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    British Mathematical Olympiad 2007/8 British Mathematical Olympiad Round 1: Friday, 30 November 2007

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    Zacky Mhd
    The document provides instructions and questions for the British Mathematical Olympiad Round 1 exam to be held on November 30, 2007. It instructs students to show all work and write in a clear mathematical style. It notes that earlier questions tend to be easier and advises focusing on completing a small number of fully solved questions rather than attempting all problems. The six exam questions cover topics like solving simultaneous equations, properties of triangles, divisibility, and recursive functions.

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    British Mathematical Olympiad 2007/8 British Mathematical Olympiad Round 1: Friday, 30 November 2007

    Uploaded by

    Zacky Mhd
    70 views1 page
    The document provides instructions and questions for the British Mathematical Olympiad Round 1 exam to be held on November 30, 2007. It instructs students to show all work and write in a clear mathematical style. It notes that earlier questions tend to be easier and advises focusing on completing a small number of fully solved questions rather than attempting all problems. The six exam questions cover topics like solving simultaneous equations, properties of triangles, divisibility, and recursive functions.

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    Past year paper

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    BM UPSR

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    The document provides instructions and questions for the British Mathematical Olympiad Round 1 exam to be held on November 30, 2007. It instructs students to show all work and write in a clear mathematical style. It notes that earlier questions tend to be easier and advises focusing on completing a small number of fully solved questions rather than attempting all problems. The six exam questions cover topics like solving simultaneous equations, properties of triangles, divisibility, and recursive functions.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    70 views1 page

    British Mathematical Olympiad 2007/8 British Mathematical Olympiad Round 1: Friday, 30 November 2007

    Uploaded by

    Zacky Mhd
    The document provides instructions and questions for the British Mathematical Olympiad Round 1 exam to be held on November 30, 2007. It instructs students to show all work and write in a clear mathematical style. It notes that earlier questions tend to be easier and advises focusing on completing a small number of fully solved questions rather than attempting all problems. The six exam questions cover topics like solving simultaneous equations, properties of triangles, divisibility, and recursive functions.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 1

    United Kingdom Mathematics Trust

    United Kingdom Mathematics Trust

    British Mathematical Olympiad


    Round 1 : Friday, 30 November 2007
    1 Time allowed 3 2 hours. 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 in rough rst, and then write up your best attempt. Do not hand in rough work. One complete solution will gain more credit than several unnished attempts. It is more important to complete a small number of questions than to try all the problems. Each question carries 10 marks. However, earlier questions tend to be easier. In general you are advised to concentrate on these problems rst. The use of rulers and compasses is allowed, but calculators and protractors are forbidden. 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 top left hand corner and your name, initials and school in the top right hand corner. Complete the cover sheet provided and attach it to the front of your script, followed by your solutions in question number order. Staple all the pages neatly together in the top left hand corner. Do not turn over until told to do so.

    2007/8 British Mathematical Olympiad Round 1: Friday, 30 November 2007


    1. Find the value of 14 + 20074 + 20084 . 12 + 20072 + 20082

    2. Find all solutions in positive integers x, y, z to the simultaneous equations x + y z = 12 2 x + y 2 z 2 = 12. 3. Let ABC be a triangle, with an obtuse angle at A. Let Q be a point (other than A, B or C) on the circumcircle of the triangle, on the same side of chord BC as A, and let P be the other end of the diameter through Q. Let V and W be the feet of the perpendiculars from Q onto CA and AB respectively. Prove that the triangles P BC and AW V are similar. [Note: the circumcircle of the triangle ABC is the circle which passes through the vertices A, B and C.] 4. Let S be a subset of the set of numbers {1, 2, 3, ..., 2008} which consists of 756 distinct numbers. Show that there are two distinct elements a, b of S such that a + b is divisible by 8. 5. Let P be an internal point of triangle ABC. The line through P parallel to AB meets BC at L, the line through P parallel to BC meets CA at M , and the line through P parallel to CA meets AB at N . Prove that AN 1 BL CM LC MA NB 8 and locate the position of P in triangle ABC when equality holds. 6. The function f is dened on the set of positive integers by f (1) = 1, f (2n) = 2f (n), and nf (2n + 1) = (2n + 1)(f (n) + n) for all n 1. i) Prove that f (n) is always an integer. ii) For how many positive integers less than 2007 is f (n) = 2n ?

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