[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 2003.pdf | Plain Text

Supported by British Mathematical Olympiad Round 2 : Tuesday, 25 February 2003 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 (3-6 April). On the final morning of the training session, students sit a paper with just 3 Olympiad-style problems, and 8 students will be selected for further training. Those selected will be expected to participate in correspondence work and to attend further training. The UK Team of 6 for this summer’s International Mathematical Olympiad (to be held in Japan, 7-19 July) will then be chosen. Do not turn over until told to do so. Supported by 2003 British Mathematical Olympiad Round 2 1. For each integer n >1, let p(n ) denote the largest prime factor of n. Determine all triples x, y, zof distinct positive integers satisfying (i) x, y, z are in arithmetic progression, and (ii) p(xyz )≤ 3. 2. Let AB Cbe a triangle and let Dbe a point on ABsuch that 4 AD =AB . The half-line ℓis drawn on the same side of ABasC, starting from Dand making an angle of θwith DAwhere θ= 6 AC B . If the circumcircle of AB Cmeets the half-line ℓat P, show that P B = 2P D . 3. Let f:N → Nbe a permutation of the set Nof all positive integers. (i) Show that there is an arithmetic progression of positive integers a, a+d, a + 2d, where d >0, such that f (a ) < f (a + d) < f (a + 2 d). (ii) Must there be an arithmetic progression a, a+d, . . . , a + 2003 d, where d >0, such that f (a ) < f (a + d) < . . . < f (a + 2003 d)? [ A permutation of Nis a one-to-one function whose image is the whole of N; that is, a function from Nto Nsuch that for al l m∈N there exists a unique n∈ N such that f(n ) = m.] 4. Let fbe a function from the set of non-negative integers into itself such that for all n≥ 0 (i) ¡ f (2 n+ 1) ¢ 2 − ¡ f (2 n)¢ 2 = 6 f(n ) + 1, and (ii) f(2 n) ≥ f(n ). How many numbers less than 2003 are there in the image of f?