By Derek Holton
The foreign Mathematical Olympiad (IMO) is an annual foreign arithmetic festival held for pre-collegiate scholars. it's also the oldest of the foreign technology olympiads, and festival for locations is very fierce. This e-book is an amalgamation of the 1st eight of 15 booklets initially produced to steer scholars meaning to contend for placement on their country's IMO group. the fabric contained during this e-book presents an advent to the most mathematical issues lined within the IMO, that are: Combinatorics, Geometry and quantity conception. furthermore, there's a certain emphasis on the right way to strategy unseen questions in arithmetic, and version the writing of proofs. complete solutions are given to all questions. notwithstanding a primary Step to Mathematical Olympiad difficulties is written from the viewpoint of a mathematician, it really is written in a fashion that makes it simply understandable to youth. This e-book can be a must-read for coaches and teachers of mathematical competitions.
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Additional info for A First Step to Mathematical Olympiad Problems
Clearly 1000 is too large to handle. Let's get some insight into things by trying for 10 instead. Well I can do it with one consecutive number. Clearly 10 adds up to 10! But I doubt that's what the question is all about. In fact, because it says “numbers” I think it really rules out one consecutive number. So we'll work on two or more numbers. Can we get 10 with two consecutive numbers? Can a + (a + 1) = 10? That would mean that 2a + 1 = 10. Hence 2a = 9, so a = 9/2. But a was supposed to be a whole number, so it can't be a fraction.
Prove the theorem. 27. Repeat Exercise 26 for 28. All numbers n > c can be written in the form n = ra + sb, where r and s have no factors in common and a and b are not negative. What is the best possible value for c in terms of r and s? Prove it. While we've been concentrating on the upper end of things, the “all n > c” part, something interesting has slipped past us at the lower end. Have a look at Table 3. Is there any pattern in all this? Are we able to say anything about those n < c for which n = 3a + sb?
Returning to αr + βs = 1 it is clear that acr + (cs = c and, for any integer n, r(ca — ns) + s((c + nr) = c. We now choose n = n' so that (c + n'r is the smallest positive (or zero) value to satisfy this last equation. Clearly 0 ≤ (c + n'r ≤ r — 1 because if (c + n'r ≥ r then (c + (n' — 1)r is positive (or zero) and is less than (c + n'r which was assumed smallest. Hence r(ca — n's) = c — s((3c + n'r) ≥ c — s(r — 1) > (r — 1)(s — 1) — s(r — 1). In other words, r(ca — n's) > —(r — 1). So ca — n's > 1/r — 1.