File Download Area
Information about "UKMT - BMO Round 2 - British Mathematical Olympiad 2015.pdf"
-
Filesize:
63.02 KB
-
Uploaded:
02/07/2020 21:01:18
-
Status:
Active
Free Educational Files Storage. Upload, share and manage your files for free. Upload your spreadsheets, documents, presentations, pdfs, archives and more. Keep them forever on this site, just simply drag and drop your files to begin uploading.
Download Urls
-
File Page Link
https://www.edufileshare.com/26f5be86ecb15e0e/UKMT_-_BMO_Round_2_-_British_Mathematical_Olympiad_2015.pdf
-
HTML Code
<a href="https://www.edufileshare.com/26f5be86ecb15e0e/UKMT_-_BMO_Round_2_-_British_Mathematical_Olympiad_2015.pdf" target="_blank" title="Download from edufileshare.com">Download UKMT - BMO Round 2 - British Mathematical Olympiad 2015.pdf from edufileshare.com</a>
-
Forum Code
[url]https://www.edufileshare.com/26f5be86ecb15e0e/UKMT_-_BMO_Round_2_-_British_Mathematical_Olympiad_2015.pdf[/url]
[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 2015.pdf | Plain Text
United Kingdom Mathematics Trust British Mathematical Olympiad Round 2 : Thursday, 29 January 2015 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 30 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 (26-30 March 2015). 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 Chiang Mai, Thailand, 8–16 July 2015) will then be chosen. Do not turn over until told to do so. United Kingdom Mathematics Trust 2014/15 British Mathematical Olympiad Round 2 1. The first term x 1 of a sequence is 2014. Each subsequent term of the sequence is defined in terms of the previous term. The iter ative formula is xn+1 =(√ 2 + 1) x n − 1 (√ 2 + 1) + x n . Find the 2015th term x 2015 . 2. In Oddesdon Primary School there are an odd number of class es. Each class contains an odd number of pupils. One pupil from each cl ass will be chosen to form the school council. Prove that the followin g two statements are logically equivalent. a) There are more ways to form a school council which includes an odd number of boys than ways to form a school council which inc ludes an odd number of girls. b) There are an odd number of classes which contain more boys t han girls. 3. Two circles touch one another internally at A. A variable chord P Q of the outer circle touches the inner circle. Prove that the l ocus of the incentre of triangle AQPis another circle touching the given circles at A. The incentre of a triangle is the centre of the unique circle which is inside the triangle and touches al l three sides. A locusis the col lection of al l points which satisfy a given condition. 4. Given two points Pand Qwith integer coordinates, we say that P sees Qif the line segment P Qcontains no other points with integer coordinates. An n-loop is a sequence of npoints P 1, P 2, . . . , P n, each with integer coordinates, such that the following conditio ns hold: a) P i sees P i+1 for 1 ≤i≤ n− 1, and P n sees P 1; b) No P i sees any P j apart from those mentioned in (a); c) No three of the points lie on the same straight line. Does there exist a 100-loop?