[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 2014.pdf | Plain Text

United Kingdom Mathematics Trust British Mathematical Olympiad Round 2 : Thursday, 30 January 2014 Time allowed Three and a half hours. Each question is worth 10 marks. Instructions •Ful l written solutions – not just answers – are required, with complete proofs of any assertions you may make. Marks awarded wil l depend on the clarity of your mathematical presentation. Work in rough first, and then draft your final version careful ly before writing up your best attempt. Rough work shouldbe handed in, but should be clearly marked. • One or two completesolutions wil l gain far more credit than partial attempts at al l four problems. • The use of rulers and compasses is al lowed, but calculators and protractors are forbidden. • Staple al l 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. • To accommodate candidates sitting in other time zones, please do not discuss any aspect of the paper on the internet until 8am GMT on Friday 31 January. In early March, twenty students eligible to rep- resent the UK at the International Mathematical Olympiad will be invited to attend the training session to be held at Trinity College, Cambridge (3–7 April 2014). At the training session, students sit a pair of IMO-style papers and eight students will be selected for further training and selection examinations. The UK Team of six for this summer’s IMO (to be held in Cape Town, South Africa, 3–13 July 2014) will then be chosen. Do not turn over until told to do so. United Kingdom Mathematics Trust 2013/14 British Mathematical Olympiad Round 2 1. Every diagonal of a regular polygon with 2014 sides is colo ured in one of ncolours. Whenever two diagonals cross in the interior, they are of different colours. What is the minimum value of nfor which this is possible? 2. Prove that it is impossible to have a cuboid for which the vo lume, the surface area and the perimeter are numerically equal. The perimeter of a cuboid is the sum of the lengths of al l its twelve edges. 3. Let a 0 = 4 and define a sequence of terms using the formula a n = a 2 n − 1 − a n− 1 for each positive integer n. a) Prove that there are infinitely many prime numbers which ar e factors of at least one term in the sequence; b) Are there infinitely many prime numbers which are factors o f no term in the sequence? 4. Let AB Cbe a triangle and Pbe a point in its interior. Let AP meet the circumcircle of AB Cagain at A′ . The points B′ and C ′ are similarly defined. Let O A be the circumcentre of B C P. The circumcentres O B and O C are similarly defined. Let O A′ be the circumcentre of B′ C ′ P . The circumcentres O B′ and O C′ are similarly defined. Prove that the lines O AO A′ , O BO B′ and O CO C′ are concurrent.