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British Math Olympiad 2003 Problems

1) The document provides instructions and 5 problems for the 2003/4 British Mathematical Olympiad Round 1 exam. Candidates are asked to show full written solutions in 3.5 hours. 2) The first problem involves solving a system of 4 simultaneous equations with 4 variables. 3) The second problem asks to prove that a triangle formed within a rectangle is isosceles. 4) The third problem involves a card game between two players and proving one player can always score at least as high as the other. 5) The last two problems involve determining the number of "wicked subsets" of a set of numbers and finding all possible values of pqr given certain divisibility conditions.

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664 views1 page

British Math Olympiad 2003 Problems

1) The document provides instructions and 5 problems for the 2003/4 British Mathematical Olympiad Round 1 exam. Candidates are asked to show full written solutions in 3.5 hours. 2) The first problem involves solving a system of 4 simultaneous equations with 4 variables. 3) The second problem asks to prove that a triangle formed within a rectangle is isosceles. 4) The third problem involves a card game between two players and proving one player can always score at least as high as the other. 5) The last two problems involve determining the number of "wicked subsets" of a set of numbers and finding all possible values of pqr given certain divisibility conditions.

Uploaded by

D Samy
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 2003/4 British Mathematical Olympiad


Round 1 : Wednesday, 3 December 2003 Round 1
1. Solve the simultaneous equations
Time allowed Three and a half hours.
Instructions • Full written solutions - not just answers - are ab + c + d = 3, bc + d + a = 5, cd + a + b = 2, da + b + c = 6,
required, with complete proofs of any assertions where a, b, c, d are real numbers.
you may make. Marks awarded will depend on the
clarity of your mathematical presentation. Work
2. ABCD is a rectangle, P is the midpoint of AB, and Q is the point
in rough first, and then draft your final version on P D such that CQ is perpendicular to P D.
carefully before writing up your best attempt. Prove that the triangle BQC is isosceles.
Do not hand in rough work.
• One complete solution will gain far more credit
3. Alice and Barbara play a game with a pack of 2n cards, on each of
than several unfinished attempts. It is more
which is written a positive integer. The pack is shuffled and the cards
important to complete a small number of questions
laid out in a row, with the numbers facing upwards. Alice starts, and
than to try all five problems. the girls take turns to remove one card from either end of the row,
• Each question carries 10 marks. until Barbara picks up the final card. Each girl’s score is the sum of
• The use of rulers and compasses is allowed, but the numbers on her chosen cards at the end of the game.
calculators and protractors are forbidden. Prove that Alice can always obtain a score at least as great as
Barbara’s.
• Start each question on a fresh sheet of paper. Write
on one side of the paper only. On each sheet of
working write the number of the question in the 4. A set of positive integers is defined to be wicked if it contains no
top left hand corner and your name, initials and three consecutive integers. We count the empty set, which contains
school in the top right hand corner. no elements at all, as a wicked set.
Find the number of wicked subsets of the set
• Complete the cover sheet provided and attach it to
the front of your script, followed by the questions {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
1,2,3,4,5 in order.
• Staple all the pages neatly together in the top left
hand corner. 5. Let p, q and r be prime numbers. It is given that p divides qr − 1,
q divides rp − 1, and r divides pq − 1.
Do not turn over until told to do so. Determine all possible values of pqr.

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