By Dmitri Fomin, Sergey Genkin, Ilia V. Itenberg
"This is a pattern of wealthy Russian mathematical tradition written by way of specialist mathematicians with nice event in operating with highschool scholars ... difficulties are on extremely simple degrees, yet construction to extra advanced and complex paintings ... [contains] options to nearly all difficulties; methodological notes for the trainer ... constructed for a mainly Russian establishment (the mathematical circle), yet simply tailored to American lecturers' wishes, either in and out the classroom." --from the Translator's notes what sort of ebook is that this? it's a ebook produced by way of a notable cultural condition within the former Soviet Union which fostered the construction of teams of scholars, academics, and mathematicians known as "mathematical circles". The paintings is based at the concept that learning arithmetic can generate an identical enthusiasm as enjoying a staff sport--without inevitably being aggressive. This booklet is meant for either scholars and lecturers who love arithmetic and need to review its quite a few branches past the bounds of college curriculum. it's also a publication of mathematical recreations and, even as, a publication containing huge theoretical and challenge fabric in major parts of what authors deliberate to be "extracurricular mathematics". The ebook is predicated on a special adventure received by means of a number of generations of Russian educators and students.
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Extra info for Mathematical Circles: Russian Experience (Mathematical World, Vol. 7)
Prove that the length of median AM in triangle ABC is not greater than half the sum of sides AB and AC. Prove also that the sum of the lengths of the three medians is not greater than the triangle's perimeter. §4. Miscellaneous problems Problem 20. A polygon is cut out of paper, and then folded in two along a straight line (see Figure 45). Prove that the perimeter of the polygon formed is not greater than the perimeter of the original polygon. 1 FIGURE 45 Problem 21. Prove that a convex polygon cannot have three sides, each of which is greater than the longest diagonal.
Prove that the perimeter of a triangle is not greater than 4/3 the sum of its medians.
Solution. Assume that there are only n prime numbers, and let us den0te them all by PI, P2, , Pn' Then the number PIP2. Pn + 1 is divisible by none of the prime ,Pn' Therefore, this natural number cannot be represented as numbers PI, P2, the product of primes, which is absurd. This contradiction completes the proof. For teachers. The problems of this section should not be given for solution at one session. They can be given in the course of an entire year of classes, or used for olympiads, various types of contests, et cetera.