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British Mathematical Olympiad Round 2 : Thursday 25 January 2018 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 26 January. Candidates sitting the paper in time zones more than 3 hours ahead of GMT must sit the paper on Friday 26 January (as defined local ly). 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 (4–9 April 2018). 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 year’s IMO (to be held in Cluj–Napoca, Romania 3–14 July 2018) will then be chosen. Do not turn over until told to do so. 2017/18 British Mathematical Olympiad Round 2 1. Consider triangle AB C. The midpoint of ACisM . The circle tangent to B C atBand passing through Mmeets the line ABagain at P. Prove that AB×B P = 2B M 2 . 2. There are nplaces set for tea around a circular table, and every place has a small cake on a plate. Alice arrives first, sits at the tab le, and eats her cake (but it isn’t very nice). Next the Mad Hatter arr ives, and tells Alice that she will have a lonely tea party, and that she must keep on changing her seat, and each time she must eat the cake i n front of her (if it has not yet been eaten). In fact the Mad Hatt er is very bossy, and tells Alice that, for i= 1 ,2 , . . . , n −1, when she moves for the i-th time, she must move a i places and he hands Alice the list of instructions a 1, a 2, . . . , a n− 1. Alice does not like the cakes, and she is free to choose, at every stage, whether to move cloc kwise or anticlockwise. For which values of ncan the Mad Hatter force Alice to eat all the cakes? 3. It is well known that, for each positive integer n, 1 3 + 2 3 + ∙ ∙ ∙ +n3 = n 2 (n + 1) 2 4 and so is a square. Determine whether or not there is a positiv e integer m such that (m + 1) 3 + ( m+ 2) 3 + ∙ ∙ ∙ + (2 m)3 is a square. 4. Let fbe a function defined on the real numbers and taking real value s. We say that fis absorbing iff(x ) ≤ f(y ) whenever x≤ yand f2018 (z ) is an integer for all real numbers z. a) Does there exist an absorbing function fsuch that f(x ) is an integer for only finitely many values of x? b) Does there exist an absorbing function fand an increasing sequence of real numbers a 1 < a 2< a 3< . . . such that f(x ) is an integer only if x= a i for some i? Note that if kis a positive integer and fis a function, then fk denotes the composition of kcopies of f. For example f3 (t) = f(f (f (t))) for all real numbers t.