 July) will be chosen immediately thereafter. lengths. For instance, 1/20, 1/8, 1/5 is such a progression,
Those selected will be expected to participate of length 3 (and common difference 3/40). Moreover, this
in further correspondence work between April is a maximal progression in S of length 3 since it cannot be
and July, and to attend a short residential extended to the left or right within S (−1/40 and 11/40 not
session in late June or early July before leaving being members of S).
for Argentina. (i) Find a maximal progression in S of length 1996.
Do not turn over until told to do so. (ii) Is there a maximal progression in S of length 1997?

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