[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 1997.pdf | Plain Text

BRITISH MATHEMATICAL OLYMPIAD Round 2 : Thursday, 27 February 1997 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. In early March, twenty students will be invited to attend the training session to be held at Trinity College, Cambridge (10-13 April). On the final morning of the training session, students sit a paper with just 3 Olympiad-style problems. The UK Team - six members plus one reserve - for this summer’s International Mathematical Olympiad (to be held in Mar del Plata, Argentina, 21-31 July) will be chosen immediately thereafter. Those selected will be expected to participate in further correspondence work between April and July, and to attend a short residential session in late June or early July before leaving for Argentina. Do not turn over until told to do so. B RITISH MATHEMATICAL OLYMPIAD 1. Let Mand Nbe two 9-digit positive integers with the property that if anyone digit of Mis replaced by the digit of N in the corresponding place (e.g., the ‘tens’ digit of M replaced by the ‘tens’ digit of N) then the resulting integer is a multiple of 7. Prove that any number obtained by replacing a digit of Nby the corresponding digit of Mis also a multiple of 7. Find an integer d >9 such that the above result concerning divisibility by 7 remains true when Mand Nare two d-digit positive integers. 2. In the acute-angled triangle AB C,C F is an altitude, with F on AB , and B Mis a median, with MonC A . Given that B M =C F and 6 M B C =6 F C A , prove that the triangle AB C is equilateral. 3. Find the number of polynomials of degree 5 with distinct coefficients from the set {1,2 ,3 ,4 ,5 ,6 ,7 ,8 } that are divisible by x2 − x+ 1. 4. The set S= {1/r :r = 1 ,2 ,3 , . . . }of reciprocals of the positive integers contains arithmetic progressions of various lengths. For instance, 1 /20 ,1 /8,1 /5 is such a progression, of length 3 (and common difference 3 /40). Moreover, this is a maximal progression inSof length 3 since it cannot be extended to the left or right within S(− 1/40 and 11 /40 not being members of S). (i) Find a maximal progression in Sof length 1996. (ii) Is there a maximal progression in Sof length 1997?